Portfolio Selection under Median and Quantile Maximization
aa r X i v : . [ q -f i n . M F ] A ug Portfolio Selection under Median and QuantileMaximization ∗ Xue Dong He † and Zhaoli Jiang ‡ and Steven Kou § August 25, 2020
Abstract
In this paper, we study a portfolio selection problem in which an agent trades a risk-free asset and multiple risky assets with deterministic mean return rates and volatilityand wants to maximize the α -quantile of her wealth at some terminal time. Becauseof the time inconsistency caused by quantiles, we consider intra-personal equilibriumstrategies. We find that among the class of time-varying, affine portfolio strategies, theintra-personal equilibrium does not exist when α > /
2, leads to zero investment inthe risky assets when α < /
2, and is a portfolio insurance strategy when α = 1 /
2. Wethen compare the intra-personal equilibrium strategy in the case of α = 1 /
2, namelyunder median maximization, to some other strategies and apply it to explain why morewealthy people invest more precentage of wealth in risky assets. Finally, we extend ourmodel to account for multiple terminal time.
Key words: quantiles; median; portfolio selection; intra-persional equilibrium; port-folio insurance ∗ He and Jiang acknowledge financial support from the General Research Fund of the Research GrantsCouncil of Hong Kong SAR (Project No. 14200917). † Corresponding Author. Room 505, William M.W. Mong Engineering Building, Department of SystemsEngineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong,Telphone: +852-39438336, Email: [email protected]. ‡ Department of Systems Engineering and Engineering Management, The Chinese University of HongKong, Shatin, N.T., Hong Kong, Email: [email protected]. § Boston University, Questrom School of Business, Rafik B. Hariri Building, 595 Commonwealth AvenueBoston, MA 02215, Email: [email protected]. Introduction
Maximization of the mean of investment return leads to excessive risk taking, so in themodern portfolio selection theory, investors are assumed to be concerned about not only themean but also the variance of investment return (Markowitz, 1952). On the other hand,median is an alternative to mean to summarize a distribution. Median is also a specialcase of quantile: The α -quantile of a distribution is defined to be the threshold such thatthe probability of observing a value beyond this threshold is equal to α , and median is the1 / α -quantile ofher utility for some α ∈ (0 , α measures the individual’srisk attitude with the riskiness of a distribution defined in terms of quantile-preservingspreads. Chambers (2009) proves that a representation of an individual’s preferences fordistributions is invariant to ordinal transformation and weakly monotonic with respect tofirst-order stochastic dominance if and only if it is the α -quantile of the individual’s utilityfor some α ∈ (0 , α -quantile of her wealth at certainterminal time. The risk-free rate and the mean return rates and volatility of the risky assetsare assumed to be deterministic, i.e. the Black-Scholes model. In addition, the agent facessome cone constraints on her portfolio, an example being the constraint of no short sales.This portfolio problem is time inconsistent in that a dynamic portfolio that, at the currenttime, maximizes the quantile of the terminal wealth does not necessarily maximizes thequantile of the terminal wealth at future time. This is in contrast to the dynamic portfolioselection problem considered by de Castro and Galvao (2019a), where the authors employ thedynamic quantile model proposed by de Castro and Galvao (2019b). Indeed, in that dynamicquantile model, quantiles are used at the beginning of each period to evaluate certain risk atthe end of the period. In other words, quantile maximization in that model is only appliedin every single period, so there is no time inconsistency caused by quantiles. In our model,however, the agent evaluates the quantile of her wealth at the terminal time, and there is apositive, continuous time period between the time of the evaluation of the quantile and theterminal time, so time inconsistency arises.Because of time inconsistency, without self-control or the help of commitment devices,the agent cannot commit her future selves to following the dynamic portfolio that maximizesthe quantile of her terminal wealth today. Following the literature on time inconsistency,we consider so-called intra-personal equilibrium portfolio strategies; see for instance Strotz(1955-1956), Ekeland and Lazrak (2006), Bj¨ork et al. (2017), He and Jiang (2019), andthe references therein. More precisely, we assume that the agent has no self control, so weregard her selves at different time to be different players in a game and seek an equilibrium inthis game. As a result, an intra-personal equilibrium is a time-consistent portfolio strategybecause at any time the agent is not willing to deviate from it and thus is able to implementit throughout the investment horizon.We focus on time-varying affine strategies under which the dollar amount invested in therisky assets are affine functions of the agent’s wealth and the intercepts and linear coefficientsare time varying. This family of strategies include many commonly used strategies, such asinvesting a certain proportion of wealth in the risky assets, investing a wealth-independentdollar amount in the risky assets, and the mixture of the above two strategies taken in3ifferent time periods. We prove that for α -quantile maximization with α > /
2, theredoes not exist an intra-personal equilibrium strategy that is time varying and affine. For α -quantile maximization with α < /
2, a time-varying affine strategy is an intra-personalequilibrium strategy if and only if it leads to zero investment in the risky assets at all time.For 1 / ξ and invests any capital in excessof ξ into Kelly portfolio and the remaining into the risk-free asset. In particular, we derivemultiple intra-personal equilibrium strategies because different portfolio insurance levels leadto different investment strategies.Our results show that in continuous-time portfolio selection, the risk attitude of the agentdoes not vary smoothly as α changes. When α > /
2, the agent is too risk seeking to takelimited risk. When α < /
2, the agent is so risk averse that she takes no risk at all. When α = 1 /
2, the agent takes nonzero, limited risk. In this case, we show that with a smallerportfolio insurance level ξ , the median of the terminal wealth becomes strictly larger at anytime and any wealth level but the agent can end up with lower wealth levels. Therefore, ξ becomes a parameter that trades off the growth of the portfolio, which is measured by themedian of the terminal wealth, and the risk of the portfolio, which is measured by the lowestlevel that the agent’s wealth in the future may touch. Thus, we can consider ξ to be thesecond parameter to represent the agent’s risk attitude.We then compare the intra-personal equilibrium strategy to fractional Kelly strategies andfind that neither of them dominates the other in terms of median of the terminal wealth. Theintra-personal equilibrium strategy, however, entails less risk than fractional Kelly strategiesbecause it implies a higher level of minimum wealth. We also compare the intra-personalequilibrium strategy to the pre-committed strategy and the naive strategy under medianmaximization. The pre-committed strategy is one that maximizes the quantile of the ter-minal wealth at the initial time, so it is optimal for the agent’s self at the initial time. Thenaive strategy is the actual strategy implemented by the agent if she were not aware of thetime-inconsistency and thus, at each time, is only able to implement in an infinitesimallyperiod of time the strategy that maximizes the quantile of the terminal wealth at that time.We find that the pre-committed strategy can lead to arbitrarily large holding of the riskyassets and leads to capped wealth, so it is less preferable to the intra-personal equilibriumstrategy. Under the naive strategy, the agent would take an infinite amount of risk aroundthe terminal time and the median of the terminal wealth is always lower than that of theintra-personal equilibrium strategy. Thus, the naive strategy is not preferable either.4s an application, we find that the intra-personal equilibrium strategy can explain anempirical finding by Wachter and Yogo (2010): Households with a higher level of wealthtend to have a larger portfolio shares in risky assets (i.e., to invest more percentage oftheir wealth in risky assets). This empirical finding cannot be explained by expected utilitymaximization with a power utility function even when the asset prices have stock volatility,stochastic return rates, or jumps, because the resulting optimal percentage of wealth investedin risky assets is independent of the agent’s wealth. The mean-variance log return modelproposed by Dai et al. (2019) cannot explain the finding either because in their model thepercentage of wealth invested in risky assets is independent of the agent’s wealth as well.Under the intra-personal equilibrium strategy in our model, which is derived in the simpleBlack-Scholes model, households with a higher level of wealth indeed have a larger portfolioshares in risky assets. The intuition is as follows: When households become older, theirwealth, on average, also becomes larger and thus farther away from the portfolio insurancelevel, so they invest more in the risky assets.Finally, we extend our model to the case in which the agent is concerned about quantileof wealth at multiple time points, e.g., the quantle of wealth in one year and the quantileof wealth at the retirement age. We set the objective function to be a weighted averageof the quantile of wealth at a finite set of different terminal times and study intra-personalequilibrium strategies. Again, we find that among the family of time-varying, affine strate-gies, intra-personal equilibrium does not exist when α > /
2, must be zero-investment inthe risky assets when α < /
2, and is a portfolio insurance strategy when α = 1 / Model
Denote the transpose of any matrix A as A ⊤ and denote the Euclidean norm of a vector a as k a k . For any nonempty interval I , denote by C ( I ) the set of continuous functions from I to certain metric space B , where B varies with and will be clear in different contexts. Forany c < d , denote by C pw ([ c, d )) the set of functions g from [ c, d ) to B such that there exists c =: t < t < · · · < t N := d and g on each [ t i − , t i ) can be continuously extended to [ t i − , t i ], i = 1 , . . . , N . In other words, C pw ([ c, d )) is the set of piece-wise continuous functions. Consider an agent who can trade a risk-free asset and m risky assets continuously in thetime period [0 , T ]. The price of the risk-free asset, denoted as S ( t ), and the price of riskyasset i , denoted as S i ( t ), i = 1 , . . . , m , follow dS ( t ) = S ( t ) r ( t ) dt, t ≥ ,dS i ( t ) = S i ( t ) (cid:2)(cid:0) b i ( t ) + r ( t ) (cid:1) dt + d X j =1 σ ij ( t ) dW j ( t ) (cid:3) , t ≥ , i = 1 , . . . , m, where W ( t ) := (cid:0) W ( t ) , ..., W d ( t ) (cid:1) ⊤ , t ≥ d -dimensional Brownian motion.Here and hereafter, we denote the transpose of any matrix A as A ⊤ and denote the Euclideannorm of a vector a as k a k . The risk-free rate r ( t ), excess mean return rate vector of the riskyassets b ( t ) := (cid:0) b ( t ) , . . . , b m ( t ) (cid:1) ⊤ , and volatility matrix of the risky assets σ ( t ) := (cid:0) σ i,j ( t ) (cid:1) satisfy the following assumption that will be in force throughout of the paper. Assumption 1 r , b , and σ are deterministic and belong to C pw ([0 , T )) . Moreover, thefollowing two non-degeneracy conditions hold: (i) b ( t ) = 0 for all t ∈ [0 , T ) and (ii) thereexists δ > such that σ ( t ) σ ( t ) ⊤ − δI is positive semi-definite for all t ∈ [0 , T ) , where I stands for the m -dimensional identity matrix. Suppose at each time t an agent invests π i ( t ) dollars in risky asset i , i = 1 , . . . , m andthe remaining of her wealth in the risk-free asset. Then, the dynamics of the agent’s wealth,7enoted as X ( t ), follow dX ( t ) = (cid:0) r ( t ) X ( t ) + π ( t ) ⊤ b ( t ) (cid:1) dt + π ( t ) ⊤ σ ( t ) dW ( t ) , t ∈ [0 , T ] , (2.1)where π ( t ) := (cid:0) π ( t ) , . . . , π m ( t ) (cid:1) ⊤ is referred to as the agent’s portfolio . Suppose that the agent is endowed with initial wealth x at time 0 and wants to maximizethe median of her terminal wealth (i.e., wealth at the end time T ). The agent faces someportfolio constraints, such as the no-short-selling constraint, represented by Qπ ( t ) ≥ , t ∈ [0 , T ) for some n -by- m matrix Q . Suppose that the agent is going to revisit the portfoliodecision at each time t ∈ [0 , T ), with the same objective of maximizing the median of theterminal wealth. In contrast to expected utility maximization, a portfolio π ( s ) , s ∈ [0 , T )that maximizes the median of the terminal wealth at time 0 does not necessarily maximizethe median of the terminal wealth at time t , leading to time-inconsistent behavior. To obtainconsistent investment behavior, we follow the literature to consider the so-called equilibriumstrategies, in which the agent is assumed to have no control of her selves in the future andthus the selves at different time can be viewed as different players in a game.Formally, we restrict ourselves to consider Markovian portfolio strategies π that aremappings from [0 , T ) × R to R m : At time t with wealth x at that time, the agent invests π ( t, x ) dollars in the risky assets. A portfolio strategy π is feasible if (i) there exists L > | π ( t, x ) − π ( t, y ) | ≤ L | x − y | , | π ( t, x ) | ≤ L (1 + | x | ), ∀ t ∈ [0 , T ) , x, y ∈ R , whichimplies that the following wealth equation ( dX π ,x ( t ) = (cid:0) r ( t ) X π ,x ( t ) + π ( s, X π ,x ( t )) ⊤ b ( t ) (cid:1) dt + π ( t, X π ,x ( t )) ⊤ σ ( t ) dW ( t ) , t ∈ [0 , T ] ,X π ,x (0) = x (2.2)has a unique strong solution, and (ii) Q π ( t, X π ,x ( t )) ≥ , t ∈ [0 , T ). Denote by Π the set offeasible portfolio strategies π .As the agent may revisit the portfolio selection problem at any intermediate time t ∈ , T ) with any wealth level x , we also consider the following SDE ( dX π t,x ( s ) = (cid:0) r ( s ) X π t,x ( s ) + π ( s, X π t,x ( s )) ⊤ b ( s ) (cid:1) ds + π ( s, X π t,x ( s )) ⊤ σ ( s ) dW ( s ) , s ∈ [ t, T ] ,X π t,x ( t ) = x. (2.3)For each π ∈ Π, the above SDE has a unique strong solution, which represents the agent’swealth process if she starts at time t with wealth x and follows π to invest. Denote by F π ( t, x, y ) := P ( X π t,x ( T ) ≤ y ) , y ∈ R (2.4)the cumulative distribution function of the terminal wealth given wealth level of x at time t , and denote by G π ( t, x, α ) := sup { y ∈ R : F π ( t, x, y ) ≤ α } , α ∈ (0 , x attime t . In particular, G π ( t, x, /
2) is the median of the terminal wealth given wealth levelof x at time t .Suppose that the agent wants to maximize the α -level quantile of her terminal wealth.In particular, when α = 1 /
2, the agent is a median maximizer. Because quantile maximiza-tion leads to time-inconsistency, we consider so-called intra-personal equilibrium. Formally,suppose that we are given a strategy ˆ π ∈ Π. Denote by X x , ˆ π t the set of reachable wealthlevels at time t from the initial wealth x at time 0 and following the strategy ˆ π ; i.e., X x , ˆ π t is defined as follows: X x , ˆ π t = int( S X ˆ π ,x ( t ) ) ∪ n x ∈ ∂ S X ˆ π ,x ( t ) : P (cid:0) X ˆ π ,x ( t ) ∈ B δ ( x ) ∩ ∂ S X ˆ π ,x ( t ) (cid:1) > δ > o , (2.5)where B δ ( x ) denotes the ball with radius δ and centered at x , S X ˆ π ,x ( t ) is the support of X ˆ π ,x ( t ) and int( S X ˆ π ,x ( t ) ), ∂ S X ˆ π ,x ( t ) denote the interior and the boundary of S X ˆ π ,x ( t ) respec-tively. Definition 1 ˆ π ∈ Π is an intra-personal equilibrium for α -level quantile maximization iffor any t ∈ [0 , T ), x ∈ X x , ˆ π t , and π = ˆ π ( t, x ) with Qπ ≥
0, there exists ǫ ∈ (0 , T − t ) such9hat G ˆ π t,ǫ,π ( t, x, α ) − G ˆ π ( t, x, α ) ≤ , ∀ ǫ ∈ (0 , ǫ ] , (2.6)where ˆ π t,ǫ,π ( s, y ) = π, s ∈ [ t, t + ǫ ) , y ∈ R , ˆ π ( s, y ) , s / ∈ [ t, t + ǫ ) , y ∈ R . (2.7)Imagine that at time t , the agent is only able to control herself for a period of length ǫ ,so she can choose to invest any dollar amount π in the risky assets in the period [ t, t + ǫ )and after that period she is expected follow certain given strategy, e.g., ˆ π . As a result, thestrategy that the agent will actually implement until the end date is ˆ π t,ǫ,π as defined by(2.7). Definition 1 then stipulates that ˆ π is an intra-personal equilibrium if at any time t with any wealth level x that is reachable at that time under ˆ π , the objective function,namely, the quantile of the terminal wealth, becomes smaller if she chooses an alternative amount π (satisfying the portfolio constraints) to invest in the risky assets, assuming thatshe is only able to control herself to invest π dollars in the risky assets in an infinitesimallysmall period of time.The above definition of equilibrium strategies is so-called regular equilibrium, whichslightly differs from the notion of weak equilibrium that is used in most studies of continuous-time time-inconsistent problems in the literature. As explained in He and Jiang (2019), thenotion of regular equilibrium is preferred to the notion of weak equilibrium because the agentcan still be willing to deviate from a weak equilibrium strategy and take a very differentalternative.Because the discount factor for the period [0 , s ], namely e − R s r ( u ) du , is a deterministicfunction of s ∈ [0 , T ], maximizing the quantile of the terminal wealth is equivalent to max-imizing the quantile of the discounted terminal wealth. Thus, for notational simplicity, inthe following presentation, we set r ≡ Let us first present Kelly’s portfolio. Assume the following, which stipulates that at eachtime, one can find a portfolio with a positive instantaneous mean return rate:10 ssumption 2
For any t ∈ [0 , T ) , the set { v ∈ R m | b ( t ) ⊤ v > , Qv ≥ } is nonempty andfor any t ∈ (0 , T ] , the set { v ∈ R m | b ( t − ) ⊤ v > , Qv ≥ } is nonempty, where b ( t − ) denotesthe left-limit of b at t . For each t ∈ [0 , T ), denote by v ∗ ( t ) the optimal solution to following problem ( min v ∈ R m v ⊤ σ ( t ) σ ( t ) ⊤ v − b ( t ) ⊤ v, subject to Qv ≥ . (3.1) Lemma 1
Suppose Assumptions 1 and 2 hold. For each fixed t ∈ [0 , T ) , v ∗ ( t ) = 0 and b ( t ) ⊤ v ∗ ( t ) = k σ ( t ) ⊤ v ∗ ( t ) k > . Consequently, v ∗ ∈ C pw ([0 , T ); R m ) , inf t ∈ [0 ,T ) k v ∗ ( t ) k > ,and inf t ∈ [0 ,T ) k σ ( t ) ⊤ v ∗ ( t ) k > . It is well known that Kelly’s portfolio, namely the one that maximizes the expectedlogarithmic utility of the terminal wealth, is π Kelly ( t, x ) = v ∗ ( t ) x, t ∈ [0 , T ) , x ∈ R . (3.2)See for instance Karatzas and Shreve (1998).We are particularly interested in the following family of affine strategies : A = (cid:8) π | π ( t, x ) = θ ( t ) + θ ( t ) x, t ∈ [0 , T ) , x ∈ R for some θ , θ ∈ C pw ([0 , T )) taking values in R m (cid:9) . Note that this family of strategies include many commonly used strategies, such as investinga certain proportion of wealth in the risky assets, investing a wealth-independent dollaramount in the risky assets, and the mixture of the above two strategies taken in differenttime periods. Note that Kelly’s strategy is an affine one.
Theorem 1
Suppose Assumptions 1 and 2 hold.(i) Suppose α = 1 / . Then, ˆ π ∈ A is an intra-personal equilibrium if and only if ˆ π ( t, x ) = v ∗ ( t )( x − ξ ) , t ∈ [0 , T ) , x ∈ R (3.3) for some constant ξ < x . ii) Suppose α ∈ (0 , / . Then, ˆ π ∈ A is an intra-personal equilibrium if and only if ˆ π ( t, x ) = θ ( t )( x − x ) , ∀ t ∈ [0 , T ] , x ∈ R (3.4) for some θ ∈ C pw ([0 , T )) .(iii) Suppose α ∈ (1 / , . Then, any ˆ π ∈ A is not an intra-personal equilibrium. Theorem 1-(i) characterizes all affine strategies that are intra-personal equilibria for me-dian maximization. The wealth process X ˆ π ,x under intra-personal equilibrium (3.3) satisfies ( d ( X ˆ π ,x ( t ) − ξ ) = ( X ˆ π ,x ( t ) − ξ ) (cid:2) v ∗ ( t ) ⊤ b ( t ) dt + v ∗ ( t ) ⊤ σ ( t ) dW ( t ) (cid:3) , t ∈ [0 , T ] ,X ˆ π ,x ( t ) = x > ξ. Therefore, X ˆ π ,x ( t ) > ξ for all t ∈ [0 , T ], and the set of reachable wealth levels at time t is ( ξ, + ∞ ) for t ∈ (0 , T ]. Thus, ξ stands for the guaranteed wealth level, or a portfolioinsurance level .By definition, revising the value of ˆ π ( t, x ) for x / ∈ X x , ˆ π t changes neither the wealth process X ˆ π ,x nor whether or not ˆ π is an intra-personal equilibrium. Therefore, any ˜ π such that ˜ π agrees with ˆ π as given by (3.3) for x ∈ X x , ˆ π t , t ∈ [0 , T ), e.g., ˜ π ( t, x ) = v ∗ ( t )( x − ξ ) + , t ∈ [0 , T ) , x ∈ R , is also an intra-personal equilibrium.Theorem 1-(i) also shows that there exist multiple intra-personal equilibria for medianmaximization, parameterized by the portfolio insurance level ξ , and the multiplicity hereis generic in that the dollar amount invested in the risky assets, ˆ π ( t, X ˆ π ,x ( t )), differs withrespect to different values of ξ . Multiplicity of intra-personal equilibria for time-inconsistentproblems has been noted in the literature both in discrete settings (see e.g., Vieille andWeibull 2009 and Cao and Werning 2018) and in continuous-time settings (see e.g., Ekelandand Lazrak 2010 and Cao and Werning 2016).Theorem 1-(ii) shows that an affine strategy ˆ π is an intra-personal equilibrium for quantilemaximization with quantile level α < / π ( t, x ) = 0 , t ∈ [0 , T ). As a result, under the equilibrium strategy, ˆ π ( t, X ˆ π ,x ( t )) = 0 , t ∈ [0 , T ), i.e., the agent does not invest in the risky assets at all. Theorem 1-(iii) shows thatfor quantile maximization with quantile level α > /
2, there does not exist an intra-personalequilibria that is affine in the wealth level.
Proposition 1
Suppose Assumptions 1 and 2 hold. Consider any strategy ˜ π such that ˜ π agrees with ˆ π as given by (3.3) for x ∈ X x , ˆ π t , t ∈ [0 , T ) and that ˜ π ( t, x ) is continuous in . Then, for any t ∈ (0 , T ) , there exists ǫ ∈ (0 , T − t ) such that G ˜ π t,ǫ,v ∗ ( t ) ( t, ξ, / > ξ = G ˆ π ( t, ξ, / for any ǫ ∈ (0 , ǫ ) , where ˜ π t,ǫ,π is defined similarly as in (2.7) with ˆ π thereinreplaced by ˜ π . In Definition 1, a portfolio strategy is an intra-personal equilibrium if at any time andany reachable wealth level, the agent is not willing to deviate from it in the sense of condition(2.6). It is reasonable to exclude wealth levels that are not reachable in the test of whethera strategy is an intra-personal equilibrium because the actions of the agent’s future selves atthose wealth levels are irrelevant from the perspective of the agent’s self today. Proposition 1shows that if we mechanically force the condition (2.6) to hold for all wealth levels, even thosethat are not reachable, then for the median maximization problem the portfolio strategieswe derive in Theorem 1 are no longer intra-personal equilibrium. Proposition 1 also showswhy we define the set of reachable wealth levels at each time t to be (2.5) rather than to bethe support of X ˆ π ,x ( t ): for the strategy (3.3), ξ is in the support of X ˆ π ,x ( t ), but it does notsatisfy the condition (2.6) and is actually not visited by the wealth process. In this section, we further discuss the intra-personal equilibria for median maximizationas given by (3.3). To highlight the dependence of this intra-personal equilibria on ξ , wedenote it as ˆ π ξ in the following. Proposition 2
For any ξ < x , G ˆ π ξ ( t, x, /
2) = ξ + ( x − ξ ) e R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds , x ∈ X x , ˆ π ξ t , t ∈ [0 , T ) . (4.1) Moreover, for any ξ < ξ < x , G ˆ π ξ ( t, x, / > G ˆ π ξ ( t, x, / for any t ∈ [0 , T ) and x ∈ X x , ˆ π ξ t ⊇ X x , ˆ π ξ t . The multiplicity of intra-personal equilibria for median maximization raises a questionof which equilibrium strategy to choose. Proposition 2 shows that with a smaller portfolio Almost all the existing literature on time inconsistency consider all states including those that are notreachable; see He and Jiang (2019) for a discussion and for the relevant references. ξ , at any time and any wealth level, the median of the terminal wealthbecomes strictly larger. On the other hand, with a smaller portfolio insurance level ξ , theagent’s wealth in the future can reach lower wealth levels. Therefore, ξ becomes a parameterthat trades off the growth of the portfolio, which is measured by the median of the terminalwealth, and the risk of the portfolio, which is measured by the lowest level the agent’swealth in the future may touch. As a comparison, the mean-variance portfolio selectionproblem features a tradeoff between the growth and risk of the portfolio that are measuredrespectively by the expectation and variance of the portfolio return.Now, imagine that an investor specifies a maximum amount of loss L , e.g., 20% of theinitial wealth, she can tolerate. Moreover, she wants to maximize the median of her terminalwealth and to have a consistent investment plan. Then, portfolio (3.3) with ξ = x − L canbe recommended to her. One of the critiques of Kelly’s strategy is that it entails too much risk, and to addressthis issue, the so-called fractional Kelly strategies have been proposed in the literature; seefor instance MacLean et al. (1992). Formally, fixing γ >
0, a γ -fractional Kelly strategy isdefined to be π γ − Kelly ( t, x ) = γv ∗ ( t ) x, t ∈ [0 , T ) , x ∈ R . (4.2)It is well known that in the market setting in the present paper, the γ -fractional Kellystrategy is the one that maximizes the expected utility of terminal wealth with a constantrelative risk aversion degree 1 /γ ; see for instance Karatzas and Shreve (1998). For γ ∈ (0 , γ -fractional Kelly strategy leads to less investment in the risky assets compared to theKelly strategy.Now, for the intra-personal equilibrium ˆ π ξ for median maximization with ξ ∈ (0 , x ), wehave ˆ π ξ ( t, x ) /x = (cid:0) ( x − ξ ) /x (cid:1) v ∗ ( t ) , x > ξ. Therefore, compared to Kelly’s strategy, ˆ π ξ implies less investment in the risky assets because( x − ξ ) /x <
1. In the following, we compare the intra-personal equilibrium ˆ π ξ with thefractional Kelly strategy in terms of their growth and risk.14 roposition 3 For any γ > , G π γ − Kelly ( t, x, /
2) = xe ( γ − γ ) R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds , x > , t ∈ [0 , T ) . (4.3) Moreover, for fixed ξ ∈ (0 , x ) , γ ∈ (0 , , and t ∈ [0 , T ) , we have a t,γ := e R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds − e R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds − e ( γ − γ ) R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds > , (4.4) and G π γ − Kelly ( t, x, / is strictly larger than (strictly smaller than, respectively) G ˆ π ξ ( t, x, / if and only if x < a t,γ ξ ( x > a t,γ ξ , respectively). Proposition 3 shows that at time t ∈ (0 , T ), neither of the intrapersonal equilibrium ˆ π ξ and the fractional Kelly strategy dominates the other in terms of median of the terminalwealth: the former implies a higher median of the terminal wealth than the latter when thewealth level is high and vice versa when the wealth level is low. At the initial time (withinitial wealth x ), which of the above two strategies imply a higher median of the terminalwealth depends on the value of ξ and γ . On the risk side, the intrapersonal equilibrium ˆ π ξ entails less risk than the fractional Kelly strategy in that the former implies a higher levelof minimum wealth. Finally, as implied by Theorem 1, the fractional Kelly strategy (exceptfor the case γ = 1) is not an intrapersonal equilibrium for median maximization; i.e., it isan inconsistent investment strategy for median maximization. Wachter and Yogo (2010) find that households with a higher level of wealth tend to havea larger portfolio shares in risky assets (i.e., to invest more percentage of their wealth in riskyassets). More precisely, in one of their studies, the authors consider a representative sample,provided by the Board of Governors of the Federal Reserve System, of approximately 3,000households. In this sample, the net worth, i.e., the wealth, and the portfolio share in riskyassets of each household is observed. The authors conduct linear regression with the log networth as the explanatory variable and the portfolio share in risky assets as the dependentvariable in the cross-section of households. The authors divide the households in the sampleinto four age groups: 26–35, 36–45, 56–65, 66–75, and find that the portfolio share in riskyassets is more positively correlated with the log net worth for elder age groups; see Table 4of Wachter and Yogo (2010).The above empirical finding cannot be explained by expected utility maximization with15 power utility function even when the asset prices have stock volatility, stochastic returnrates, or jumps, because the resulting optimal percentage of wealth invested in risky assetsis independent of the agent’s wealth. The mean-variance log return model proposed by Daiet al. (2019) cannot explain the finding either because in their model the percentage of wealthinvested in risky assets is independent of the agent’s wealth as well. Wachter and Yogo (2010)develop a life-cycle consumption and portfolio choice model to explain this empirical finding.The intra-personal equilibrium under median maximization is consistent with the empir-ical finding in Wachter and Yogo (2010). Suppose that the households follow ˆ π ξ for some ξ > x also becomeslarger on average, so the portfolio shares in risky assets, which is ( x − ξ ) /x under ˆ π ξ , alsobecome larger. To confirm the above intuition, we conduct a numerical analysis in thefollowing.Suppose that there are 3,000 households in the market, indexed by j = 1 , . . . , j is endowed with initial wealth x ,j and has a portfolio insurance level α , e.g., 60%, proportion of her initial wealth, i.e., ξ j = αx ,j , j = 1 , . . . , , j are( x ,j − ξ j ) /x ,j = 1 − α , which are the same for different households. Now, imagine thatafter t years, household j ’s investment in risky assets generate a gross return rate R t,j , soher wealth becomes X t,j = ξ j + ( x ,j − ξ j ) R t,j and thus her portfolio shares in risky assetsbecomes p t,j := ξ j + ( x ,j − ξ j ) R t,j − ξ j ξ j + ( x ,j − ξ j ) R t,j = (1 − α ) R t,j α + (1 − α ) R t,j . Note that after t = 10, 20, 30, and 40 years, the households’ age become 36–45, 46–55,56–65, and 66–75, respectively. Thus, for each t ∈ { , , , } , we follow Wachter andYogo (2010) to run linear regression with ln X t,j as the explanatory variable and p t,j as thedependent variable.We simulate the initial wealth of the 3,000 households by setting x ,j to be the j -thsample of ¯ x e ̺U , where U is a standard normal random variable. Thus, ln ¯ x and ̺ representsrespectively the average log net worth and the standard deviation of the log net worth acrosshouseholds with in the age group 26–35. We use the sample provided by Survey of ConsumerFinances that tracks the wealth of US households every three years from 1989 to 2016 toestimate ¯ x and ̺ . Following the study in Wachter and Yogo (2010), we exclude householdswith non-positive net worth or with no risky-asset holding from the sample. Using the data The sample is available at .
16n 2016, we obtain the following estimates: ¯ x = 61811 . ̺ = 0 . − α to be in the range 40%–60%, so we choose three values of α : 40%, 50% and60%.We simulate the gross return rate R t,j of the 3,000 households from the following distribu-tion: (1+ µt ) e − ̟ t + ̟ √ tZ , where Z is a standard normal random variable that is independentof U , µ = 4%, and ̟ >
0. In other words, we set the average excess return rate per yearacross households to be 4%, and ̟ measures the standard deviation of the annual log returnrate across households. Because we do not have the data to estimae ̟ , we simply choosethree values of ̟ in the following: 0.65%,0.70%,0.75%.Finally, we run linear regression with X t,j to be the explanatory variable and p t,j tobe the dependent variable to obtain the coefficient of ln X t,j . We repeat the simulationfor 2,000 times and report the mean and standard deviation (in parentheses) of the coef-ficient of ln X t,j in Table 1, where different rows and columns refer to different values of t ∈ { , , , } (corresponding to age 36–45, 46–55, 56–65, and 66–75, respectively) and ̟ ∈ { . , . , . } . We can see that the coefficient is indeed more positive for largerwith t , which is consistent with the empirical finding in Wachter and Yogo (2010). Moreover,we report in the last row of Table 1 the sensitivity of portfolio shares in risky assets withrespect to net worth in the empirical study of Wachter and Yogo (2010, Table 4), and itshows that our model can generate quantitatively comparable sensitives as well. When facing time inconsistency, some individuals may commit their future selves tofollow the plans they set up today that are optimal under today’s decision criteria, and suchplans are called pre-committed strategies. For instance, one can delegate her investment toa portfolio manager and asks the manager to maximize her decision criterion today. In thefollowing, we compare the intra-personal equilibrium ˆ π ξ under median maximization, whichis a rational choice of an agent who is not able to commit her future selves to followingher plan today, with the pre-committed strategy under median maximization. To facilitatethe comparison, we assume the same portfolio insurance level ξ in the derivation of thepre-committed strategy. 176–35 36–45 46–55 56–65 66–75 α = 40% 0 1.26 (0.09) 2.28 (0.11) 3.02 (0.11) 3.52 (0.11) ̟ = 0 . α = 50% 0 1.23 (0.11) 2.41 (0.13) 3.38 (0.15) 4.13 (0.14) α = 60% 0 1.06 (0.11) 2.27 (0.15) 3.41 (0.17) 4.39 (0.18) α = 40% 0 1.45 (0.10) 2.60 (0.12) 3.40 (0.11) 3.92 (0.12) ̟ = 0 . α = 50% 0 1.41 (0.11) 2.74 (0.14) 3.82 (0.15) 4.62 (0.16) α = 60% 0 1.22 (0.12) 2.60 (0.16) 3.88 (0.18) 4.96 (0.19) α = 40% 0 1.65 (0.11) 2.93 (0.13) 3.79 (0.12) 4.32 (0.12) ̟ = 0 . α = 50% 0 1.61 (0.12) 3.10 (0.15) 4.28 (0.16) 5.12 (0.16) α = 60% 0 1.39 (0.12) 2.95 (0.17) 4.37 (0.19) 5.53 (0.19)Wachter and Yogo (2010) 0.52 1.84 3.88 4.32Table 1: Sensitivity (in percentage) of portfolio shares in risky assets p t,j with respect to wealthln X t,j in the cross-section of households. The second to sixth columns refer the age group 26–35,36–45, 46–55, 56–65, 66–75, respectively, which correspond to t = 0, 10, 20, 30, and 40,respectively. The number of households is 3,000, and their initial net worth is simulated from¯ x e ̺U with ¯ x = 61811 . ̺ = 0 . U is a standard normal random variable. Thevalue of α is set to be 40%, 50%, and 60%. The gross return rate of the households are simulatedfrom (1 + µt ) e − ̟ t + ̟ √ tZ with µ = 4% and ̟ to one of the three values: =0.65%, 0.70%, and0.75%, where Z is a standard normal random variable independent of U . The numbers inparentheses are standard error of the estimates of the sensitivities. The last row reports thesensitivity of portfolio shares in risky assets with respect to net worth in the empirical study ofWachter and Yogo (2010, Table 4). roposition 4 (i) The pre-committed strategy π , pc that maximizes the time-0 median ofthe terminal wealth is π , pc ( t, x ) = ∆ , pc ( t, x ) v ∗ ( t )( x − ξ ) , t ∈ [0 , T ) , x > ξ, (4.5)∆ , pc ( t, x ) : = 1 qR Tt k σ ( τ ) ⊤ v ∗ ( τ ) k dτ × Φ ′ (cid:0) d ( t, z ( t, x )) (cid:1) Φ (cid:0) d ( t, z ( t, x )) (cid:1) ,d ( t, z ) : = − R Tt k σ ( τ ) ⊤ v ∗ ( τ ) k dτ + z qR Tt k σ ( τ ) ⊤ v ∗ ( τ )) k dτ with z ( t, x ) uniquely determined by x − ξ Φ (cid:0) d ( t, z ( t, x )) (cid:1) = x − ξ Φ (cid:18) − qR T k σ ( τ ) ⊤ v ∗ ( τ ) k dτ (cid:19) =: k ∗ . Moreover, X π , pc ,x ( T ) = ξ + k ∗ R T v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) ≥ , and ∆ , pc ( t, x ) is strictly decreasing, continuous in x and satisfies lim x ↓ ξ ∆ , pc ( t, x ) = + ∞ , lim x ↑ ξ + k ∗ ∆ , pc ( t, x ) = 0 . (ii) G π , pc (0 , x , /
2) = ξ + k ∗ , and G π , pc ( t, x, /
2) = ξ + k ∗ , x ∈ [ x , ξ + k ∗ ) ,ξ, x ∈ ( ξ, x ) , t ∈ (0 , T ) . In addition, G π , pc (0 , x , / > G ˆ π ξ (0 , x , / , and for any t ∈ (0 , T ) , G π , pc ( t, x, / ≥ G ˆ π ξ ( t, x, / , x ≤ x ≤ ξ + ( x − ξ )˜ a t ,< G ˆ π ξ ( t, x, / , x ∈ ( ξ, x ) ∪ ( ξ + ( x − ξ )˜ a t , ξ + k ∗ ) , (4.6)˜ a t := e − R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds / Φ − sZ T k σ ( τ ) ⊤ v ∗ ( τ ) k dτ ∈ (cid:0) , k ∗ / ( x − ξ ) (cid:1) . π ξ , the agent’s dollar amount invested inthe risky assets is proportional to the distance between the current wealth and portfolioinsurance level, and the proportion is independent of the current wealth level. For the pre-committed portfolio strategy, however, this proportion depends on the current wealth leveland can become arbitrarily large when the wealth approaches the portfolio insurance level.In terms of the wealth process, under the intra-personal equilibrium, the agent has potentialto attain arbitrarily high wealth levels in the future, but under the pre-committed portfoliostrategy, the wealth in the future is capped at certain level ξ + k ∗ .The pre-committed portfolio strategy obviously implies a higher level of time-0 medianof the terminal wealth than the intra-personal equilibrium. After the initial time, i.e., attime t ∈ (0 , T ), however, the pre-committed portfolio strategy results in smaller median ofthe terminal wealth than the intra-personal equilibrium when the wealth level at that timeis very low or very high; see (4.6).It can be costly to implement pre-committed strategies; for instance, portfolio delegationusually incurs some management fees. In some situations, individuals can be unaware of thetime-inconsistency or wrongly believe that they can commit their future selves to the plan setup today. As a result, they may keep re-optimizing and updating their plans over time. Inthe extreme case, at each instant an agent can only implement her plan for an infinitesimallysmall time period and re-optimizes and updates the plan afterwards. The resulting strategythat is actually implemented by the agent over time is called the naive strategy . Proposition 5
The naive portfolio strategy π na under median maximization is given by π na ( t, x ) = ∆ na ( t ) v ∗ ( t )( x − ξ ) , (4.7) where ∆ na ( t ) := 1 qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds × Φ ′ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) Φ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) . (4.8) Moreover, ∆ na ( t ) > , t ∈ [0 , T ) and lim t ↑ T ∆ na ( t ) = + ∞ . Furthermore, for any fixed t ∈ [0 , T ) and x > ξ , denoting by G π na ( t, x, / the limit of the median, conditional ontime- t wealth level of x , of the wealth at time τ as τ goes to T , we have G π na ( t, x, /
2) = ξ . Proposition 5 shows that under the naive strategy, the dollar amount invested in the20isky assets is also proportional to the distance between the current wealth level and theportfolio insurance level. Moreover, the proportion is always strictly larger than 1, implyinghigher risky asset holdings than the intra-personal equilibrium. The proportion even goes toinfinity when it is near the terminal time, showing that under the naive strategy, the agentwould take an infinite amount of risk around the terminal time.Because under the naive strategy the agent invests infinite amount of money in riskyassets around the terminal time, the terminal wealth of the naive strategy is not well defined.We, however, can still study the wealth around the terminal time, in particular the medianof the wealth at time τ when τ is very close to the terminal time. It turns out that the limitof the median exists when τ goes to the terminal time T , and the limit is ξ . This showsthat in terms of the median of terminal wealth, the naive strategy always underperforms theintra-personal equilibrium. The reason is because under the naive strategy the agent takesan infinite amount of risk around the terminal time, which significantly reduces the medianof the terminal wealth. Proposition 6
The intra-personal equilibrium ˆ π ξ for median maximization is the optimalportfolio strategy that maximizes E [ln( X π ( T ) − ξ )] . Proposition 6 shows that the intra-personal equilibrium ˆ π ξ under median maximizationis the same as the portfolio that maximizes the expected utility of the terminal wealth withutility function log( x − ξ ). This utility function is a special case of the so-called hyperbolicabsolute risk aversion (HARA); see for instance Section 6 of Merton (1971) and Kim andOmberg (1996). It is worth emphasizing that we are only able to prove the above equivalencebetween the intra-personal equilibrium under median maximization and expected utilitymaximization with a logarithmic utility function in the market setting in the present paper. Due to different life objectives, such as education, kids, and retirement, investors mayconcern their wealth at multiple time points that respectively correspond to those objectives;see for instance Sironi (2016). In the following, we generalize our model in Section 2 toaccount for multiple objectives.Suppose that the agent is concerned about the her wealth at not only the terminaltime T but also some intermediate moments. More precisely, consider multiple time points21 =: T < T < · · · < T N := T . At time t ∈ [ T n − , T n ) with wealth x , the agent’sdecision criterion at that time is a weighted average of the α -level quantile of her wealth at T n , T n +1 , . . . , T N , i.e., is J π ( t, x, α ) = i = N X i = n w n,i G π ( t, x, α ; T i ) , (5.1)where G π ( t, x, α ; T i ) stands for the α -level quantile of X π t,x ( T i ), w n,i ≥ , i = n, . . . , N areconstants and satisfy P Ni = n w n,i = 1, and w n,N > Definition 2 ˆ π ∈ Π is an intra-personal equilibrium for multiple-time-point α -level quantilemaximization if for any n = 1 , . . . , N , t ∈ [ T n − , , T n ), x ∈ X x , ˆ π t , and π = ˆ π ( t, x ) with Qπ ≥
0, there exists ǫ ∈ (0 , T − t ) such that J ˆ π t,ǫ,π ( t, x, α ) − J ˆ π ( t, x, α ) ≤ , ∀ ǫ ∈ (0 , ǫ ] , (5.2)where ˆ π t,ǫ,π is given by (2.7). Theorem 2
Suppose Assumptions 1 and 2 hold.(i) Suppose α = 1 / . Then, ˆ π ∈ A is an intra-personal equilibrium for multiple-time-point α -level quantile maximization if and only if ˆ π is given by (3.3) for some constant ξ < x .(ii) Suppose α ∈ (0 , / . Then, ˆ π ∈ A is an intra-personal equilibrium for multiple-time-point α -level quantile maximization if and only if it is given by (3.4) for some θ ∈ C pw ([0 , T )) .(iii) Suppose α ∈ (1 / , . Then, any ˆ π ∈ A is not an intra-personal equilibrium formultiple-time-point α -level quantile maximization. Theorem 2 shows that the intra-personal equilibrium for multiple-time-point quantilemaximization is the same as for single-time-point quantile maximization. In particular,for median maximization, although the decision criterion jumps at each time point T i , theportfolio insurance level remains constant over time.22 Conclusions
Median is a popular alternative to mean as a summary statistic of a distribution. In thispaper, we studied portfolio selection under the α -quantile maximization, particularly undermedian maximization when α is set to be 1 /
2. We considered an agent who trades a risk-free asset and multiple risky assets continuously in time with an objective of maximizing the α -quantile of her wealth at certain terminal time, and the mean return rates and volatilityof the assets are assumed to be deterministic. Because of time inconsistency, we consideredintra-personal equilibrium strategies.We found that in the class of time-varying, affine portfolio strategies, the intra-personalequilibrium does not exist when α > / α < /
2. For the case of α = 1 /
2, namely the case of median maximization, a time-varying affine strategy is an intra-personal equilibrium strategy if and only if it is a portfolioinsurance strategy. Different choices of the portfolio insurance level then induce differentintra-personal equilibria and can be interpreted as different degrees of risk attitude of theagent.We compared the intra-personal equilibrium strategy under median maximization tofractional Kelly strategies, the pre-committed strategy, and the naive strategy under medianmaximization, and found that the intra-personal equilibrium strategy is better than the oth-ers. We also showed that the intra-personal equilibrium strategy can explain why householdswith a higher level of wealth tend to invest more percentage of their wealth in risky assets.Finally, we extended our model to the case in which the agent is concerned about median ofwealth at multiple time points and derived similar results.
A Proofs
A.1 Proof of Theorem 1
We introduce some notations to be used in the following proof.For any interval [ a, b ) and open set O in R l , denote by C , ∞ ([ a, b ) × O ) the set of functions g ( t, z ) from [ a, b ) × O to R such that its derivatives with respect to z of any order exist andare continuous in ( t, z ) on [ a, b ) × O and by C , ∞ ([ a, b ) × O ) the set of functions g ( t, z ) from[ a, b ) × O to R such that its first-order derivative with respect to t and its derivatives withrespect to z of any order exist and are continuous in ( t, z ) on [ a, b ) × O .For any x ∈ R l and δ ≥
0, denote by B ℓδ ( x ) := { y ∈ R l | k y − x k ≤ δ } .23he proof of Theorem 1 is divided into three parts: In Section A.1.1, we prove a cruciallemma. In Section A.1.2, we prove the sufficiency part of the theorem: ˆ π as given by (3.3) and(3.4) are equilibrium strategies for respective α ’s. In Section A.1.3, we prove the remainingpart of the theorem. Because the proof is involved, we present it by summarizing importantintermediate steps of the proofs as lemmas and relegate all proofs in Section A.1.4. A.1.1 Calculation of DerivativesLemma 2
Consider any ˆ π ∈ A , i.e., ˆ π ( t, x ) = θ ( t ) + θ ( t ) x, t ∈ [0 , T ) , x ∈ R with θ , θ ∈ C pw ([0 , T )) , and define t ∗ : = inf { t ∈ [0 , T ) : θ ( s ) = θ ( s ) = 0 , ∀ s ∈ [ t, T ) } , (A.1) t ∗ : = inf { t ∈ [0 , t ∗ ) : θ ( s ) + ξθ ( s ) = 0 , ∀ s ∈ [ t, t ∗ ) and some ξ ∈ R } . (A.2) Then, if t ∗ < t ∗ , there exists unique ξ ∈ R such that θ ( s ) + ξθ ( s ) = 0 , ∀ s ∈ [ t ∗ , t ∗ ) . Denote S ˆ π t = ∅ , t ∈ [0 , t ∗ ) , S ˆ π t = { ξ } , t ∈ [ t ∗ , T ) . (A.3) Then, for any t ∈ [0 , t ∗ ) , there exists η ∈ (0 , t ∗ − t ) such that for any x ∈ R \ S ˆ π t , α ∈ (0 , ,and π ∈ R m , G ˆ π t,ǫ,π ( t, x, α ) is continuous in ǫ ∈ [0 , η ) , F ˆ π y ( t, x, G ˆ π ( t, x, α )) > , and lim ǫ ↓ G ˆ π t,ǫ,π ( t, x, α ) − G ˆ π ( t, x, α ) ǫ = − A π F ˆ π ( t, x, G ˆ π ( t, x, α )) F ˆ π y ( t, x, G ˆ π ( t, x, α )) = ϕ ˆ π t,x,α ( ˆ π ( t, x )) − ϕ ˆ π t,x,α ( π ) F ˆ π y ( t, x, G ˆ π ( t, x, α )) , (A.4) where A π is applied to F ˆ π ( t, x, y ) as a function of ( t, x ) with A π f ( t, x ) := f t ( t, x )+ b ( t ) ⊤ πf x ( t, x )+ k σ ( t ) ⊤ π k f xx ( t, x ) and ϕ ˆ π t,x,α ( v ) := F ˆ π x ( t, x, G ˆ π ( t, x, α )) b ( t ) ⊤ v + 12 F ˆ π xx ( t, x, G ˆ π ( t, x, α )) k σ ( t ) ⊤ v k . (A.5)Lemma 2 provides the derivative G ˆ π t,ǫ,π ( t, x, α ) in ǫ = 0. For t ∈ [ t ∗ , T ), it is obvious that F ˆ π ( t, x, y ) = x ≤ y and thus is not differentiable in x and y . For t ∈ [0 , t ∗ ), we have desireddifferentiability except at singular points x ∈ S ˆ π t .24 .1.2 SufficiencyProposition 7 ˆ π as given by (3.3) is an equilibrium strategy for α = 1 / and ˆ π as givenby (3.4) is an equilibrium strategy for α ∈ (0 , / . A.1.3 NecessityLemma 3
Consider ˆ π ∈ A and define t ∗ , t ∗ , ξ , and S ˆ π t as in Lemma 2. Suppose that ˆ π isan intra-personal equilibrium strategy for a given α ∈ (0 , . Then, for any t ∈ [0 , t ∗ ) and x ∈ X x , ˆ π t \ S ˆ π t , we have F ˆ π xx ( t, x, G ˆ π ( t, x, α )) > , F ˆ π x ( t, x, G ˆ π ( t, x, α )) < , and ˆ π ( t, x ) = − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) v ∗ ( t ) . (A.6)Lemma 3 proves a necessary condition for ˆ π ∈ A to be an equilibrium strategy. Lemma 4
Consider ˆ π ∈ A , recall t ∗ , t ∗ , and ξ as defined in Lemma 2, and define t := sup { s ∈ [0 , T ] : θ ( τ ) + θ ( τ ) x = 0 , ∀ τ ∈ [0 , s ] } . (A.7) Then, t ≥ t ∗ if and only if t = T . Moreover, if ˆ π is an intra-personal equilibrium strategyfor α ∈ [1 / , , then t ∗ = T and t = 0 . The differentiability result in Lemma 2 applies to t ∈ [0 , t ∗ ) only. On the other hand, for t ∈ [0 , t ], X x , ˆ π t = { x } is a singleton, so we cannot obtain too much information about theproperty of an equilibrium strategy. Thus, we expect to conduct analysis of the equilibriumstrategy for t ∈ ( t, t ∗ ), and Lemma 4 provides some properties of the interval ( t, t ∗ ). Lemma 5
Consider ˆ π ∈ A , recall t ∗ , t ∗ , ξ , and S ˆ π t as defined in Lemma 2 and t as definedin Lemma 4. Suppose t < t ∗ and ˆ π is an intra-personal equilibrium strategy for a given α ∈ (0 , . Then,(i) For each t ∈ ( t, T ] , X x , ˆ π t is either ( x ( t ) , + ∞ ) for some x ( t ) ∈ R , or ( −∞ , ¯ x ( t )) forsome ¯ x ( t ) ∈ R , or R . Moreover, X x , ˆ π t is increasing in t ∈ [0 , T ] .(ii) There exists a , a ∈ C ([ t, t ∗ ]) taking values in R such that − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) = a ( t ) + a ( t ) x > , t ∈ ( t, t ∗ ) , x ∈ X x , ˆ π t \ S ˆ π t , (A.8)ˆ π ( t, x ) = (cid:0) a ( t ) + a ( t ) x (cid:1) v ∗ ( t ) , ( t, x ) ∈ ( t, t ∗ ) × R . (A.9)25 onsequently, the following PDE holds: ( G ˆ π t ( t, x, α ) + G ˆ π x ( t, x, α ) ρ ( t ) (cid:0) a ( t ) + a ( t ) x (cid:1) = 0 , t ∈ ( t, t ∗ ) , x ∈ X x , ˆ π t \ S ˆ π t , lim t ↑ t ∗ ,x ′ → x G ˆ π t ( t, x ′ , α ) = x, x ∈ R , (A.10) where ρ ( t ) := 2 b ( t ) ⊤ v ∗ ( t ) − k σ ( t ) ⊤ v ∗ ( t ) k . (A.11)Lemma 5 shows that if ˆ π ∈ A is an intra-personal equilibrium, then it must take theform (A.9) in the time interval ( t, t ∗ ) and the quantile of the terminal wealth under ˆ π ∈ A satisfies the PDE (A.10). Note that this PDE is a linear transportation equation defined ona possibly strict subset of ( t, t ∗ ) × R . In order to identify the equilibrium strategy ˆ π , weneed to solve a and a , so it is crucial to solve the transportation equation. Lemma 6
Fix τ < τ , ¯ c ∈ R , c, γ , γ ∈ C ([ τ , τ ]) taking values in R , and α , α ∈ R .Define ˆ g ( t, x ) := α + α (cid:20)Z τ t γ ( s ) e R τ s γ ( τ ) dτ ds + xe R τ t γ ( s ) ds (cid:21) , t ∈ [ τ , τ ] × R . (A.12) Denote X ,t := (¯ c, + ∞ ) , X ,t := ( −∞ , ¯ c ) , and X ,t := ( c ( t ) , + ∞ ) , t ∈ [ τ , τ ] . For each i = 1 , , , define D i = { ( t, x ) | x ∈ X i,t , t ∈ [ τ , τ ) } , ¯ D i = { ( t, x ) | x ∈ X i,t , t ∈ [ τ , τ ] } , denote by C , ( ¯ D i ) the set of real-valued functions that are continuous on ¯ D i and differentiableon D i , and consider ( g t ( t, x ) + g x ( t, x )( γ ( t ) + γ ( t ) x ) = 0 , ( t, x ) ∈ D i ,g ( τ , x ) = α + α x, x ∈ X i,τ . (A.13) (i) If g ∈ C , ( ¯ D ) is the solution to (A.13) with i = 1 , then there exists x ≥ ¯ c such that g ( t, x ) = ˆ g ( t, x ) , ( t, x ) ∈ [ τ , τ ] × ( x, + ∞ ) .(ii) If g ∈ C , ( ¯ D ) is the solution to (A.13) with i = 2 , then there exists ¯ x ≤ ¯ c such that g ( t, x ) = ˆ g ( t, x ) , ( t, x ) ∈ [ τ , τ ] × ( −∞ , ¯ x ) . iii) If c ( t ) is decreasing in t ∈ [ τ , τ ] , γ ( t )+ γ ( t ) x > for all ( t, x ) ∈ D , and g ∈ C , ( ¯ D ) is the solution to (A.13) with i = 3 , then g ( t, x ) = ˆ g ( t, x ) , ( t, x ) ∈ D . Lemma 6 solves the linear transportation equation with affine terminal condition. Thesolution is also an affine function of x for each t . Lemma 7
Consider ˆ π ∈ A , recall t ∗ , t ∗ , ξ , and S ˆ π t as defined in Lemma 2 and t as definedin Lemma 4. Suppose t < t ∗ and ˆ π is an intra-personal equilibrium for a given α ∈ (0 , ,and recall a and a as defined in Lemma 5. Define β ( t, α ) = e R t ∗ t a ( s ) ρ ( s ) ds , β ( t, α ) := 12 Z t ∗ t a ( s ) ρ ( s ) β ( s, α ) ds, t ∈ [ t, t ∗ ] . (A.14) Fix any t ∈ ( t, t ∗ ) .(i) There exists s ∈ [ t, t ∗ ) such that a ( s ) = 0 . Consequently, Z t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds > . (A.15) (ii) If X x , ˆ π t ⊇ ( x t , + ∞ ) for certain x t ∈ R , then there exists c t > max( x t , ξ ) such that G ˆ π ( s, x, α ) = β ( s, α ) x + β ( s, α ) (A.16) for all ( s, x ) ∈ [ t, t ∗ ] × [ c t , + ∞ ) . Moreover, Z t ∗ t a ( s ) (cid:0) − a ( s ) (cid:1) k σ ( s ) ⊤ v ∗ ( s ) k ds + Φ − ( α ) sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds = 0 . (A.17) (iii) If X x , ˆ π t ⊇ ( −∞ , ¯ x t ) for certain ¯ x t ∈ R , then there exists ¯ c t < min( ξ, ¯ x t ) such that (A.16) holds all ( s, x ) ∈ [ t, t ∗ ] × ( −∞ , ¯ c t ) . Moreover, Z t ∗ t a ( s ) (cid:0) − a ( s ) (cid:1) k σ ( s ) ⊤ v ∗ ( s ) k ds + Φ − (1 − α ) sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds = 0 . (A.18)Lemma 5 already proved that an equilibrium strategy ˆ π ∈ A must take the form (A.9)with a and a undetermined. Lemma 7 provides a necessary condition for a , which is then27sed in the following Proposition 8 to characterize equilibrium strategies for α = 1 / α = 1 / Proposition 8
For α ∈ (0 , / , ˆ π ∈ A is an intra-personal equilibrium if and only if ˆ π isgiven by (3.4) for some θ ∈ C pw ([0 , T )) taking values in R m . For α ∈ (1 / , , any ˆ π ∈ A isnot an intra-personal equilibrium. Proposition 9
For α = 1 / , ˆ π ∈ A is an intra-personal equilibrium if and only if it isgiven by (3.3) for some ξ < x . A.1.4 Detailed Proofs
Proof of Lemma 2
Suppose t ∗ < t ∗ and fix any t ∈ [ t ∗ , t ∗ ). By Proposition 13-(i), there exists η ∈ (0 , t ∗ − t ) such that F ˆ π ( s, x, y ) ∈ C , ∞ ([ t, t + η ] × R \ ( ξ, ξ )), F ˆ π t ( s, x, y ) ∈ C , ∞ ([ t, t + η ] × R \ ( ξ, ξ )), F ˆ π t is bounded on [ t, t + η ] × R \ ( ξ, ξ ), and the derivatives of F ˆ π and F ˆ π t with respect to x and y of any order are bounded on [ t, t + η ] × R \ B δ ( ξ ) for any δ > π ∈ R m . For any fixed x ∈ R , y = ξ , noting that F ˆ π t,ǫ,π ( t, x, y ) = E [ F ˆ π ( t + ǫ, X πt,x ( t + ǫ ) , y )] and recalling that for any δ > F ˆ π t and F ˆ π x are bounded on [ t, t + η ] × R \ B δ ( ξ ), wederive by the dominated convergence theorem that for any ǫ ∈ [0 , η ),lim ǫ → ǫ sup x ∈ R ,y ∈ R \ B δ ( ξ ) (cid:12)(cid:12) F ˆ π t,ǫ,π ( t, x, y ) − F ˆ π t,ǫ ,π ( t, x, y ) (cid:12)(cid:12) = 0 . The above, together with the continuity of F ˆ π t,ǫ,π ( t, x, y ) in y = ξ as implied by Proposition13-(iii), yields that F ˆ π t,ǫ,π ( t, x, y ) is continuous in ( ǫ, y ) with ǫ ∈ [0 , η ) and y = ξ . Now, forany x = ξ , Proposition 14-(ii) shows that for any α ∈ (0 , F ˆ π t,ǫ,π ( t, x, G ˆ π t,ǫ,π ( t, x, α )) = α , G ˆ π t,ǫ,π ( t, x, α ) = ξ , and F ˆ π t,ǫ,π y ( t, x, G ˆ π t,ǫ,π ( t, x, α )) >
0. By Proposition 13-(iii), F ˆ π t,ǫ,π y ( t, x, y )is continuous in y = ξ . The implicit function theorem then yields that G ˆ π t,ǫ,π ( t, x, α ) iscontinuous in ǫ ∈ [0 , η ).Fixing x ∈ R , because G ˆ π t, ,π ( t, x, α ) = G ˆ π ( t, x, α ) = ξ and because G ˆ π t,ǫ,π ( t, x, α ) iscontinuous in ǫ , there exists δ > | G ˆ π t,ǫ,π ( t, x, α ) − ξ | > δ for sufficiently small ǫ .Because the derivatives of F ˆ π and F ˆ π t with respect to x and y of any orders are bounded on[ t, t + η ) × R \ B δ ( ξ ), there exists L > s ∈ [ t,η ) (cid:12)(cid:12) A π F ˆ π ( s, x, y ) − A π F ˆ π ( s, x, y ) (cid:12)(cid:12) ≤ L (1 + | x | + | x | ) | y − y | , x ∈ R , ( y , y ) ∈ [ ξ + δ, + ∞ ) ∪ ( −∞ , ξ − δ ] . (A.19)28ecause E (cid:2) sup s ∈ [ t,η ) | X πt,x ( s ) | p (cid:3) < + ∞ for any p ≥
1, we conclude that from (A.19) that forsufficiently small ǫ > E " sup s ∈ [ t,η ) (cid:12)(cid:12) A π F ˆ π ( s, X πt,x ( s ) , G ˆ π t,ǫ,π ( t, x, α )) − A π F ˆ π ( s, X πt,x ( s ) , G ˆ π ( t, x, α )) (cid:12)(cid:12) ≤ L ′ | G ˆ π t,ǫ,π ( t, x, α ) − G ˆ π ( t, x, α ) | (A.20)for some constant L ′ >
0. On the other hand, we have F ˆ π t,ǫ,π (cid:0) t, x, G ˆ π t,ǫ,π ( t, x, α ) (cid:1) − F ˆ π (cid:0) t, x, G ˆ π t,ǫ,π ( t, x, α ) (cid:1) = E [ F ˆ π ( t + ǫ, X πt,x ( t + ǫ ) , G ˆ π t,ǫ,π ( t, x, α ))] − F ˆ π (cid:0) t, x, G ˆ π t,ǫ,π ( t, x, α ) (cid:1) = A π F ˆ π ( t, x, G ˆ π ( t, x, α )) ǫ + E h Z t + ǫt (cid:16) A π F ˆ π ( s, X πt,x ( s ) , G ˆ π t,ǫ,π ( t, x, α )) − A π F ˆ π ( s, X πt,x ( s ) , G ˆ π ( t, x, α )) (cid:17) ds i + E (cid:20)Z t + ǫt (cid:16) A π F ˆ π ( s, X πt,x ( s ) , G ˆ π ( t, x, α )) − A π F ˆ π ( t, x, G ˆ π ( t, x, α )) (cid:17) ds (cid:21) . Combining the above with (A.20), recalling that F ˆ π t , F ˆ π x , and F ˆ π xx are bounded on [ t, t + η ) × R \ B δ ( ξ ), noting that E (cid:2) sup s ∈ [ t,η ) | X πt,x ( s ) | p (cid:3) < + ∞ for any p ≥
1, and applying thedominated convergence theorem, we conclude thatlim ǫ ↓ F ˆ π t,ǫ,π (cid:0) t, x, G ˆ π t,ǫ,π ( t, x, α ) (cid:1) − F ˆ π (cid:0) t, x, G ˆ π t,ǫ,π ( t, x, α ) (cid:1) ǫ = A π F ˆ π ( t, x, G ˆ π ( t, x, α )) . (A.21)Because F ˆ π t,ǫ,π ( t, x, G ˆ π t,ǫ,π ( t, x, α )) = α for any ǫ ∈ [0 , η ), we conclude that0 = F ˆ π t,ǫ,π ( t, x, G ˆ π t,ǫ,π ( t, x, α )) − F ˆ π ( t, x, G ˆ π t,ǫ,π ( t, x, α )) ǫ + F ˆ π ( t, x, G ˆ π t,ǫ,π ( t, x, α )) − F ˆ π ( t, x, G ˆ π ( t, x, α )) ǫ . Because F ˆ π y is continuous on [ t, t + η ) × R \ ( ξ, ξ ), G ˆ π ( t, x, α ) = ξ , and F ˆ π y ( t, x, y ) > y with F ˆ π ( t, x, y ) ∈ (0 , A ˆ π F ˆ π ( s, x, y ) = 0 on [ t, t + η ) × R \ ( ξ, ξ )).The last equality in (A.4) then follows.Suppose t ∗ > t ∈ [0 , t ∗ ). By Proposition 13-(ii), there exists η ∈ (0 , t ∗ − t )29uch that F ˆ π ( s, x, y ) ∈ C , ∞ ([ t, t + η ) × R , F ˆ π t ( s, x, y ) ∈ C , ∞ ([ t, t + η ) × R , the deriva-tives of F ˆ π with respect to x and y of any order are bounded on [ t, t + η ) × R , andsup s ∈ [ t,t + η ) ,y ∈ R | ∂ i + j F ˆ π t ∂x i ∂y j ( s, x, y ) | is of polynomial growth in x for any i, j ∈ N . The remainingproof then follows the same line as for the case t ∈ [ t ∗ , t ∗ ). (cid:3) Proof of Proposition 7
We first consider the case α = 1 /
2. It is straightforward to see that d ( X ˆ π ,x ( s ) − ξ ) = ( X ˆ π ,x ( s ) − ξ ) (cid:2) v ∗ ( s ) ⊤ b ( s ) ds + v ∗ ( s ) ⊤ σ ( s ) dW ( s ) (cid:3) , s ∈ [0 , T )and X ˆ π ,x (0) − ξ = x − ξ >
0. Because v ∗ ( s ) = 0 and σ ( s ) σ ( s ) ⊤ is positive definition, s ∈ [0 , T ), we conclude X x , ˆ π = { x } and X x , ˆ π t = ( ξ, + ∞ ), t ∈ (0 , T ). For every t ∈ [0 , T ),by the definition of v ∗ ( t ), Qv ∗ ( t ) ≥
0. Because X ˆ π ,x ( t ) > ξ , we immediately conclude Q ˆ π ( t, X ˆ π ,x ( t )) ≥
0. In addition, v ∗ ( t ) is bounded in t ∈ [0 , T ). Thus, ˆ π ∈ Π.Now, fix any t ∈ [0 , T ) and x ∈ X x , ˆ π t . Straightforward calculation shows that F ˆ π x ( t, x, G ˆ π ( t, x, / − φ (0) qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds ( x − ξ ) < ,F ˆ π xx ( t, x, G ˆ π ( t, x, / − ( x − ξ ) − F ˆ π x ( t, x, G ˆ π ( t, x, / > . Then, because v ∗ ( t ) is the optimal solution to (3.1), ˆ π ( t, x ) = v ∗ ( t )( x − ξ ) is the uniqueoptimal solution of ( min π ∈ R m ϕ ˆ π t,x,α ( π )subject to Qπ ≥ , (A.22)with α = 1 /
2, where ϕ ˆ π t,x,α is defined by (A.5). As a result, for any π ∈ R m with Qπ ≥ π = ˆ π ( t, x ), Lemma 2 yields thatlim ǫ ↓ G ˆ π t,ǫ,π ( t, x, / − G ˆ π ( t, x, / ǫ < . Thus, ˆ π is an equilibrium strategy.Next, we consider the case α < /
2. It is straightforward to see that X ˆ π ,x ( t ) ≡ x andthus X x , ˆ π t = { x } , t ∈ [0 , T ]. As a result, ˆ π ( t, X ˆ π ,x ( t )) = 0 , t ∈ [0 , T ). In addition, θ ( t ) isbounded in t ∈ [0 , T ). Thus, ˆ π ∈ Π.Fix any t ∈ [0 , T ) and π ∈ R m with π = ˆ π ( t, x ) = 0. For any ǫ ∈ (0 , T − t ), straightfor-30ard calculation leads to X ˆ π t,ǫ,π t,x ( T ) − x = (cid:20)Z t + ǫt b ( s ) ⊤ πds + Z t + ǫt π ⊤ σ ( s ) dW ( s ) (cid:21) e R Tt + ǫ [ b ( τ ) ⊤ θ ( τ ) − k σ ( τ ) ⊤ θ ( τ ) k ] dτ + R Tt + ǫ θ ( τ ) ⊤ σ ( τ ) dW ( τ ) . (A.23)As a result, F ˆ π t,ǫ,π ( t, x , x ) = P ( X ˆ π t,ǫ,π t,x ( T ) ≤ x ) = P (cid:18)Z t + ǫt b ( s ) ⊤ πds + Z t + ǫt π ⊤ σ ( s ) dW ( s ) ≤ (cid:19) = Φ − R t + ǫt b ( s ) ⊤ πds qR t + ǫt k σ ( s ) ⊤ π k ds . (A.24)Because α < /
2, the right-hand side of the above is strictly larger than α when ǫ is suf-ficiently small. As a result, G ˆ π t,ǫ,π ( t, x , α ) ≤ x for sufficiently small ǫ . Thus, ˆ π is anequilibrium strategy. (cid:3) Proof of Lemma 3
For any t ∈ [0 , t ∗ ) and x ∈ X x , ˆ π t and for any t ∈ [ t ∗ , t ∗ ) and x ∈ X x , ˆ π t with x = ξ , Lemma 2 implies that ϕ ˆ π t,x,α ( ˆ π ( t, x )) ≤ ϕ ˆ π t,x,α ( π ) , ∀ π = ˆ π ( t, x ) with Qπ ≥ F ˆ π x ( t, x, G ˆ π ( t, x, α )) <
0. On the other hand, because ofAssumption 2, we can find v ∈ R m with Qv ≥ b ( t ) ⊤ v >
0. Then Q ( λv ) ≥ λ >
0. If F ˆ π xx ( t, x, G ˆ π ( t, x, α )) ≤
0, we consider λv for sufficiently large, positivescalar λ so that λv = ˆ π ( t, x ) and ϕ ˆ π t,x,α ( ˆ π ( t, x )) > ϕ ˆ π t,x,α ( λv ), which contradicts (A.25).Thus, we must have F ˆ π xx ( t, x, G ˆ π ( t, x, α )) >
0. Then, (A.25) immediately implies that − ˆ π ( t, x ) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) /F ˆ π x ( t, x, G ˆ π ( t, x, α ))is the optimizer of (3.1), i.e., (A.6) holds. (cid:3) Proof of Lemma 4 If t = T , then it is obvious that t ≥ t ∗ . On the other hand, by thedefinition of t ∗ , θ ( τ ) + θ ( τ ) x = 0 , τ ∈ [ t ∗ , T ). Thus, t ≥ t ∗ implies t = T .Next, we fix α ∈ [1 / , t ∗ < T . Then, for any t ∈ [ t ∗ , T ) and x ∈ X x , ˆ π t , we have ˆ π ( t, x ) = 0. Choose any π ∈ R m with b ( t ) ⊤ π > π ≥
0. Note that such π exists due to Assumption 2. Then, we must have π = 0 = ˆ π ( t, x ).For any ǫ ∈ (0 , T − t ), straightforward calculation leads to X ˆ π t,ǫ,π t,x ( T ) = x + Z t + ǫt b ( s ) ⊤ πds + Z t + ǫt π ⊤ σ ( s ) dW ( s ) . Because α ≥ / b ( t ) ⊤ π >
0, and b is right-continuous, straightforward calculation showsthat for sufficiently small ǫ > G ˆ π t,ǫ,π ( t, x, α ) > x = G ˆ π ( t, x, α ). This contradicts theassumption that ˆ π is an equilibrium strategy. Thus, we must have t ∗ = T .Next, for the sake of contradiction, suppose t >
0. Then, X ˆ π ,x ( s ) = x and thus X x , ˆ π s = { x } for all s ∈ [0 , t ). Recall t ∗ as defined in (A.2) and θ ( s ) + θ ( s ) ξ = 0 , s ∈ [ t ∗ , t ∗ ) = [ t ∗ , T )for certain uniquely determined ξ ∈ R . When t ∗ > t ∈ [0 , t ∗ ∧ t ) andwhen ξ = x we can choose any t ∈ [0 , t ). In either case, Lemma 3 can apply to this particular t together with x ∈ X x , ˆ π t , leading to F ˆ π x ( t, x , G ˆ π ( t, x , α )) < F ˆ π xx ( t, x , G ˆ π ( t, x , α )) > π ( t, x ) = − F ˆ π x ( t, x , G ˆ π ( t, x , α )) F ˆ π xx ( t, x , G ˆ π ( t, x , α )) v ∗ ( t ) . The above is a contradiction because ˆ π ( t, x ) = 0 and v ∗ ( t ) = 0.When t ∗ = 0 and ξ = x , we have t = T and thus X x , ˆ π s = { x } and X ˆ π s,x ( T ) = X ˆ π ,x ( T )for all s ∈ [0 , t ). Fix any t ∈ [0 , T ) and choose any π ∈ R m with b ( t ) ⊤ π > Qπ ≥ π can be found due to Assumption 2. For any ǫ ∈ (0 , T − t ), (A.24) holds. Thus,because α ≥ /
2, for sufficiently small ǫ > F ˆ π t,ǫ,π ( t, x , x ) < α . In addition, because π = 0, F ˆ π t,ǫ,π ( t, x , y ) is strictly increasing and continuous in y for any ǫ ∈ (0 , T − t ).Consequently, we conclude that G ˆ π t,ǫ,π ( t, x , α ) > x = G ˆ π ( t, x , α ) for sufficiently small ǫ >
0. This contradicts the assumption that ˆ π is an equilibrium policy. The proof thencompletes. (cid:3) Proof of Lemma 5
Part (i) is an immediate consequence of Proposition 12-(iii) and (iv).We prove part (ii) in the following.Fix any t ∈ ( t, t ∗ ), Lemma 4 implies that − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) v ∗ ( t ) = ˆ π ( t, x ) = θ ( t ) + θ ( t ) x, x ∈ X x , ˆ π t \ S ˆ π t . (A.26)32ultiplying v ∗ ( t ) ⊤ on both sides of the above equality and noting that v ∗ ( t ) = 0, we conclude − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) = k v ∗ ( t ) k − (cid:0) v ∗ ( t ) ⊤ θ ( t ) + v ∗ ( t ) ⊤ θ ( t ) x (cid:1) , x ∈ X x , ˆ π t \ S ˆ π t . Then, (cid:20)(cid:18) − F ˆ π x ( t, x ′ , G ˆ π ( t, x ′ , α )) F ˆ π xx ( t, x ′ , G ˆ π ( t, x ′ , α )) (cid:19) − (cid:18) − F ˆ π x ( t, x ′′ , G ˆ π ( t, x ′′ , α )) F ˆ π xx ( t, x ′′ , G ˆ π ( t, x ′′ , α )) (cid:19)(cid:21) / ( x ′ − x ′′ )does not depend on the choice of x ′ , x ′′ ∈ X x , ˆ π t \ S ˆ π t with x ′ = x ′′ , and we denote this commonvalue by a ( t ). Because X x , ˆ π t is a nonempty open interval and S ˆ π t is either the empty set ora singleton, we can always find x ′ , x ′′ ∈ S ˆ π t with x ′ = x ′′ and thus a ( t ) is well defined. Then,fixing any ¯ x ∈ X x , ˆ π t \ S ˆ π t , we have − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) − a ( t ) x = − F ˆ π x ( t, ¯ x, G ˆ π ( t, ¯ x, α )) F ˆ π xx ( t, ¯ x, G ˆ π ( t, ¯ x, α )) − a ( t )¯ x, x ∈ X x , ˆ π t \ S ˆ π t . It is obvious that the right-hand side does not depend on the choice of ¯ x , and we denote itby a ( t ). Consequently, − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π xx ( t, x, G ˆ π ( t, x, α )) = a ( t ) + a ( t ) x, x ∈ X x , ˆ π t \ S ˆ π t . Combining the above with (A.26), we immediately conclude that θ ( t ) = a ( t ) v ∗ ( t ) and θ ( t ) = a ( t ) v ∗ ( t ).Next, we prove that a , a ∈ C ([ t, t ∗ ]). For any fixed t ∈ ( t, t ∗ ), because X x , ˆ π t is anonempty interval, we can find distinct x ′ = x ′′ ∈ X x , ˆ π t \{ ξ } ⊆ X x , ˆ π t \ S ˆ π t , t ∈ [ t , t ∗ ). Then,we have a ( t ) = (cid:20)(cid:18) − F ˆ π x ( t, x ′ , G ˆ π ( t, x ′ , α )) F ˆ π xx ( t, x ′ , G ˆ π ( t, x ′ , α )) (cid:19) − (cid:18) − F ˆ π x ( t, x ′′ , G ˆ π ( t, x ′′ , α )) F ˆ π xx ( t, x ′′ , G ˆ π ( t, x ′′ , α )) (cid:19)(cid:21) / ( x ′ − x ′′ ) , t ∈ [ t , t ∗ ) . By Proposition 13-(iii), F ˆ π x ( t, x, G ˆ π ( t, x, α )) and F ˆ π xx ( t, x, G ˆ π ( t, x, α )) are continuous in t ∈ [0 , t ∗ ) for any x = ξ . As a result, a ∈ C ([ t , t ∗ )). Similarly, a ∈ C ([ t , t ∗ ); R ). Because t isarbitrary, we have a , a ∈ C (( t, t ∗ )). For any t ∈ ( t, t ∗ ), because θ ( t ) = a ( t ) v ∗ ( t ), we have a ( t ) = θ ( t ) ⊤ v ∗ ( t ) / k v ∗ ( t ) k . Because the limits of v ∗ ( t ) and θ ( t ) exist when t converges to t and because the former limit is not zero, we conclude that the limit of a ( t ) as t goes to t exists. Similarly, the limit of a ( t ) as t goes to t ∗ exists. Thus, we can extend the definitionof a to the domain [ t, t ∗ ] and a ∈ C ([ t, t ∗ ]). Similarly, we can show that a ∈ C ([ t, t ∗ ]).33inally, for any t ∈ ( t, t ∗ ) and x ∈ X x , ˆ π t \ S ˆ π t , Proposition 13-(i) and (ii) yield that F ˆ π t ( t, x, G ˆ π ( t, x, α )) + F ˆ π x ( t, x, G ˆ π ( t, x, α )) b ( t ) ⊤ ˆ π ( t, x ) + 12 F ˆ π xx ( t, x, G ˆ π ( t, x, α )) k σ ( t ) ˆ π ( t, x ) k = 0 . Combining the above with (A.8) and (A.9) and noting that F ˆ π x ( t, x, G ˆ π ( t, x, α )) = 0 andthus a ( t ) + a ( t ) x = 0, we obtain F ˆ π t ( t, x, G ˆ π ( t, x, α )) + 12 F ˆ π x ( t, x, G ˆ π ( t, x, α )) ρ ( t ) (cid:0) a ( t ) + a ( t ) x (cid:1) = 0 . Proposition 14-(ii) and (iii) show that G ˆ π x ( t, x, α ) = − F ˆ π x ( t, x, G ˆ π ( t, x, α )) F ˆ π y ( t, x, G ˆ π ( t, x, α )) , G ˆ π t ( t, x, α ) = − F ˆ π t ( t, x, G ˆ π ( t, x, α )) F ˆ π y ( t, x, G ˆ π ( t, x, α )) . As a result, we derive G ˆ π t ( t, x, α ) + 12 G ˆ π x ( t, x, α ) ρ ( t ) (cid:0) a ( t ) + a ( t ) x (cid:1) = 0 . Proposition 14-(iv) shows that for any x ∈ R ,lim t ↑ t ∗ ,x ′ → x G ˆ π ( t, x ′ , α ) = x. The proof then completes. (cid:3)
Proof of Lemma 6
We prove (i) first. Because γ , γ ∈ C ([ τ , τ ]), there exists L > s ∈ [ τ ,τ ] | γ i ( s ) | ≤ L , i = 0 ,
1. Set x := e L ( τ − τ ) (cid:0) ¯ c + Le L ( τ − τ ) ( τ − τ ) (cid:1) . Fix any t ∈ [ τ , τ ) and x > x , consider ϕ ( s ) = g ( s, h ( s ) x + h ( s )) , s ∈ [ t, τ ] , where h ( s ) = Z st e R sτ γ ( z ) dz γ ( τ ) dτ, h ( s ) = e R st γ ( τ ) dτ , s ∈ [ t, τ ] . h ′ ( s ) x + h ′ ( s ) = γ ( s ) + γ ( s ) (cid:0) h ( s ) x + h ( s ) (cid:1) , s ∈ [ t, τ ] . (A.27)For any s ∈ [ t, τ ], h ( s ) x + h ( s ) ≥ e − L ( τ − τ ) x − Le L ( τ − τ ) ( τ − τ ) > e − L ( τ − τ ) x − Le L ( τ − τ ) ( τ − τ ) = ¯ c, where the first inequality is the case because sup s ∈ [ τ ,τ ] | γ i ( s ) | ≤ L , i = 0 , x > x . Therefore, ϕ ( s ) , s ∈ [ t, τ ] is well defined and isdifferentiable in s ∈ [ t, τ ). Moreover, applying the chain rule, we derive ϕ ′ ( s ) = g t ( s, h ( s ) x + h ( s )) + g x ( s, h ( s ) x + h ( s )) (cid:0) h ′ ( s ) x + h ′ ( s ) (cid:1) = g t ( s, h ( s ) x + h ( s )) + g x ( s, h ( s ) x + h ( s )) (cid:0) h ( s ) γ ( s ) x + γ ( s ) + h ( s ) γ ( s ) (cid:1) = 0 , s ∈ [ t, τ ) , where the second equality follows from (A.27) and the third follows from the differentiableequation satisfied by g . As a result, g ( t, x ) = ϕ ( t ) = ϕ ( τ ) = g ( τ , h ( τ ) x + h ( τ )) = α + α ( h ( τ ) x + h ( τ )) = ˆ g ( t, x ) , where the fourth equality is the case due to the terminal condition satisfied by g at τ .Part (ii) can be proved similarly. For (iii), because γ ( t )+ γ ( t ) x ≥ t, x ) ∈ D andbecause c ( t ) is decreasing, for each fixed ( t, x ) ∈ D , (A.27) shows that h ′ ( s ) x + h ′ ( s ) ≥ h ( s ) x + h ( s ) is increasing in s ∈ [ t, τ ] and thus ( s, h ( s ) x + h ( s )) ∈ D , s ∈ [ t, τ ). Then,following the same proof as in part (i) of the lemma, we conclude that g ( t, x ) = ˆ g ( t, x ). (cid:3) Proof of Lemma 7
Recall the form of the strategy ˆ π as in (A.9). Suppose X x , ˆ π t ⊇ ( x t , + ∞ )for certain x t ∈ R . Then, ξ / ∈ [ x ′ t , + ∞ ) for certain x ′ t ≥ x t and (A.10) holds in the region[ t, t ∗ ] × ( x ′ t , + ∞ ). Because G ˆ π ∈ C ([ t, t ∗ ] × ( x ′ t , + ∞ )) (due to Proposition 14-(ii)) and becausethere exists a partition of [ t, t ∗ ], t =: τ < τ < · · · < τ N = t ∗ , such that a , a , ρ ∈ C ([ τ i − , τ i ])and G ˆ π ∈ C , ∞ ([ τ i − , τ i ) × ( x ′ t , + ∞ )) (due to Proposition 14-(iii)) i = 1 , . . . , N , by applyingLemma 6 in [ τ i − , τ i ) sequentially, we conclude that there exists c t > max( x t , ξ ) such that(A.16) holds all ( s, x ) ∈ [ t, t ∗ ] × [ c t , + ∞ ). Similarly, in the case X x , ˆ π t ⊇ ( −∞ , ¯ x t ) for certain¯ x t ∈ R , we can find ¯ c t < min( ξ, ¯ x t ) such that (A.16) holds all ( s, x ) ∈ [ t, t ∗ ] × ( −∞ , ¯ c t ).Next, we prove (i). For the sake of contradiction, suppose a ( s ) = 0 , s ∈ [ t, t ∗ ). Then,35e must have t ∗ = t ∗ ; otherwise θ ( s ) = − ξθ ( s ) = − ξa ( s ) v ∗ ( s ) = 0 for s ∈ [ t ∗ , t ∗ ),contradicting the definition of t ∗ . Moreover, by the definition of t ∗ , for any s ∈ [ t, t ∗ ), thereexists τ ∈ [ s, t ∗ ) with θ ( τ ) = 0 and thus a ( τ ) = 0. Also note that X ˆ π s,x ( T ) = x + Z t ∗ s a ( τ ) v ∗ ( τ ) ⊤ b ( τ ) dτ + Z t ∗ s a ( τ ) v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) , which is a normal random variable. Thus, we have G ˆ π ( s, x, α ) = x + Z t ∗ s a ( τ ) v ∗ ( τ ) ⊤ b ( τ ) dτ + Φ − ( α ) sZ t ∗ s k σ ( τ ) ⊤ v ∗ ( τ ) k a ( τ ) dτ . (A.28)Lemma 5-(i) shows that it is either the case in which X x , ˆ π t = ( x t , + ∞ ) for certain x t ∈ R ,or the case in which X x , ˆ π t = ( −∞ , ¯ x t ) for certain ¯ x t ∈ R , or the case X x , ˆ π t = R . In eithercase, as we already proved, there exists a nonempty open interval I ⊂ R such that (A.16)holds for any s ∈ [ t, t ∗ ) and x ∈ I . Comparing (A.16) and (A.28), we conclude β ( s, α ) = Z t ∗ s a ( τ ) v ∗ ( τ ) ⊤ b ( τ ) dτ + Φ − ( α ) sZ t ∗ s k σ ( τ ) ⊤ v ∗ ( τ ) k a ( τ ) dτ ,β ( s, α ) = 1 , s ∈ [ t, t ∗ ) . The above immediately yields that a ( s ) = 0 , s ∈ [ t, t ∗ ) and − Φ − ( α ) sZ t ∗ s k σ ( τ ) ⊤ v ∗ ( τ ) k a ( τ ) dτ = Z t ∗ s a ( τ ) v ∗ ( τ ) ⊤ b ( τ ) dτ − Z t ∗ s a ( τ ) ρ ( τ ) dτ = 12 Z t ∗ s a ( τ ) k σ ( τ ) ⊤ v ∗ ( τ ) k dτ, s ∈ [ t, t ∗ ) , (A.29)where the second equality is due to the definition of ρ . When α = 1 /
2, because k σ ( s ) ⊤ v ∗ ( s ) k > , s ∈ [0 , T ), (A.29) implies that a ( s ) = 0 , s ∈ [ t, t ∗ ), which is a contradiction. When α = 1 /
2, taking square and then taking derivative with respect to s on both sides of (A.29),and noting that k σ ( s ) ⊤ v ∗ ( s ) k > , s ∈ [0 , T ) and that a ( s ) > , s ∈ [ t, t ∗ ) because of (A.8)and a ( s ) = 0, s ∈ [ t, t ∗ ), we derive the following integral equation (cid:0) Φ − ( α ) (cid:1) a ( s ) − Z t ∗ s a ( τ ) k σ ( τ ) ⊤ v ∗ ( τ ) k dτ = 0 , s ∈ [ t, t ∗ ) . g ( s ) := R t ∗ s a ( τ ) k σ ( τ ) ⊤ v ∗ ( τ ) k dτ = 0 satisfies (cid:0) Φ − ( α ) (cid:1) g ′ ( s ) + 12 k σ ( s ) ⊤ v ∗ ( s ) k g ( s ) = 0 , s ∈ [ t, t ∗ ) , g ( t ∗ ) = 0 . Because (cid:0) Φ − ( α ) (cid:1) = 0 and k σ ( s ) ⊤ v ∗ ( s ) k is bounded in s ∈ [ t, t ∗ ), we derive g ( s ) = 0 , s ∈ [ t, t ∗ ], i.e., a ( s ) = 0 , s ∈ [ t, t ∗ ], which is a contradiction.Next, we suppose X x , ˆ π t ⊇ ( x t , + ∞ ) for certain x t ∈ R and prove (A.17). Straightforwardcalculation yields that X ˆ π t,x ( T ) = x ˜ Z ( T ; t ) + ˜ Z ( T ; t ) . where d ˜ Z ( s ; t ) = ˜ Z ( s ; t ) (cid:2) b ( s ) ⊤ θ ( s ) ds + θ ( s ) ⊤ σ ( s ) dW ( s ) (cid:3) , s ∈ [ t, T ] , ˜ Z ( t ; t ) = 1 . (A.30) d ˜ Z ( s ; t ) = (cid:16) b ( s ) ⊤ θ ( s ) + b ( s ) ⊤ θ ( s ) ˜ Z ( s ; t ) (cid:17) ds + (cid:16) θ ( s ) ⊤ σ ( s ) + θ ( s ) ⊤ σ ( s ) ˜ Z ( s ; t ) (cid:17) dW ( s ) ,s ∈ [ t, T ] , ˜ Z ( t ; t ) = 0 . (A.31)For any x > max( c t , G Z ( α ) , α ∈ (0 ,
1) and G Y ( x, α ) , α ∈ (0 ,
1) the right-continuous quantile functions of ˜ Z ( T ; t ) and Y ˆ π t,x ( T ) := X ˆ π t,x ( T ) /x , respectively. We alreadyproved part (i) of the lemma, which implies that ˜ Z ( T ; t ) is a non-degenerate lognormal ran-dom variable and thus its quantile function is continuous. Also note that lim x ↑ + ∞ Y ˆ π t,x ( T ) =˜ Z ( T ; t ) almost surely. As a result,lim x ↑ + ∞ G ˆ π ( t, x, α ) x = lim x ↑ + ∞ G Y ( x, α ) = G Z ( α ) , ∀ α ∈ (0 , . By computing G Z ( α ) from (A.30) and recalling (A.16), we then derive (A.17).Finally, (A.18) can be proved similarly. (cid:3) Proof of Proposition 8
Suppose that ˆ π ∈ A is an equilibrium strategy for certain α = 1 / t ∗ as defined in Lemma 2 and t as defined in Lemma 4. We prove that it cannot bethe case that t < t ∗ .For the sake of contradiction, suppose t < t ∗ . Combining Lemma 5-(i) and Lemma 7, weconclude that it is either the case in which (A.17) holds for any t ∈ ( t, t ∗ ) or the case in which(A.18) holds for any t ∈ ( t, t ∗ ). Denote λ = Φ − ( α ) in the first case and λ = Φ − (1 − α ) in37he second case. Cauchy’s inequality implies (cid:12)(cid:12)(cid:12)(cid:12)Z t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ sZ t ∗ t k σ ( s ) ⊤ v ∗ ( s ) k ds sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds. Combining the above with (A.17) and (A.18), we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds − λ sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sZ t ∗ t k σ ( s ) ⊤ v ∗ ( s ) k ds sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds. Because of (A.15), we can divide qR t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds on both sides of the aboveinequality to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)sZ t ∗ t a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sZ t ∗ t k σ ( s ) ⊤ v ∗ ( s ) k ds. Sending t to t ∗ on both sides of the above inequality and noting that λ = 0 because α = 1 / t ≥ t ∗ . On the other hand, Lemma 4 shows that if ˆ π ∈ A is anequilibrium strategy for α ∈ (1 / , t = 0 and t ∗ = T . Thus, any ˆ π ∈ A isnot an equilibrium strategy for α ∈ (1 / , α ∈ (0 , / t ≥ t ∗ implies t = T , which is the case if and only if ˆ π ( s, x ) = 0 , s ∈ [0 , T ), i.e., ˆ π takes the form in(3.4). On the other hand, we already proved that any ˆ π as given by (3.4) is an equilibriumstrategy. The proof then completes. (cid:3) Proof of Proposition 9
Suppose that ˆ π ∈ A is an equilibrium strategy for certain α = 1 / t ∗ as defined in Lemma 2 and t as defined in Lemma 4. Lemma 4 shows that t = 0and t ∗ = T . Then, Lemma 7-(ii) and (iii) imply that12 Z t ∗ t a ( s ) (cid:0) − a ( s ) (cid:1) k σ ( s ) ⊤ v ∗ ( s ) k ds = 0 , t ∈ (0 , t ∗ ) , which implies that a ( t )(1 − a ( t )) = 0 , t ∈ (0 , t ∗ ). Lemma 7-(i) shows for any t ∈ (0 , t ∗ ), a ( s ) = 0 for some s ∈ [ t, t ∗ ). Then we must have a ( t ) = 1 , t ∈ (0 , T ) because a ∈ ([0 , T ]; R ) as shown by Lemma 5-(ii). Then, (A.9) implies thatˆ π ( t, x ) = ( a ( t ) + x ) v ∗ ( t ) , t ∈ [0 , T ) , x ∈ R . (A.32)What remains is to prove that a ( t ) is constant in t ∈ [0 , T ].According to Lemma 5-(i), (A.8), and a ( t ) = 1 , t ∈ (0 , T ), we have X x , ˆ π t = ( x ( t ) , + ∞ )for some x ( t ) ∈ R . Because of (A.32), Proposition 12-(iv) shows that a ( t ) is increasingin t ∈ (0 , T ] and X x , ˆ π t = ( − a ( t ) , + ∞ ) , t ∈ (0 , T ]. By the definition of t ∗ and ξ , we have t ∗ = inf { t ∈ [0 , T ] : a ( s ) = a ( T ) , s ∈ [ t, T ] } and ξ = − a ( T ). Because a ( t ) is increasingin t ∈ (0 , T ], we conclude that X x , ˆ π t \ S ˆ π t = ( − a ( t ) , + ∞ ) , t ∈ (0 , T ]. Now, Lemma 5-(ii) andLemma 6-(iii) yield that (A.16) holds for any x > − a ( t ) , t ∈ (0 , T ].Now, for the sake of contradiction, suppose a ( t ) is not constant in t ∈ [0 , T ], which isequivalent to t ∗ >
0. Fix any t ∈ (0 , t ∗ ). Then, a ( s ) is not a constant in s ∈ [ t, t ∗ ], whichmeans that da ( s ) defines a positive measure on [ t, T ]. Applying Itˆo’s lemma, recalling(A.32), Lemma 1 and ˜ Z as in (A.30), and noting that a ∈ C ([0 , T ]) and a ≡
1, we derive d a ( s ) + X ˆ π t,x ( s )˜ Z ( s ; t ) ! = da ( s )˜ Z ( s ; t ) , s ∈ [ t, T ) , x ∈ R . Then, we obtain X ˆ π t, − a ( t ) ( T ) = − a ( T ) + Z Tt ˜ Z ( T ; t )˜ Z ( s ; t ) da ( s ) , = − a ( T ) + Z Tt e R Ts ( b ( τ ) ⊤ v ∗ ( τ ) − k σ ( τ ) ⊤ v ∗ ( τ ) k dτ ) dτ + R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) da ( s ) , (A.33)where the second equality is the case due to the definition of ˜ Z and because a ( s ) = 1 andthus θ ( s ) = v ∗ ( s ) , s ∈ [0 , T ). Then, recalling t ∗ = T , β , β as defined in (A.14), and that ρ ( s ) = 2 b ( s ) ⊤ v ∗ ( s ) − k σ ( s ) ⊤ v ∗ ( s ) k , we derive X ˆ π t, − a ( t ) ( T ) + a ( t ) β ( t, / − β ( t, / − a ( T ) + Z Tt e R Ts ρ ( τ ) dτ + R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) da ( s )+ a ( t ) e R Tt ρ ( s ) ds − Z Tt a ( s ) ρ ( s ) e R Ts ρ ( τ ) dτ ds = Z Tt e R Ts ρ ( τ ) dτ + R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) da ( s ) − Z Tt e R Ts ρ ( τ ) dτ da ( s ) , (A.34)39here second equality is the case because d (cid:16) a ( s ) e R Ts ρ ( τ ) dτ (cid:17) = e R Ts ρ ( τ ) dτ da ( s ) − ρ ( s ) e R Ts ρ ( τ ) dτ a ( s ) ds. Next, we prove P (cid:0) X ˆ π t, − a ( t ) ( T ) ≤ − a ( t ) β ( t, /
2) + β ( t, / (cid:1) < / . (A.35)According to (A.34), one can see that (A.35) is equivalent to P (cid:18)Z Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) ≤ (cid:19) < / , (A.36)where M ( s ) := R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) and dµ ( s ) := e R Ts ρ ( τ ) dτ da ( s ), s ∈ [0 , T ]. Because da ( s ) is a positive, atomless measure on [ t, T ], so is dµ ( s ). In the following, we first provethat P (cid:18)Z Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) > , Z Tt M ( s ) ds ≤ (cid:19) > . (A.37)To this end, we first construct a path m ( s ) , s ∈ [ t, T ] of the stochastic process M ( s ) , s ∈ [0 , T ]such that Z Tt (cid:0) e m ( s ) − (cid:1) dµ ( s ) > , Z Tt m ( s ) dµ ( s ) < . (A.38)The construction is as follows: First, there exists ζ ∈ R such that R Tt ( T − s )( ζ − s ) dµ ( s ) = 0.Because e x − > x for any x = 0 and because dµ ( s ) is a positive, atomless measureon [ t, T ], we have R Tt (cid:0) e ( T − s )( ζ − s ) − (cid:1) dµ ( s ) >
0. Thus, there exists ζ < ζ such that R Tt (cid:0) e ( T − s )( ζ − s ) − (cid:1) dµ ( s ) >
0. As a result, Z Tt ( T − s )( ζ − s ) dµ ( s ) = Z Tt ( T − s )( ζ − s ) dµ ( s ) + ( ζ − ζ ) Z Tt ( T − s ) dµ ( s ) < , where the inequality is the case because ζ < ζ and R Tt ( T − s ) dµ ( s ) >
0. Thus, setting m ( s ) := ( T − s )( ζ − s ) , s ∈ [0 , T ], (A.38) holds. As a result, there exists ǫ > Z Tt (cid:0) e m ( s ) − ǫ − (cid:1) dµ ( s ) > , Z Tt ( m ( s ) + ǫ ) dµ ( s ) < . (A.39)40efine Y ( s ) := M ( t ) − M ( s ) = R st v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) , s ∈ [ t, T ] and f ( s ) := m ( t ) − m ( s ) = ( T − t )( ζ − t ) − ( T − s )( ζ − s ) , s ∈ [0 , T ] , so f can be considered to be a path of Y . Defining an absolutely continuous function w ( s ) , s ∈ [0 , T ] by w ( t ) = 0 , w ′ ( s ) = σ ( s ) ⊤ v ∗ ( s ) k σ ( s ) ⊤ v ∗ ( s ) k f ′ ( s ) , s ∈ [ t, T ] , so we have f ( s ) = R st v ∗ ( z ) ⊤ σ ( z ) w ′ ( z ) dz, s ∈ [0 , T ]. Then, Lemma 2 of He and Jiang (2020)shows that f ( s ) , s ∈ [ t, T ] is in the support of Y ( s ) , s ∈ [ t, T ], with the latter being viewedas a random variable taking values in the space of continuous functions on [ t, T ] with themaximum norm. As a result, for any ǫ >
0, in particular the one satisfying (A.39), we have P sup s ∈ [ t,T ] | M ( s ) − m ( s ) | < ǫ ! = P sup s ∈ [ t,T ] | Y ( T ) − Y ( s ) − ( f ( T ) − f ( s )) | < ǫ ! ≥ P sup s ∈ [ t,T ] | Y ( s ) − f ( s ) | < ǫ/ ! > . Combining the above with (A.39), we derive P (cid:18)Z Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) > , Z Tt M ( s ) ds ≤ (cid:19) ≥ P sup s ∈ [ t,T ] | M ( s ) − m ( s ) | < ǫ ! > , i.e., (A.37) is proved.Now, recalling that (A.37) and noting that R Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) ≥ R Tt M ( s ) dµ ( s ) because e x − ≥ x for any x ∈ R , we derive P (cid:18)Z Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) ≤ (cid:19) = P (cid:18)Z Tt M ( s ) dµ ( s ) ≤ (cid:19) − P (cid:18)Z Tt M ( s ) dµ ( s ) ≤ , Z Tt (cid:0) e M ( s ) − (cid:1) dµ ( s ) > (cid:19) < P (cid:18)Z Tt M ( s ) dµ ( s ) ≤ (cid:19) = 1 / , where the last equality is the case because R Tt M ( s ) dµ ( s ) is a normal random variable withzero mean. Thus, (A.35) is proved. 41ecause t < t ∗ , Proposition 14-(ii) shows that G ˆ π ( t, x, α ) is continuous in ( x, α ) ∈ R × (0 , G ˆ π ( t, − a ( t ) , ) − [ β ( t, ) − a ( t ) β ( t, )] >
0, and there exists δ t > G ˆ π ( t, x, / − (cid:0) β ( t, /
2) + xβ ( t, / (cid:1) > , ∀ x ∈ ( − a ( t ) − δ t , − a ( t ) + δ t ) . (A.40)Because have shown that (A.16) holds for any x > − a ( t ), we arrive at contradiction. Then, a ( t ) must be a constant on [0 , T ], i.e., ˆ π must be given by (3.3) for some ξ < x .On the other hand, we already proved that ˆ π as given by (3.3) for any ξ < x is anequilibrium strategy. The proof then completes. (cid:3) A.2 Proof of Theorem 2
Similar to the proof of Theorem 1, we present the proof of Theorem 2 by summarizingimportant intermediate steps of the proof as lemmas and relegate all proofs in Section A.2.1.
Lemma 8
Fixed T ∈ [0 , T ) . Consider ˆ π ∈ A and for any x ∈ R and t ∈ [ T , T ] , denote by X x,T , ˆ π t the set of reachable states of X ˆ π T ,x ( t ) . Then, for any x ∈ X x , ˆ π T , we have X x ,T , ˆ π t ⊆ X x , ˆ π t for any t ∈ [ T , T ] . Lemma 8 shows that any state that is reachable at given future time by a wealth equationstarting from an intermediate time and state that is reachable from the initial wealth is alsoreachable from the initial wealth. As a result, if ˆ π is an intrapersonal equilibrium at theinitial time, it is also an intrapersonal equilibrium at intermediate time. Proposition 10
For α ∈ (0 , / , ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-point for α -level quantile maximization if and only if ˆ π is given by (3.4) for some θ ∈ C pw ([0 , T )) taking values in R m . For α ∈ (1 / , , any ˆ π ∈ A is not an intra-personalequilibrium for multiple-time-point α -level quantile maximization. Proposition 10 shows that Theorem 2-(ii) and (iii) are true. In the following, we dealwith the remaining case α = 1 /
2, i.e. the case of median maximization. To this end, for anyˆ π ∈ A , define τ ∗ : = inf { t ∈ [0 , T N − ) : θ ( s ) = θ ( s ) = 0 , ∀ s ∈ [ t, T N − ) } , (A.41) τ ∗ : = inf { t ∈ [0 , τ ∗ ) : θ ( s ) + ξ θ ( s ) = 0 , ∀ s ∈ [ t, τ ∗ ) and some ξ ∈ R } . (A.42)42hen, there exists unique ξ ∈ R such that θ ( s ) + ξ θ ( s ) = 0 , ∀ s ∈ [ τ ∗ , τ ∗ ). Denote˜ S ˆ π t = ∅ , t ∈ [0 , τ ∗ ) , ˜ S ˆ π t = { ξ } , t ∈ [ τ ∗ , T N − ) . (A.43)Recall t ∗ , t ∗ , and ξ as defined in Lemma 2. Recalling that S ˆ π t = ∅ , ∀ t ∈ [0 , t ∗ ), we derive that S ˆ π t ⊆ ˜ S ˆ π t , ∀ t ∈ [0 , T N − ) because we have ξ = ξ and t ∗ = τ ∗ in the case t ∗ < T N − . Lemma 9
Suppose ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-point me-dian maximization. Then, the left end of X x , ˆ π T N − is finite and ˆ π ( t, x ) = v ∗ ( t )( x − ξ ) , t ∈ [ T N − , T ) , x ∈ R (A.44) for some constant ξ that satisfies ξ < x, ∀ x ∈ X x , ˆ π T N − . (A.45) Recall t ∗ , t ∗ , ξ , and S ˆ π t as defined in Lemma 2 and τ ∗ , τ ∗ , and ξ as defined in (A.41) and (A.42) . Then, τ ∗ = T N − , t ∗ = T , and t ∗ ≤ T N − . Lemma 9 shows that if ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-pointmedian maximization, it must be a portfolio insurance strategy in the period [ T N − , T ), andthe portfolio insurance level must be lower than any wealth level that is reachable at time T N − .Denote by F π ( t, x, y ; s ) := P ( X π t,x ( s ) ≤ y ) , y ∈ R the cumulative distribution function ofthe wealth at time s given wealth level of x at time t ≤ s . Lemma 10
Recall t as defined in Lemma 4 and ˜ S ˆ π t as in (A.43) . Suppose ˆ π ∈ A is an intra-personal equilibrium for multiple-time-point median maximization. For any t ∈ [ T N − , T N − ) , x ∈ X x , ˆ π t \ ˜ S ˆ π t , denoting λ ˆ π ( t, x ) = (1 − w N − ,N ) F ˆ π x ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) + w N − ,N F ˆ π x ( t, x, G ˆ π ( t, x, )) F ˆ π y ( t, x, G ˆ π ( t, x, )) ,λ ˆ π ( t, x ) = (1 − w N − ,N ) F ˆ π xx ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) + w N − ,N F ˆ π xx ( t, x, G ˆ π ( t, x, )) F ˆ π y ( t, x, G ˆ π ( t, x, )) , e have λ ˆ π ( t, x ) > , λ ˆ π ( t, x ) < , and ˆ π ( t, x ) = − λ ˆ π ( t, x ) λ ˆ π ( t, x ) v ∗ ( t ) . (A.46) Moreover, we have t ≤ T N − . Lemma 10 shows that any intra-personal equilibrium ˆ π ∈ A must take a particular formas in (A.46) in the period [ T N − , T N − ): the allocation across the risky assets in this periodis the same as the Kelly strategy. Lemma 11
Suppose ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-pointmedian maximization problem. Recall ξ and ˜ S ˆ π t as defined in (A.42) and (A.43) , respectively.Then,(i) For each t ∈ ( T N − , T N − ] , X x , ˆ π t = ( x ( t ) , + ∞ ) for some x ( t ) < x , and x ( t ) is decreas-ing in t ∈ ( T N − , T N − ] .(ii) There exists ˜ a , ˜ a ∈ C ([ T N − , T N − ]) taking values in R such that − λ ˆ π ( t, x ) λ ˆ π ( t, x ) = ˜ a ( t ) + ˜ a ( t ) x > , t ∈ ( T N − , T N − ) , x ∈ X x , ˆ π t \ ˜ S ˆ π t , (A.47)ˆ π ( t, x ) = (cid:0) ˜ a ( t ) + ˜ a ( t ) x (cid:1) v ∗ ( t ) , ( t, x ) ∈ ( T N − , T N − ) × R . (A.48) Consequently, g ˆ π ( t, x ) := w N − ,N G ˆ π ( t, x, ) + (1 − w N − ,N ) G ˆ π ( t, x, ; T N − ) satisfiesfollowing PDE: g ˆ π t ( t, x ) + g ˆ π x ( t, x ) ρ ( t ) (cid:0) ˜ a ( t ) + ˜ a ( t ) x (cid:1) = 0 , t ∈ ( T N − , T N − ) , x ∈ X x , ˆ π t \ ˜ S ˆ π t , lim t ↑ T N − ,x ′ → x g ˆ π ( t, x ′ ) = w N − ,N (cid:18) ( x − ξ ) e R TTN − ρ ( s ) ds + ξ (cid:19) + (1 − w N − ,N ) x, x ∈ R \{ ξ } , (A.49) where ρ ( t ) is as defined in (A.11) . Lemma 12
Suppose ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-pointmedian maximization problem. Recall ξ and ˜ S ˆ π t as defined in (A.42) and (A.43) , respectively,and ˜ a , ˜ a , and x ( t ) as defined in Lemma 11. Define ˜ β ( t ) : = e R TN − t ˜ a ( s ) ρ ( s ) ds , ˜ β ( t ) := 12 Z T N − t ˜ a ( s ) ρ ( s ) ˜ β ( s ) ds, t ∈ [ T N − , T N − ] . (A.50)44 ix any t ∈ ( T N − , T N − ) .(i) There exists s ∈ [ t, T N − ) such that ˜ a ( s ) = 0 . Consequently, Z T N − t ˜ a ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds > . (A.51) (ii) There exists c t > max( x ( t ) , ξ, ξ ) such that w N − ,N G ˆ π ( s, x, /
2) + (1 − w N − ,N ) G ˆ π ( s, x, / T N − )= w N − ,N (cid:0) β ( s, / x + β ( s, / (cid:1) + (1 − w N − ,N ) (cid:0) ˜ β ( s ) x + ˜ β ( s ) (cid:1) (A.52) holds for all ( s, x ) ∈ [ t, T N − ] × [ c t , + ∞ ) , where β ( s, / , β ( s, / are as defined in (A.14) with a ( s ) = − ξ, a ( s ) = 1 , s ∈ [ T N − , T ) and a ( s ) = ˜ a ( s ) , a ( s ) = ˜ a ( s ) , s ∈ [ T N − , T N − ) . Moreover, Z T N − t ˜ a ( s ) (cid:0) − ˜ a ( s ) (cid:1) k σ ( s ) ⊤ v ∗ ( s ) k ds = 0 . (A.53) Proposition 11
Suppose α = 1 / . Then, ˆ π ∈ A is an intra-personal equilibrium formultiple-time-point α -level quantile maximization if and only if ˆ π is given by (3.3) for someconstant ξ < x . A.2.1 Detailed Proofs
Proof of Lemma 8 If t ≥ T , then X ,x ( T ) = x and thus X x , ˆ π T = { x } . For any t ∈ [ T , T ],we have X ˆ π T ,x ( t ) = X ˆ π ,x ( t ) and thus X x ,T , ˆ π t = X x , ˆ π t . In the following, we consider thecase in which t < T .Fix x ∈ X x , ˆ π T and t ∈ [ T , T ]. If θ ( s ) + θ ( s ) x = 0 for any s ∈ [ T , t ), then Proposition12-(i) and (ii) show that X x ,T , ˆ π t = { x } ⊆ X x , ˆ π T ⊆ X x , ˆ π t . If θ ( s ) + θ ( s ) x = 0 forsome s ∈ [ T , t ), then Proposition 12-(iii) and (iv) show that X ˆ π ,x ( t ) and X ˆ π T ,x ( t ) havedensity functions, so X x , ˆ π t and X x ,T , ˆ π t are the interiors of the support of X ˆ π ,x ( t ) and X ˆ π T ,x ( t ), respectively. It is well known that the support of X ˆ π T ,x ( t ) is contained in thesupport of X ˆ π ,x ( t ) for each x in the support of X ˆ π ,x ( T ). As a result, X x ,T , ˆ π t ⊆ X x , ˆ π t forany x ∈ X x , ˆ π T . (cid:3) Proof of Proposition 10
For α ∈ (0 , / π as given by (3.4) is an intra-personal equilibrium for multiple-time-point α -level45uantile maximization.Suppose that ˆ π ∈ A is an intra-personal equilibrium for the multiple-time-point α -level quantile maximization. For α ∈ (1 / , x ∈ X x , ˆ π T N − . According to Lemma8, ˆ π ( t, x ) , t ∈ [ T N − , T ) , x ∈ R is an intra-personal equilibrium for the α -level quantilemaximization with time interval [ T N − , T ] and initial wealth x at time T N − . According toTheorem 1, however, ˆ π cannot be an intrapersonal equilibrium so we derive a contradiction.Thus, there does not exist intra-personal equilibrium in A for the multiple-time-point α -levelquantile maximization with α ∈ (1 / , α ∈ (0 , / t ∗ as defined in Lemma 2, t as defined in Lemma 4. We claim thatit is either the case t ∗ = 0 or the case t = T and, consequently, (3.4) holds. For the sake ofcontradiction, suppose t ∗ > t < T . Then, the definition of t ∗ and t implies that t < t ∗ .Moreover, there exist two target dates T i − < T i , such that t ∗ ∈ ( T i − , T i ], i ∈ { , , . . . , N } .Fix any T ′ i − ∈ (max { T i − , t } , t ∗ ). For any t ∈ [ T ′ i − , t ∗ ), ǫ ∈ (0 , t ∗ − t ), s ∈ [ t ∗ , T ], and x ∈ R ,we have X ˆ π t,x ( s ) = X ˆ π t,x ( t ∗ ), X ˆ π t,ǫ,π t,x ( s ) = X ˆ π t,ǫ,π t,x ( t ∗ ). For any x ′ ∈ X x , ˆ π T ′ i − , by Lemma 8, ˆ π isan intra-personal equilibrium for the α -level quantile maximization for the period [ T ′ i − , t ∗ ]and with initial wealth x ′ at time T ′ i − . Then, by Theorem 1, ˆ π ( t, x ) = θ ( t )( x − x ′ ) , ∀ t ∈ [ T ′ i , t ∗ ) , x ∈ R for some θ ∈ C pw ([ T ′ i − , t ∗ )). Because t < T ′ i − , Proposition 12-(iii) and (iv)imply that X x , ˆ π T ′ i − is a nonempty open interval. Then, taking arbitrary x ′ = x ′′ in X x , ˆ π T ′ i − , wehave ˆ π ( t, x ) = θ ( t )( x − x ′ ) and ˆ π ( t, x ) = θ ( t )( x − x ′′ ) for all t ∈ [ T ′ i − , t ∗ ) and x ∈ R , so θ ( t ) = 0 for any t ∈ [ T ′ i − , t ∗ ). This, however, contradicts, the definition of t ∗ . (cid:3) Proof of Lemma 9
Fix any x N − ∈ X x , ˆ π T N − . According to Lemma 8, ˆ π is an intra-personalequilibrium for the α -level quantile maximization in the period [ T N − , T ] and with initialwealth x N − at time T N − . Theorem 1 then yields that (A.44) holds for some constant ξ < x N − . Because x N − ∈ X x , ˆ π T N − is arbitrary, we derive (A.45) holds and, consequently, theleft end of X x , ˆ π T N − is finite.Next, it is straightforward to see from (A.44) that t ∗ = T and t ∗ ≤ T N − . For thesake of contradiction, suppose τ ∗ < T N − . Then, similar to the proof in Lemma 4, for any t ∈ [ τ ∗ ∨ T N − , T N − ), x ∈ X x , ˆ π t , and π ∈ R m with b ( t ) ⊤ π > Qπ ≥
0, there exists ǫ > G π t,ǫ,π ( t, x, ; T N − ) > x = G ˆ π ( t, x, ; T N − ) for any ǫ ∈ (0 , ǫ ]. Recall t asdefined in Lemma 4. If t ≥ τ ∗ , then t ≥ T N − and, consequently, X x , ˆ π T N − = { x } . Then, (A.45)implies that ξ < x . As a result, for any t ∈ [ τ ∗ ∨ T N − , T N − ), X x , ˆ π t \ S ˆ π t = { x } is nonempty.If t < τ ∗ , then for any t ∈ [ τ ∗ ∨ T N − , T N − ), Proposition 12-(iii) and (iv) show that X x , ˆ π t isa nonempty open interval and thus X x , ˆ π t \ S ˆ π t is nonempty. Therefore, whether or not t ≥ τ ∗ ,for any t ∈ [ τ ∗ ∨ T N − , T N − ), we can always find some x ∈ X x , ˆ π t \ S ˆ π t , and Proposition 13-(iv)46nd Proposition 14-(ii) show that F ˆ π x ( t, x, G ˆ π ( t, x, α )) <
0. By the definition of τ ∗ , we haveˆ π ( t, x ) = 0, so we conclude from (A.4) that there exists π ∈ R m with b ( t ) ⊤ π > Qπ ≥ ǫ ∈ (0 , ǫ ] such that G π t,ǫ,π ( t, x, ) > G ˆ π ( t, x, ) , ∀ ǫ ∈ (0 , ǫ ]. Recall that we alreadyshowed that for such π , G π t,ǫ,π ( t, x, ; T N − ) > x = G ˆ π ( t, x, ; T N − ) for sufficiently small ǫ .As a result, ˆ π is not an intra-personal equilibrium, which is a contradiction. Thus, we musthave τ ∗ = T N − . (cid:3) Proof of Lemma 10
For each α ∈ (0 , ϕ ˆ π t,x,α ( v ) : = F ˆ π x ( t, x, G ˆ π ( t, x, α ; T N − ); T N − ) b ( t ) ⊤ v + 12 F ˆ π xx ( t, x, G ˆ π ( t, x, α ; T N − ); T N − ) k σ ( t ) ⊤ v k . (A.54)For any t ∈ [ T N − , T N − ) and x ∈ X x , ˆ π t \ ˜ S ˆ π t , Lemma 2 with [ t, T ] therein replaced by [ t, T N − ]yield thatlim ǫ ↓ G π t,ǫ,π ( t, x, α ; T N − ) − G ˆ π ( t, x, α ; T N − ) ǫ = ˜ ϕ ˆ π t,x,α ( ˆ π ( t, x )) − ˜ ϕ ˆ π t,x,α ( π ) F ˆ π y ( t, x, G ˆ π ( t, x, α ; T N − ); T N − ) . Combining the above with Lemma 2, we havelim ǫ ↓ J ( t, x ; π t,ǫ,π ) − J ( t, x ; ˆ π ) ǫ = (1 − w N − ,N ) ˜ ϕ ˆ π t,x,α ( ˆ π ( t, x )) − ˜ ϕ ˆ π t,x,α ( π ) F ˆ π y ( t, x, G ˆ π ( t, x, α ; T N − ); T N − )+ w N − ,N ϕ ˆ π t,x,α ( ˆ π ( t, x )) − ϕ ˆ π t,x,α ( π ) F ˆ π y ( t, x, G ˆ π ( t, x, α )) . Now, we set α = . Then, we havelim ǫ ↓ J ( t, x ; π t,ǫ,π ) − J ( t, x ; ˆ π ) ǫ = ˆ ϕ ˆ π t,x ( ˆ π ( t, x )) − ˆ ϕ ˆ π t,x ( π ) , where ˆ ϕ ˆ π t,x ( v ) := λ ˆ π ( t, x ) b ( t ) ⊤ v + 12 λ ˆ π ( t, x ) k σ ( t ) ⊤ v k . As a result, because ˆ π is an intra-personal equilibrium, we haveˆ ϕ ˆ π t,x ( ˆ π ( t, x )) ≤ ˆ ϕ ˆ π t,x ( π ) , ∀ π = ˆ π ( t, x ) with Qπ ≥ . (A.55)47imilar to the proof of Lemma 3, one can show that λ ˆ π ( t, x ) < λ ˆ π ( t, x ) >
0. Then,(A.55) immediately implies that − ˆ π ( t, x ) λ ˆ π ( t, x ) /λ ˆ π ( t, x ) is the optimizer of (3.1), i.e.,(A.46) holds.Next, we prove t ≤ T N − . For the sake of contradiction, suppose t > T N − . By thedefinition of t , X ˆ π ,x ( s ) = x and thus X x , ˆ π s = { x } for all s ∈ [0 , t ]. When τ ∗ > T N − ,we choose any t ∈ [ T N − , τ ∗ ∧ t ∧ T N − ) and when τ ∗ ≤ T N − and ξ = x , we choose any t ∈ [ T N − , t ∧ T N − ). For the above selected t , because x = ξ and X x , ˆ π t = { x } , we have x ∈ X x , ˆ π t \ ˜ S ˆ π t . As a result, λ ˆ π ( t, x ) < λ ˆ π ( t, x ) >
0, and ˆ π ( t, x ) = − λ ˆ π ( t,x ) λ ˆ π ( t,x ) v ∗ ( t ). Theabove is a contradiction because t < t and thus ˆ π ( t, x ) = 0 and because v ∗ ( t ) = 0.The remaining case is when τ ∗ ≤ T N − and ξ = x . In this case, we have t ≥ T N − ,so X ˆ π ,x ( s ) = x and X x , ˆ π s = { x } for all s ∈ [0 , t ]. Fix t ∈ [ T N − , T N − ). For any π ∈ R m with b ( t ) ⊤ π > Qπ ≥
0, there exists ǫ ∈ (0 , T N − − t ), such that forany ǫ ∈ (0 , ǫ ], F π t,ǫ,π ( t, x , x ; T N − ) = Φ (cid:18) − R t + ǫt b ( s ) ⊤ πds √ R t + ǫt k σ ( s ) ⊤ π k ds (cid:19) < . In addition, because π = 0, F π t,ǫ,π ( t, x , y ; T N − ) is strictly increasing and continuous in y for any ǫ ∈ (0 , ǫ ].Consequently, we conclude that G π t,ǫ,π ( t, x , ; T N − ) > x = G ˆ π ( t, x , ; T N − ) for any ǫ ∈ (0 , ǫ ], where the equality is the case because X ˆ π ,x ( T N − ) = x . In other words, the medianof wealth at time T N − is improved by taking any strategy π with b ( t ) ⊤ π > Qπ ≥ π with b ( t ) ⊤ π > Qπ ≥ π also improves the median of wealth at time T because this implies thatˆ π is not an intra-personal equilibrium and thus we have a contradiction.If ξ = ξ , then we must have t = T because t > T N − , ξ = x , and (A.44) holds. Inthis case, for any π ∈ R m with b ( t ) ⊤ π > Qπ ≥
0, there exists ǫ ∈ (0 , ǫ ] such that F π t,ǫ,π ( t, x , x ) = Φ (cid:18) − R t + ǫt b ( s ) ⊤ πds √ R t + ǫt k σ ( s ) ⊤ π k ds (cid:19) < for any ǫ ∈ (0 , ǫ ]. In addition, because π = 0, F π t,ǫ,π ( t, x , y ) is strictly increasing and continuous in y for any ǫ ∈ (0 , ǫ ]. As a result, G π t,ǫ,π ( t, x , ) > x = G ˆ π ( t, x , ) , ∀ ǫ ∈ (0 , ǫ ], where the equality is the case because t = T and thus X ˆ π ,x ( T ) = x .If ξ = ξ , then we must have t ∗ = T N − . As a result, Proposition 13-(iv) and Proposi-tion 14-(ii) imply that F ˆ π x ( t, x , G ˆ π ( t, x , α )) <
0. Because ˆ π ( t, x ) = 0, we conclude from(A.4) that there exists ǫ ∈ (0 , ǫ ] and π ∈ R m with b ( t ) ⊤ π > Qπ ≥ G π t,ǫ,π ( t, x, ) > G ˆ π ( t, x, ) , ∀ ǫ ∈ (0 , ǫ ]. (cid:3) Proof of Lemma 11
Proposition 12-(i) shows that X x , ˆ π t is increasing in t ∈ [0 , T ]. Combiningthe above with (A.45), we immediately derive part (i) of the Lemma.Next, we prove part (ii). Recall that Lemma 9 shows τ ∗ = T N − and t ∗ = T and that48emma 10 shows t ≤ T N − . Then, following the same proof as the one of Lemma 5, wecan derive (A.47) and (A.48) from (A.46). Because τ ∗ = T N − and t ∗ = T and because S ˆ π t ⊆ ˜ S ˆ π t , ∀ t ∈ [0 , T N − ), we derive from Propositions 13 and 14 that for t ∈ (0 , T N − ) and x ∈ X x , ˆ π t \ ˜ S ˆ π t , F ˆ π t ( t, x, G ˆ π ( t, x, / F ˆ π x ( t, x, G ˆ π ( t, x, / b ( t ) ⊤ ˆ π ( t, x )+ 12 F ˆ π xx ( t, x, G ˆ π ( t, x, / k σ ( t ) ˆ π ( t, x ) k = 0 ,F ˆ π t ( t, x, G ˆ π ( t, x, / T N − ); T N − ) + F ˆ π x ( t, x, G ˆ π ( t, x, / T N − ); T N − ) b ( t ) ⊤ ˆ π ( t, x )+ 12 F ˆ π xx ( t, x, G ˆ π ( t, x, / T N − ); T N − ) k σ ( t ) ˆ π ( t, x ) k = 0 . Multiplying by w N − ,N /F ˆ π y ( t, x, G ˆ π ( t, x, )) and (1 − w N − ,N ) /F ˆ π y ( t, x, G ˆ π ( t, x, ; T N − ); T N − )to both sides of the first and second equations, respectively, in the above and summing themup, we obtain w N − ,N F ˆ π t ( t, x, G ˆ π ( t, x, )) F ˆ π y ( t, x, G ˆ π ( t, x, )) + (1 − w N − ,N ) F ˆ π t ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, ; T N − ); T N − )+ λ ˆ π ( t, x ) b ( t ) ⊤ ˆ π ( t, x ) + 12 λ ˆ π ( t, x ) k σ ( t ) ˆ π ( t, x ) k = 0 . Combining the above with (A.47) and (A.48), we obtain w N − ,N F ˆ π t ( t, x, G ˆ π ( t, x, )) F ˆ π y ( t, x, G ˆ π ( t, x, )) + (1 − w N − ,N ) F ˆ π t ( t, x, G ˆ π ( t, x, ; T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, ; T N − ); T N − )+ 12 λ ˆ π ( t, x ) ρ ( t ) (cid:0) ˜ a ( t ) + ˜ a ( t ) x (cid:1) = 0 . (A.56)Proposition 14-(ii) and (iii) yield that G ˆ π x ( t, x, /
2) = − F ˆ π x ( t, x, G ˆ π ( t, x, / F ˆ π y ( t, x, G ˆ π ( t, x, / , G ˆ π t ( t, x, /
2) = − F ˆ π t ( t, x, G ˆ π ( t, x, / F ˆ π y ( t, x, G ˆ π ( t, x, / ,G ˆ π x ( t, x, / T N − ) = − F ˆ π x ( t, x, G ˆ π ( t, x, / T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, / T N − ); T N − ) ,G ˆ π t ( t, x, / T N − ) = − F ˆ π t ( t, x, G ˆ π ( t, x, / T N − ); T N − ) F ˆ π y ( t, x, G ˆ π ( t, x, / T N − ); T N − ) . Plugging above into (A.56), we derive the differential equation in (A.49). Because τ ∗ = T N − ,Proposition 14-(iv) yields that lim t ↑ T N − ,x ′ → x G ˆ π ( t, x ′ , / T N − ) = x, ∀ x ∈ R . Because t ∗ =49 , Proposition 14-(iv) yields thatlim t ↑ T N − ,x ′ → x G ˆ π ( t, x ′ , /
2) = G ˆ π ( T N − , x, /
2) = ( x − ξ ) e R TTN − ( ρ ( s ) / ds + ξ, ∀ x = ξ, where the where the second equality is the case due to (A.44). As a result, we derive theboundary condition in (A.49). (cid:3) Proof of Lemma 12
We prove part (i) first. For the sake of contradiction, suppose ˜ a ( s ) =0 , s ∈ [ t, T N − ). Because τ ∗ = T N − by Lemma 9, for any s ∈ [ t, T N − ), there exists τ ∈ [ s, T N − ) with θ ( τ ) = 0 and thus ˜ a ( τ ) = 0. As a result, for any x ∈ X x , ˆ π t , X ˆ π t,x ( T N − ) = x + Z T N − t ˜ a ( τ ) v ∗ ( τ ) ⊤ b ( τ ) dτ + Z T N − t ˜ a ( τ ) v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ )is a non-degenerate normal random variable, which, together with Lemma 8, implies that X x , ˆ π T N − ⊇ X x,t, ˆ π T N − = R . On the other hand, (A.45) implies that the left end of X x , ˆ π T N − is finite,so we arrive at contradiction.Next, we prove part (ii). Because X x , ˆ π t = ( x ( t ) , + ∞ ) by Lemma 11 and because X x , ˆ π s is increasing in s , there exists x t such that [ x t , + ∞ ) ⊂ X x , ˆ π s \{ ξ, ξ } , ∀ s ∈ [ t, T N − ]. As aresult, (A.49) holds in the region [ t, T N − ] × ( x t , + ∞ ). Similar to the proof of Lemma 7, wecan apply Lemma 6 to conclude that there exists c t > max( x ( t ) , ξ, ξ ) such that g ˆ π ( s, x ) = w N − ,N (cid:0) β ( s, / x + β ( s, / (cid:1) + (1 − w N − ,N ) (cid:0) ˜ β ( s ) x + ˜ β ( s ) (cid:1) for all ( s, x ) ∈ [ t, T N − ] × [ c t , + ∞ ). Consequently, (A.52) holds.Finally, the proof of (A.53) is the same as the proof of (A.17) in Lemma 7. (cid:3) Proof of Proposition 11
By Lemma 12-(ii), we have12 Z T N − t ˜ a ( s ) (cid:0) − ˜ a ( s ) (cid:1) k σ ( s ) ⊤ v ∗ ( s ) k ds = 0 , ∀ t ∈ ( T N − , T N − ) , which implies that ˜ a ( t )(1 − ˜ a ( t )) = 0 , t ∈ ( T N − , T N − ). Lemma 12-(i) shows for any t ∈ ( T N − , T N − ), ˜ a ( s ) = 0 for some s ∈ [ t, T N − ). Then, we must have ˜ a ( t ) = 1 , t ∈ ( T N − , T N − ) because ˜ a ∈ C ([ T N − , T N − )) as shown by Lemma 11-(ii). Then, (A.48) impliesthat ˆ π ( t, x ) = (˜ a ( t ) + x ) v ∗ ( t ) , t ∈ [ T N − , T N − ) , x ∈ R . (A.57)50efine ˆ a by setting ˆ a ( t ) = ˜ a ( t ) , t ∈ [ T N − , T N − ) and ˆ a ( t ) = − ξ, t ∈ [ T N − , T ). Then, wehave ˆ π ( t, x ) = (ˆ a ( t ) + x ) v ∗ ( t ) , t ∈ [ T N − , T ) , x ∈ R . Next, we prove that ˆ a ( t ) = − ξ for any t ∈ [ T N − , T ).Lemma 11-(i) shows that X x , ˆ π t = ( x ( t ) , + ∞ ) , t ∈ ( T N − , T N − ] for some decreasing func-tion x on ( T N − , T N − ]. Moreover, we derive from (A.45) that x ( T ) ≥ ξ . Because theclosure of X x , ˆ π T N − , which is equal to [ x ( T ) , + ∞ ), is the support of X ˆ π ,x ( T N − ), we have P ( X ˆ π ,x ( T N − ) ≥ ξ ) ≥ P ( X ˆ π ,x ( T N − ) ≥ x ( T )) = 1. Because ˆ π ( s, x ) = ( − ξ + x ) v ∗ ( s ) , s ∈ [ T N − , T ) and because x ( T ) ≥ ξ , we conclude that for any t ∈ ( T N − , T ], P ( X ˆ π ,x ( t ) ≥ ξ ) = 1and thus, by Proposition 12, X x , ˆ π t = ( x ( t ) , + ∞ ) for some x ( t ) ∈ R .Fix any t ∈ ( T N − , T ) and x ∈ X x , ˆ π t \{− ˆ a ( t ) } . Lemma 8 implies that X x ,t , ˆ π t ⊆ X x , ˆ π t = ( x ( t ) , + ∞ ) , ∀ t ∈ [ t , T ]. Proposition 12-(iv) then shows that ˆ a must be increasingon ( t , T ) and X x ,t , ˆ π t = ( − ˆ a ( t ) , + ∞ ) , ∀ t ∈ ( t , T ); otherwise the lower end of X x ,t , ˆ π t would be −∞ . For each t ∈ ( t , T ), because X x ,t , ˆ π t ⊆ X x , ˆ π t = ( x ( t ) , + ∞ ), we derive − ˆ a ( t ) ≥ x ( t ). Moreover, Proposition 12-(i) shows that X x ,t , ˆ π s is increasing in s ≥ t , so wehave { x } = X x ,t , ˆ π t ⊆ X x ,t , ˆ π s = ( − ˆ a ( s ) , + ∞ ) , ∀ s ∈ ( t , T N − ). Sending s in the above to t and recalling that ˆ a is right-continuous on [ T N − , T ), we conclude x ≥ − ˆ a ( t ). Because x ∈ X x , ˆ π t \{− ˆ a ( t ) } = ( x ( t ) , + ∞ ) \{− ˆ a ( t ) } is arbitrary, we derive that x ( t ) ≥ − ˆ a ( t ).Because t is arbitrary, we conclude that ˆ a is increasing on ( T N − , T ) and that x ( t ) = − ˆ a ( t )and thus X x , ˆ π t = ( − ˆ a ( t ) , + ∞ ) for all t ∈ ( T N − , T ).By the definition of τ ∗ and ξ , we have τ ∗ = inf { t ∈ [0 , T N − ] : ˆ a ( s ) = − ξ , s ∈ [ t, T N − ] } .For t ∈ ( T N − , τ ∗ ), ˜ S ˆ π t = ∅ and thus X x , ˆ π t \ ˜ S ˆ π t = ( − ˆ a ( t ) , + ∞ ), and for t ∈ [ τ ∗ , T N − ),˜ S ˆ π t = { ξ } = {− ˆ a ( t ) } and thus X x , ˆ π t \ ˜ S ˆ π t = ( − ˆ a ( t ) , + ∞ ). Also recall that ˆ a is increasingon ( T N − , T ) and is equal to ξ on [ T N − , T ), so − ˆ a ( t ) ≥ ξ for all t ∈ [ T N − , T N − ). As aresult, Lemma 11-(ii) and Lemma 6-(iii) yield that (A.52) holds for any x > − ˆ a ( s ) and s ∈ ( T N − , T N − ).Recall that t ∗ ≤ T N − and in the following, we prove that t ∗ ≤ T N − . For the sakeof contradiction, suppose that t ∗ ∈ ( T N − , T N − ]. Fix any t ∈ ( T N − , t ∗ ). Because ˆ a isincreasing in ( T N − , T ) and continuous on [ T N − , T N − ) and on [ T N − , T ), following the samecalculation of (A.33) in the proof of Proposition 9, we derive X ˆ π t, − ˆ a ( t ) ( T ) = − ˆ a ( T ) + Z Tt e R Ts ( b ( τ ) ⊤ v ∗ ( τ ) − k σ ( τ ) ⊤ v ∗ ( τ ) k dτ ) dτ + R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) d ˆ a ( s ) . X ˆ π t, − ˆ a ( t ) ( T ) − ( β ( t, / − ˆ a ( t ) β ( t, / Z Tt e R Ts ρ ( τ ) dτ + R Ts v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) d ˆ a ( s ) − Z Tt e R Ts ρ ( τ ) dτ d ˆ a ( s ) . (A.58)Because t < t ∗ , d ˆ a ( s ) defines a positive measure on ( t, T ). Similar to the proof in Proposition9, we can prove that there exists δ t > G ˆ π ( t, x, / − (cid:0) β ( t, /
2) + xβ ( t, / (cid:1) > , ∀ x ∈ ( − ˆ a ( t ) − δ t , − ˆ a ( t ) + δ t ) . (A.59)We first consider the case in which τ ∗ ≤ T N − . Then, because τ ∗ = T N − as shown inLemma 9, we conclude ˆ a ( t ) = − ξ , ∀ t ∈ [ T N − , T N − ). Because t ∗ > T N − and ξ ≥ ξ , wemust have ξ > ξ and t ∗ = T N − . Then, straightforward calculation yields G ˆ π ( t, x, / T N − ) = ξ + ( x − ξ ) e R TN − t ρ ( s ) ds = ˜ β ( t ) x + ˜ β ( t ) , x > ξ , t ∈ [ T N − , T N − ) . Combining the above with (A.59), recalling that (A.52) holds for any x > − ˆ a ( s ) and s ∈ ( T N − , T N − ), and noting that w N − ,N ∈ (0 , t ∗ ≤ T N − .Next, we consider the case in which τ ∗ > T N − . Fix any t ∈ ( T N − , τ ∗ ). Then, ˆ a ( s ) is notconstant in s ∈ [ t, τ ∗ ], which means that d ˆ a ( s ) defines a positive measure on [ t, τ ∗ ]. Similarto the derivation of (A.40), we can prove that there exists ˜ δ t >
0, such that G ˆ π ( t, x, / T N − ) − (cid:0) ˜ β ( t ) + x ˜ β ( t ) (cid:1) > , ∀ x ∈ ( − ˆ a ( t ) − ˜ δ t , − ˆ a ( t ) + ˜ δ t ) . Combining the above with (A.59), recalling that (A.52) holds for any x > − ˆ a ( s ) and s ∈ ( T N − , T N − ), and noting that w N − ,N ∈ (0 , t ∗ ≤ T N − .Having proved that t ∗ ≤ T N − and thus ˆ a ( s ) = − ξ, ∀ s ∈ [ T N − , T ), we conclude thatˆ π ( t, x ) = v ∗ ( t )( x − ξ ) , t ∈ [ T N − , T ) , x ∈ R . Applying Lemmas 9–12 and the above proof tothe periods [ T i − , T i ), i = N − , N − , . . . , π ( t, x ) =( x − ξ ) v ∗ ( t ) , t ∈ [0 , T ) , x ∈ R for some ξ < x . The proof then completes. (cid:3) .3 Other Proofs Proof of Lemma 1
By Assumption 1, there exists 0 =: t < t < · · · < t N := T such that σ and b are continuous on [ t i − , t i ) and can be continuously extended to [ t i − , t i ], i = 1 , . . . , N .Thus, in the following, we only need to fix i and consider the continuous extension of b and σ on [ t i − , t i ].Fix any t ∈ [ t i − , t i ]. It is obvious that the optimal solution of (3.1) uniquely ex-ists. Moreover, by Assumption 2, there exists v ∈ R m with Qv ≥ b ( t ) ⊤ v > ǫ > v ǫ := ǫv satisfies Qv ǫ ≥ v ⊤ ǫ σ ( t ) σ ( t ) ⊤ v ǫ − b ( t ) ⊤ v ǫ = ǫ (cid:0) v ⊤ σ ( t ) σ ( t ) ⊤ v ǫ − b ( t ) ⊤ v (cid:1) <
0, so the optimal solution of (3.1), namely v ∗ ( t ), cannot be0. The Lagrange dual theory implies the follow equations: σ ( t ) σ ( t ) ⊤ v ∗ ( t ) − b ( t ) − Q ⊤ λ = 0 ,λ ⊤ Qv ∗ ( t ) = 0 ,λ ≥ , Qv ∗ ( t ) ≥ . (A.60)Multiplying by v ∗ ( t ) ⊤ from left on both sides of the first equation of (A.60) and recallingthe second equation of (A.60), we immediately conclude that b ( t ) ⊤ v ∗ ( t ) = k σ ( t ) ⊤ v ∗ ( t ) k .Moreover, because v ∗ ( t ) = 0 and σ ( t ) σ ( t ) ⊤ is positive definite, we have k σ ( t ) ⊤ v ∗ ( t ) k > v ∗ ∈ C ([ t i − , t i ]). As a result, inf t ∈ [ t i − ,t i ] k v ∗ ( t ) k > t ∈ [ t i − ,t i ] k σ ( t ) ⊤ v ∗ ( t ) k > (cid:3) Proof of Proposition 1
Fix any t ∈ (0 , T ). Because X x , ˆ π s = ( ξ, + ∞ ) for s ∈ (0 , T ), because˜ π agrees with ˆ π for x ∈ X x , ˆ π s , s ∈ [0 , T ), and because ˜ π ( s, x ) is continuous in x , we have X ˜ π t,ξ ( s ) = ξ, s ∈ [ t, T ]. As a result, G ˜ π ( t, ξ, /
2) = ξ . Recall that v ∗ ( t ) ⊤ b ( t ) >
0, so thereexists ǫ ∈ ( T − t ) such that v ∗ ( t ) ⊤ b ( s ) > , s ∈ [ t, t + ǫ ]. Then, for any ǫ ∈ (0 , ǫ ), we have P (cid:16) X ˜ π t,ǫ,v ∗ ( t ) t,ξ ( T ) > ξ (cid:17) ≥ P (cid:16) X v ∗ ( t ) t,ξ ( t + ǫ ) > ξ (cid:17) > / , where the first inequality is the case because on (cid:8) X v ∗ ( t ) t,ξ ( t + ǫ ) > ξ (cid:9) , we have X ˜ π t,ǫ,v ∗ ( t ) t,ξ ( T ) = X ˆ π t + ǫ,X v ∗ ( t ) t,ξ ( t + ǫ ) ( T ) > ξ and the second inequality is the case because R t + ǫt v ∗ ( t ) ⊤ b ( s ) ds >
0. Asa result, G ˜ π t,ǫ,v ∗ ( t ) ( t, ξ, / > ξ due to the definition of right-continuous quantile functions. (cid:3) roof of Proposition 2 Straightforward calculation leads to (4.3). For any s ∈ [0 , T ),by Assumption 2, we have k σ ( s ) ⊤ v ∗ ( s ) k = b ( s ) ⊤ v ∗ ( s ) >
0. Because γ ∈ (0 , γ − γ ∈ (0 , a t,γ is well defined and strictly larger than 1.Finally, the comparison of G ˆ π ξ ( t, x, /
2) and G π γ − Kelly ( t, x, /
2) is straightforward. (cid:3)
Proof of Proposition 4
Given a portfolio insurance level ξ , we consider more generallythe pre-committed strategy planned by the agent at time t with wealth level x t ; i.e., theportfolio strategy that maximizes the median, conditional at time t with wealth level x t , ofthe terminal wealth at time T . Recall that the risk-free rate r is set to be zero for simplicityin our discussion.According to Section 4 of He and Zhou (2011), the problem of finding the pre-committedstrategy is equivalent to the following:Max X G X (1 / E t (cid:2)(cid:0) ˆ ρ ( T ) / ˆ ρ ( t ) (cid:1) X (cid:3) ≤ x t , X ≥ ξ, (A.61)where X stands for a possible wealth profile attained at time T , G X (1 /
2) stands for themedian of X , E t stands for the expectation conditional on the information at time t , andˆ ρ ( s ) : = e − R s k ˆ θ ( τ ) k dτ − R s ˆ θ ( τ ) ⊤ dW ( τ ) , s ∈ [ t, T ] , ˆ θ ( s ) : = argmin θ ∈ R d {k θ k : σ ( s ) θ − b ( s ) ∈ K ∗ } , s ∈ [ t, T ) , where K := { π ∈ R m : Qπ ≥ } and K ∗ is the dual cone of K , i.e., K ∗ := { y ∈ R m : y ⊤ π ≥ , ∀ π ∈ K } . It is known that ˆ ρ ( t ) / ˆ ρ ( s ) , s ∈ [ t, T ] is the wealth process of the optimalportfolio under expected logarithmic utility maximization with initial wealth at time t to be1, and this optimal portfolio is indeed the Kelly portfolio (3.2); see for instance Karatzas andShreve (1998). Therefore, we immediately conclude that ˆ θ ( s ) = σ ( s ) ⊤ v ∗ ( s ) and consequentlyˆ ρ ( s ) : = e − R s k σ ( τ ) ⊤ v ∗ ( τ ) k dτ − R s v ∗ ( τ ) ⊤ σ ( τ ) dW ( τ ) , s ∈ [ t, T ] . (A.62)By changing of variable, ˜ X = X − ξ , problem (A.61) is equivalent to (C.1) with ρ =ˆ ρ ( T ) / ˆ ρ ( t ), m to be a Dirac measure at 1 /
2, and ˜ x = x t − ξ . Then, Corollary 1 yieldsthat the optimal solution to (A.61) is X ∗ = ξ + k ∗ t ˆ ρ ( T ) / ˆ ρ ( t ) ≤ β ∗ t , where β ∗ t := F − ρ ( T ) / ˆ ρ ( t ) (1 / k ∗ t := x t − ξ E [ˆ ρ ( T ) / ˆ ρ ( t ) ˆ ρ ( T ) / ˆ ρ ( t ) ≤ β ∗ t ] . Denoting Z ( t ) := R t v ∗ ( s ) ⊤ σ ( s ) dW ( s ) , t ∈ [0 , T ], we havelog( ˆ ρ ( t )) = − Z ( t ) − R t k σ ( s ) ⊤ v ∗ ( s ) k ds . As a result, we can rewrite the optimal solution54o (A.61) as follows: X ∗ = ξ + x t − ξ E h e − ( Z ( T ) − Z ( t )) − R Tt k σ ( s ) ⊤ v ∗ ( s ) k ds Z ( T ) − Z ( t ) ≥ i Z ( T ) − Z ( t ) ≥ = ξ + x t − ξ Φ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) Z ( T ) − Z ( t ) ≥ . Denote by π t, pc the pre-committed strategy, i.e., the strategy that attains terminal wealthis equal to X ∗ . Then, for any s ∈ [ t, T ], we have X π t, pc t,x t ( s ) = E s (cid:2)(cid:0) ˆ ρ ( T ) / ˆ ρ ( s ) (cid:1) X ∗ (cid:3) = ξ + x t − ξ Φ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) E s h e − ( Z ( T ) − Z ( s )) − R Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Z ( T ) − Z ( t ) ≥ i = ξ + x t − ξ Φ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) Φ ( d ( s, Z ( s ) − Z ( t ))) (A.63)where d ( s, z ) := z − R Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ qR Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ . By Itˆo’s lemma and straightforward calculation, we obtain π t, pc (cid:0) s, X π t, pc t,x t ( s ) (cid:1) X π t, pc t,x t ( s ) − ξ = 1 qR Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Φ ′ ( d ( s, Z ( s ) − Z ( t )))Φ ( d ( s, Z ( s ) − Z ( t ))) v ∗ ( s ) , s ∈ [ t, T ) . (A.64)For each s ∈ ( t, T ], d ( s, z ) is continuous, strictly increasing in z andlim z ↓−∞ d ( s, z ) = −∞ , lim z ↑ + ∞ d ( s, z ) = + ∞ .
55s a result, X π t, pc t,x t ( s ) is continuous, strictly increasing in Z ( s ) − Z ( t ), andlim Z ( s ) − Z ( t ) ↓−∞ X π t, pc t,x t ( s ) = ξ, lim Z ( s ) − Z ( t ) ↑ + ∞ X π t, pc t,x t ( s ) = ξ + x t − ξ Φ (cid:18) − qR Tt k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) . Denote the inverse function of the relation between X π t, pc t,x t ( s ) and Z ( s ) − Z ( t ) as z t ( s, X π t, pc t,x t ( s )).Then, we have π t, pc ( s, x ) x − ξ = 1 qR Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Φ ′ ( d ( s, z t ( s, x )))Φ ( d ( s, z t ( s, x ))) v ∗ ( s ) , s ∈ [ t, T ) , x ∈ ( ξ, ξ + k ∗ t ) . Moreover, (cid:0) Φ ′ ( d ) / Φ( d ) (cid:1) ′ = [ − d Φ ′ ( d )Φ( d ) − Φ ′ ( d ) ] / Φ( d ) <
0, where the inequality is thecase because − d Φ( d ) < Φ ′ ( d ) for any d ∈ R . As a result,∆ t, pc ( s, x ) := 1 qR Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Φ ′ ( d ( s, z t ( s, x )))Φ ( d ( s, z t ( s, x )))is strictly decreasing in x ,lim x ↓ ξ ∆ t, pc ( s, x ) = lim d ↓−∞ qR Ts k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Φ ′ ( d )Φ ( d ) = + ∞ , and, similarly, lim x ↑ ξ + k ∗ t ∆ t, pc ( s, x ) = 0.Specializing the above at t = 0, we complete the proof of part (i) of the Proposition. Forpart (ii), Corollary 1 yields that the optimal median of the terminal wealth is k ∗ , i.e., G π , pc (0 , x , /
2) = ξ + k ∗ = ξ + x − ξ Φ (cid:18) − qR T k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) . Consider the function f ( x ) := Φ( − x ) − e − x / , x ≥
0. We have f ′ ( x ) = e − x / (cid:0) x − (2 π ) − / (cid:1) ,so f is strictly decreasing on [0 , (2 π ) − / ] and strictly increasing on [(2 π ) − / , + ∞ ). Because f (0) = − / < x ↑ + ∞ f ( x ) = 0, we conclude that Φ( − x ) < e − x / for any x ≥
0. Asa result, G π , pc (0 , x , / > G ˆ π ξ (0 , x , / t ∈ (0 , T ) and x ∈ ( ξ, ξ + k ∗ ), we have P t ( X π , pc ,x ( T ) = ξ ) = 1 − P t ( X π , pc ,x ( T ) = ξ + k ∗ ) = P t ( Z ( T ) < P t ( Z ( T ) − Z ( t ) < − Z ( t )) > / , Z ( t ) < , ≤ / , Z ( t ) ≥ . Also note that Z ( t ) ≥ X π , pc ,x ( t ) ≥ ξ + x − ξ Φ (cid:18) − qR T k σ ( s ) ⊤ v ∗ ( s ) k ds (cid:19) Φ ( d ( t, x . Therefore, G π , pc ( t, x, /
2) = ξ + k ∗ , x ∈ [ x , ξ + k ∗ ) ,ξ, x ∈ ( ξ, x ) . Then, Proposition 2 yields (4.6) immediately. Recall that Φ( − x ) < e − x / for any x ≥
0, so1 < e − R T k σ ( s ) ⊤ v ∗ ( s ) k ds Φ (cid:18) − qR T k σ ( τ ) ⊤ v ∗ ( τ ) k dτ (cid:19) ≤ ˜ a t < (cid:18) − qR T k σ ( τ ) ⊤ v ∗ ( τ ) k dτ (cid:19) = k ∗ x − ξ . The proof then completes. (cid:3)
Proof of Proposition 5
By definition, π na ( t, x ) = π t, pc ( t, X π t, pc t,x ( t )) = 1 qR Tt k σ ( τ ) ⊤ v ∗ ( τ ) k dτ Φ ′ ( d ( t, d ( t, v ∗ ( t )( x − ξ ) , where the second equality is the case due to (A.64). Recall that d ( t,
0) = − qR Tt k σ ( τ ) ⊤ v ∗ ( τ ) k dτ ,we immediately obtain (4.7).Because Φ ′ ( d ) > d Φ( − d ) for any d >
0, we conclude that ∆ na ( t ) > , t ∈ [0 , T ). Further-more, it is easy to see that lim t → T ∆ na ( t ) = + ∞ .57inally, fix any t ∈ [0 , T ) and x > ξ and consider τ ∈ ( t, T ). Then, we have X π na t,x ( τ ) = ξ + ( x − ξ ) e R τt ∆ na ( s ) b ( s ) ⊤ v ∗ ( s ) ds − R τt ∆ na ( s ) k σ ( s ) ⊤ v ∗ ( s ) k ds + R τt ∆ na ( s ) v ∗ ( s ) ⊤ σ ( s ) dW ( s ) = ξ + ( x − ξ ) e R τt ( ∆ na ( s ) − ∆ na ( s ) ) k σ ( s ) ⊤ v ∗ ( s ) k ds + R τt ∆ na ( s ) v ∗ ( s ) ⊤ σ ( s ) dW ( s ) , where the second equality is the case due to Lemma 1. As a result, the median of X π na t,x ( τ ),denoted as G π na ( t, x, / τ ), is G π na ( t, x, / τ ) = ξ + ( x − ξ ) e R τt ( ∆ na ( s ) − ∆ na ( s ) ) k σ ( s ) ⊤ v ∗ ( s ) k ds . Because inf s ∈ [0 ,T ) k σ ( s ) ⊤ v ∗ ( s ) k > Z Tt ∆ na ( s ) ds < + ∞ , Z Tt ∆ na ( s ) ds = + ∞ . As a result, lim τ ↑ T G π na ( t, x, / τ ) = ξ . (cid:3) Proof of Proposition 6
It is a standard result in portfolio selection that ˆ π ξ maximizes E [ln( X π ( T ) − ξ )]; see for instance Karatzas and Shreve (1998). (cid:3) B Distributional Properties of Linear SDEs
In the section, we provide some distributional properties of the wealth equation for a givenstrategy in A . Assume the risk-free rate to be zero. Then, the wealth equation becomes ( dX ( t ) = ( θ ( t ) + θ ( t ) X ( t )) ⊤ b ( t ) dt + ( θ ( t ) + θ ( t ) X ( t )) ⊤ σ ( t ) dW ( t ) , t ∈ [0 , T ] ,X (0) = x ∈ R , (B.1)where θ , θ , b, σ ∈ C pw ([0 , T )) and σ ( t ) ⊤ σ ( t ) is positive definite for any t ∈ [0 , T ]. Recall theset of reachable states of X ( t ) as defined in Section 2.3 and denote it as X t . The support of X ( t ) is then the closure of X t . The following proposition proves some properties of X t . Proposition 12
Denote t = inf { s ∈ [0 , T ] : θ ( s ) + x θ ( s ) = 0 } , ¯ t = inf { s ∈ [0 , T ] : Z s k θ ( z ) + ˜ h ( z ) θ ( z ) k dz > } , here ˜ h ( t ) := − θ ( t ) ⊤ σ ( t ) σ ( t ) ⊤ θ ( t ) k σ ( t ) ⊤ θ ( t ) k θ ( t ) =0 , t ∈ [0 , T ] ,x ∗ ( t ) := Z t b ( s ) ⊤ θ ( s ) e R ts b ( z ) ⊤ θ ( z ) dz ds + x e R t b ( s ) ⊤ θ ( s ) ds , t ∈ [0 , T ] . Then, the following are true:(i) X t is increasing in t ∈ [0 , T ] .(ii) For any t ∈ [0 , t ] , X ( t ) = x ∗ ( t ) = x and X t = { x } .(iii) For any t ∈ (¯ t, T ] , X t = R and X ( t ) has a probability density that is positive on R .(iv) Suppose t < ¯ t and fix any t ∈ ( t, ¯ t ] . X t is an open, unbounded interval and X ( t ) hasa probability density that is positive on X t . Moreover, if θ ( s ) = 0 , ∀ s ∈ ( t, t ) , thefollowing are true:(a) Suppose there exist s , s ∈ ( t, t ) , such that ˜ h ( s ) < x ∗ ( s ) , ˜ h ( s ) > x ∗ ( s ) . Then X t = R .(b) Suppose ˜ h ( s ) ≤ x ∗ ( s ) for any s ∈ ( t, t ) . If ˜ h is not decreasing on ( t, t ) , then X t = R . If ˜ h is decreasing in on ( t, t ) , then ˜ h ( s ) < x ∗ ( s ) for any s ∈ ( t, t ) and X t = (˜ h ( t − ) , + ∞ ) , where ˜ h ( t − ) := lim s ↑ t ˜ h ( s ) .(c) Suppose ˜ h ( s ) ≥ x ∗ ( s ) for any s ∈ ( t, t ) . If ˜ h is not increasing on ( t, t ) , then X t = R . If ˜ h is increasing on ( t, t ) , then ˜ h ( s ) > x ∗ ( s ) for any s ∈ ( t, t ) and X t = ( −∞ , ˜ h ( t − )) , where ˜ h ( t − ) := lim s ↑ t ˜ h ( s ) .Proof See Corollary 4 and Theorem 3 in He and Jiang (2020). (cid:3)
Next, we consider the transition probability of ( X ( t )) t ≥ . For each t ∈ [0 , T ) and x ∈ R ,define F ( t, x, y ) := P ( X ( T ) ≤ y | X ( t ) = x ) , y ∈ R ,G ( t, x, α ) := sup { y ∈ R : F ( t, x, y ) ≤ α } , α ∈ (0 , X ( t )) t ≥ . For any function g ( t, x, y )59hat is differentiable in t and twice differentiable in x , define A g ( t, x, y ) = g t ( t, x, y ) + ( θ ( t ) + θ ( t ) x ) ⊤ b ( t ) g x ( t, x, y )+ 12 k σ ( t ) ⊤ ( θ ( t ) + θ ( t ) x ) k g xx ( t, x, y ) , where g t , g x , and g xx denote respectively the first-order derivative of g with respect to t ,first- and second-order derivatives of g with respect to x .We introduce some notations. For any interval [ c, d ) and open set O in R l , denote by C , ∞ ([ a, b ) × O ) the set of functions g ( t, z ) from [ a, b ) × O to R such that its derivativeswith respect to z of any order exist and are continuous in ( t, z ) on [ a, b ) × O , denote by C , ∞ ([ a, b ) × O ) the set of functions g ( t, z ) from [ a, b ) × O to R such that its first-orderderivative with respect to t and its derivatives with respect to z of any order exist and arecontinuous in ( t, z ) on [ a, b ) × O . Denote by C , ∞ pw ([ a, b ) × O ) the set of functions g ( t, z ) from[ a, b ) × O to R l such that there exists a = t < t < . . . t N = b with g ∈ C , ∞ ([ t i − , t i ) × O ), i = 1 , . . . , N . Denote by N = N ∪ { } , where N is the set of positive integers.The following proposition provides a complete picture of the transition probability dis-tribution F ( t, x, y ). Proposition 13
Define t ∗ := inf { t ∈ [0 , T ) : θ ( s ) = θ ( s ) = 0 , ∀ s ∈ [ t, T ) } with the convention that inf ∅ = T and t ∗ := inf { t ∈ [0 , t ∗ ) : there exists ξ ∈ R such that θ ( s ) + ξθ ( s ) = 0 , ∀ s ∈ [ t, t ∗ ) } with the convention that inf ∅ = t ∗ . Then, when t ∗ < t ∗ , there exists unique ξ ∈ R such that ˜ c ∗ ( s ) + ξc ( s ) = 0 , ∀ s ∈ [ t ∗ , t ∗ ) . Consider any partitions t < t < · · · < t m = t ∗
See Corollary 3 and Theorem 2 in He and Jiang (2020). (cid:3)
Finally, the following proposition shows some properties of the transition quantile G ( t, x, α ): Proposition 14
Suppose the same conditions as assumed in Proposition 13 hold and denote D := { ( t, x ) | t ∈ [0 , t ∗ ) , x ∈ R } ∪ { ( t, x ) | x = ξ, t ∈ [ t ∗ , t ∗ ) } . Then, the following are true: i) For any ( t, x ) ∈ (cid:0) [0 , T ] × R (cid:1) \D and α ∈ (0 , , G ( t, x, α ) = x .(ii) For any ( t, x ) ∈ D and α ∈ (0 , , G ( t, x, α ) is uniquely determined by F ( t, x, G ( t, x, α )) = α. Moreover, G ( t, x, α ) is continuous in ( t, x, α ) on D × (0 , and satisfies G ( t, x, α ) = ξ, ∀ x = ξ, t ∈ [ t ∗ , t ∗ ) , α ∈ (0 , ,F y ( t, x, G ( t, x, α )) > , ∀ ( t, x ) ∈ D , α ∈ (0 , . Furthermore, G ( t, x, α ) is infinitely differentiable in ( x, α ) for any ( t, x, α ) ∈ D × (0 , ,and its derivatives are continuous in ( t, x, α ) on D × (0 , . In particular, we have G x ( t, x, α ) = − F x ( t, x, G ( t, x, α )) F y ( t, x, G ( t, x, α )) , ( t, x ) ∈ D , α ∈ (0 , . (iii) G ∈ C , ∞ (cid:0) [ t i − , t i ) × R × (0 , (cid:1) for all i = 1 , . . . , m and G ∈ C , ∞ (cid:0) { ( t, x ) | x = ξ, t ∈ [ t i − , t i ) } × (0 , (cid:1) for all i = m + 1 , . . . , n . In particular, G t ( t, x, α ) = − F t ( t, x, G ( t, x, α )) F y ( t, x, G ( t, x, α )) , ( t, x ) ∈ D , α ∈ (0 , . (iv) For any x ∈ R and α ∈ (0 , , lim t ↑ t ∗ , ( x ′ ,α ′ ) → ( x,α ) G ( t, x ′ , α ′ ) = G ( t ∗ , x, α ) = x. Proof
See Corollaries 3 and 2 in He and Jiang (2020). (cid:3)
C A Static Portfolio Selection Problem
In this Section, we consider an optimization problem that is related to portfolio selectionunder median maximization. This problem is a generalization of the portfolio selection underYaari (1987)’s dual theory of decision under risk studied by He and Zhou (2011).62 .1 Model and Solution
Consider the following problemMax V ( ˜ X ) = R [0 , G ˜ X ( z ) m ( dz )Subject to E [ ρ ˜ X ] ≤ ˜ x, ˜ X ≥ , (C.1)where ˜ X and ρ live on a probability space (Ω , F , P ) with ρ to be strictly positive andintegrable, ˜ x > G ˜ X stands for the quantile function of ˜ X with G ˜ X (0) :=essinf ˜ X and G ˜ X (1) := esssup ˜ X , and m is a probability measure on [0 , m ( · ) admits a density φ ( · ) on [0 , V ( X ) becomes the Yaari’s dual preferencemeasure (Yaari, 1987). In this case, problem (C.1) has been investigated by He and Zhou(2011) when φ ( · ) satisfies a monotonicity condition. On the other hand, He et al. (2015)consider a mean-risk portfolio choice problem which is similar to (C.1) but with an additionalconstraint standing for the mean target of the agent. Assumption 3 ρ is positive and has no atom, i.e., its CDF is continuous, and E [ ρ ] ≤ . Denote ρ := essinf ρ ≥ δ := E [ ρ ] ≤
1, and F − ρ as the quantile function of ρ . Because ρ > F − ρ ( z ) > , z ∈ (0 , G ( · ) ∈ G R [0 , G ( z ) m ( dz )Subject to R F − ρ (1 − z ) G ( z ) dz ≤ ˜ x, G (0) ≥ , (C.2)where G is the set of quantile functions, i.e., G := { G ( · ) : [0 , → R ∪ {±∞}| G ( · ) is finite-valued, right-continuous,and increasing in (0 ,
1) and G (0) = lim z ↓ G ( z ) , G (1) = lim z ↑ G ( z ) } . According to the general theory in He and Zhou (2011), problems (C.1) and (C.2) sharethe same optimal value and are equivalent in terms of the existence and uniqueness of theoptimal solution. Furthermore, if G ∗ ( · ) is optimal to (C.2), then G ∗ (1 − F ρ ( ρ )) is optimal to(C.1). Therefore, we only need to solve problem (C.2).Note that the objective function of (C.2) is linear in the decision variable G ( · ). Therefore,we only need to optimize over the extremal points of the set of feasible quantiles to problem(C.2). Jin and Zhou (2008, Proposition D.3) show that the extremal points are contained in63he set of binary quantile functions S := n G ( · ) | G ( z ) = a + k z ∈ [ c, for some c ∈ (0 ,
1) and a, k ≥ Z F − ρ (1 − z ) G ( z ) dz ≤ ˜ x o . Thus, we only need to consider Max G ( · ) ∈ S R [0 , G ( z ) m ( dz ) . (C.3)Indeed, we can show that problems (C.2) and (C.3) have the same optimal value and,consequently, if G ∗ ( · ) is optimal to problem (C.3), then it is also optimal to (C.2).Denote f ( a, k, c ) as the objective function of problem (C.3) with G ( z ) = a + k z ∈ [ c, .Straightforward calculation shows that f ( a, k, c ) = a + km ([ c, a + km ([ c, aδ + k R c F − ρ (1 − z ) dz ≤ ˜ x, a, k ≥ , c ∈ (0 , . (C.4)It is obvious that the optimal k must bind the first constraint in the above. Consequently,we only need to maximize ˜ f ( a, c ) := ζ ( c )˜ x + [1 − δζ ( c )] a over c ∈ (0 ,
1) and a ∈ [0 , ˜ x/δ ], where ζ ( c ) := m ([ c, R c F − ρ (1 − z ) dz . One can see thatmax a ∈ [0 , ˜ x/δ ] ˜ f ( a, c ) = ζ ( c )˜ x + max(1 − δζ ( c ) ,
0) ˜ xδ = max(1 /δ, ζ ( c ))˜ x. Therefore, we only need to maximize max(1 /δ, ζ ( c )) over c ∈ (0 , γ ∗ := sup c ∈ (0 , ζ ( c ) . c ∈ (0 , max(1 /δ, ζ ( c )) = max(1 /δ, γ ∗ ). Theorem 3
The optimal value of problem (C.1) is max(1 /δ, γ ∗ )˜ x . Furthermore,(i) If γ ∗ = + ∞ , there exist β n → ρ such that ˜ X n := k n ρ ≤ β n with k n := ˜ x/ E [ ρ ρ ≤ β n ] isfeasible and approaches the infinite optimal value when n goes to infinity.(ii) If γ ∗ ≤ /δ , ˜ X ∗ ≡ ˜ x/δ is optimal to problem (C.1) .(iii) If /δ < γ ∗ = sup c ∈ (0 , ζ ( c ) < + ∞ and the supremum is attained at c ∗ ∈ (0 , , then ˜ X ∗ := k ∗ ρ ≤ β ∗ with β ∗ := F − ρ (1 − c ∗ ) and k ∗ := ˜ x E [ ρ ρ ≤ β ∗ ] is optimal to problem (C.1) .(iv) If /δ < γ ∗ = sup c ∈ (0 , ζ ( c ) < + ∞ and the supremum is not attainable, then theoptimal solution to problem (C.1) does not exist. Furthermore, for any β n := F − ρ (1 − c n ) with ζ ( c n ) → γ ∗ as n → ∞ , ˜ X n := k n ρ ≤ β n with k n := ˜ x/ E [ ρ ρ ≤ β n ] is feasible andapproaches the optimal value when n goes to infinity.Proof Problem (C.4) has optimal value max(1 /δ, γ ∗ )˜ x , so does problem (C.1). Consequently,in case (i), the optimal value of problem (C.1) is infinite. Furthermore, there exist c n suchthat ζ ( c n ) → γ ∗ = + ∞ . Because m ([ c, ≤ c ∈ [0 ,
1] and R c F − ρ (1 − z ) dz iscontinuous, positive, and strictly decreasing in c ∈ [0 , ζ ( c ) is bounded in [0 , − δ ] for any δ >
0. Consequently, we must have c n →
1. Define β n := F − ρ (1 − c n ) and ˜ X n := k n ρ ≤ β n .It is straightforward to see that ˜ X n is feasible and approaches the infinite optimal value.In case (ii), the optimal value of problem (C.1) is ˜ x/δ . It is obvious that ˜ X ∗ = ˜ x/δ achieves this optimal value.In case (iii), the optimizer c ∗ of ζ ( · ), a ∗ := 0, and k ∗ := ˜ x R c ∗ F − ρ (1 − z ) dz are optimal toproblem (C.4). As a result, G ∗ ( z ) = a ∗ + k ∗ z ∈ [ c ∗ , is optimal to problem (C.2), so ˜ X ∗ = G ∗ (1 − F ρ ( ρ )) is optimal to problem (C.1).In case (iv), the optimal value of problem (C.2) is ˜ xγ ∗ . We show that the optimalsolution to problem (C.2) does not exist. Suppose G ( · ) is optimal to problem (C.2). Then, G ( · ) cannot be a constant on (0 , G ( z ) = a, z ∈ (0 , G (0) = G (0+) = a and G (1) = G (1 − ) = a . Then, Z [0 , G ( z ) m ( dz ) = a ≤ ˜ x/δ < ˜ xγ ∗ , where the first inequality is the case because G ( · ) must satisfy the budget constraint. Thus,we conclude that G ( · ) cannot be a constant on (0 , ν ( · ) as the measure induced by G ( · ) on (0 , ν ((0 , z ]) := G ( z ) − G (0) forany z ∈ (0 , G ( · ) is nonconstant on (0 , ν ( · ) has nonzero measure on (0 , Z [0 , G ( z ) m ( dz ) = G (0) + Z [0 , Z (0 ,z ] ν ( ds ) m ( dz ) = G (0) + Z (0 , Z [ s, m ( dz ) ν ( ds )= G (0) + Z (0 , m ([ s, ν ( ds ) = G (0) + Z (0 , ζ ( s ) Z s F − ρ (1 − z ) dzν ( ds ) < G (0) + γ ∗ Z (0 , Z s F − ρ (1 − z ) dzν ( ds )= G (0) + γ ∗ (cid:18)Z F − ρ (1 − z ) G ( z ) dz − G (0) Z F − ρ (1 − z ) dz (cid:19) ≤ γ ∗ ˜ x, where the first inequality is the case because ζ ( s ) < γ ∗ , s ∈ (0 ,
1) and ν has nonzero measureon (0 , R F − ρ (1 − z ) G ( z ) dz ≤ ˜ x and γ ∗ > ζ (0) = R F − ρ (1 − z ) dz = 1 /δ . Therefore, G ( · ) cannot attain the optimal value, so theoptimal solution to problem (C.2) does not exist. Consequently, problem (C.1) does notadmit optimal solutions either.Finally, it is straightforward to see that ˜ X n := k n ρ ≤ β n is feasible and approaches theoptimal value as n → + ∞ . (cid:3) We have completely solved problem (C.1) and the solution depends critically on thequantity γ ∗ . When γ ∗ = + ∞ , the optimal value is infinite, so problem (C.1) is ill-posed.This result generalizes Theorem 3.4 in He and Zhou (2011). Furthermore, the asymptoticallyoptimal strategy taken by the agent in this case is X n = k n { ρ ≤ β n } with β n → ρ . In thisstrategy, the probability of having nonzero payoffs is nearly zero, so the strategy is extremelyrisky.When γ ∗ ≤ /δ , the optimal payoff is constant, indicating the strategy of investing allmoney in the risk-free account, an extremely conservative strategy.When γ ∗ = sup c ∈ (0 , ζ ( c ) > /δ and the supremum is not attainable, problem (C.1) is alsoill-posed and the asymptotically optimal strategy is X n = k n { ρ ≤ β n } with β n := F − ρ (1 − c n )and ζ ( c n ) → γ ∗ . If m ([ c, , m admits a density,then ζ ( c ) is continuous in [0 , c n → β n → ρ . Asa result, the probability of X n taking nonzero payoffs is nearly zero, showing that it is anextremely risky strategy. 66inally, when γ ∗ = ζ ( c ∗ ) ∈ (1 /δ, + ∞ ) for some c ∗ ∈ (0 , k ∗ in good market scenarios ( ρ ≤ β ∗ ) or receiving nothing in bad marketscenarios ( ρ > β ∗ ). This result generalizes Theorem 3.7 in He and Zhou (2011) where m isassumed to admit a density and the density is assumed to satisfy a monotonicity condition.Typically, γ ∗ > /δ , so the optimal solution, if exists, must be a digital option. Indeed,lim c ↓ ζ ( c ) = m ([0 , / R F − ρ (1 − z ) dz = 1 /δ , showing that γ ∗ ≥ /δ always holds. Iffurthermore, m has zero measure in a neighbourhood of 0, e.g., in [0 , ǫ ] for some ǫ >
0, then γ ∗ ≥ ζ ( ǫ ) = m ([ ǫ, / R ǫ F − ρ (1 − z ) dz > /δ . Therefore, γ ∗ ≤ /δ only when the agentimposes significant weight on the quantiles of the terminal payoff at levels near zero, i.e.,only when the agent is so conservative that she wants to minimize the maximum potentialloss in the downside.When 1 /δ < γ ∗ = ζ ( c ∗ ) < + ∞ for some c ∗ ∈ (0 , ρ . Note that when the initial wealth ˜ x increases, β ∗ does notchange, so the probability of the digital option being “in the money” at the terminal timedoes not change. On the other hand, the payoff k ∗ of the digital option when it is in themoney increases. This feature is totally different from the optimal payoff of a goal-reachingagent (Browne, 1999). Indeed, the optimal payoff of a goad-reaching agent is also digital.However, the probability of this option being in the money increases with respect to theinitial wealth but the in-the-money payoff does not depend on the initial wealth. Similarobservations are also made in He and Zhou (2011). Corollary 1
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