Consistent Investment of Sophisticated Rank-Dependent Utility Agents in Continuous Time
aa r X i v : . [ q -f i n . M F ] J un Consistent Investment of Sophisticated Rank-DependentUtility Agents in Continuous Time
Ying Hu ∗ Hanqing Jin † Xun Yu Zhou ‡ June 4, 2020
Abstract
We study portfolio selection in a complete continuous-time market where the prefer-ence is dictated by the rank-dependent utility. As such a model is inherently time incon-sistent due to the underlying probability weighting, we study the investment behavior ofsophisticated consistent planners who seek (subgame perfect) intra-personal equilibriumstrategies. We provide sufficient conditions under which an equilibrium strategy is areplicating portfolio of a final wealth. We derive this final wealth profile explicitly, whichturns out to be in the same form as in the classical Merton model with the market price ofrisk process properly scaled by a deterministic function in time. We present this scalingfunction explicitly through the solution to a highly nonlinear and singular ordinary differ-ential equation, whose existence of solutions is established. Finally, we give a necessaryand sufficient condition for the scaling function to be smaller than 1 corresponding to aneffective reduction in risk premium due to probability weighting. ∗ Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France; Email: [email protected]. Par-tially supported by Lebesgue Center of Mathematics Investissementsdavenirprogram-ANR-11-LABX-0020-01,by ANR CAESARS (Grant No. 15-CE05-0024) and by ANR MFG (GrantNo. 16-CE40-0015-01). † Mathematical Institute and Oxford–Nie Financial Big Data Lab, University of Oxford, Oxford OX2 6GG,UK; Email: [email protected]. The research of this author was partially supported by research grants fromOxford–Nie Financial Big Data Lab and Oxford–Man Institute of Quantitative Finance. ‡ Department of IEOR, Columbia University, New York, NY 10027, USA; Email: [email protected] research of this author was supported through start-up grants at both University of Oxford and ColumbiaUniversity, and research funds from Oxford–Nie Financial Big Data Lab, Oxford-Man Institute of QuantitativeFinance, and Nie Center for Intelligent Asset Management. eywords : Rank-dependent utility, probability weighting, portfolio selection, con-tinuous time, time inconsistency, intra-personal equilibrium strategy, market price of risk The classical expected utility theory (EUT) is unable to explain many puzzling phenomenaand paradoxes, such as, just to name a few, the Allais paradox, the co-existence of risk-averse and risk-seeking behavior of a same individual, and the disposition effect in financialinvestment. Rank-dependent utility theory (RDUT), first proposed by Quiggin (1982), hasbeen developed to address some of these puzzles and has thus far been widely considered asone of the most prominent alternative theories on preferences and choices. In addition toa concave outcome utility function as in EUT, RDUT features a probability weighting (ordistortion) function whose slopes give uneven weights on random outcomes when calculatingthe mean. The presence of probability weighting is supported by a large amount of experimentaland empirical studies, many of which find that such a weighting function typically displays an“inverse S-shape” (namely, it is first concave and then convex in its domain); see, e.g. Tverskyand Kahneman (1992), Wu and Gonzalez (1996) and Tanaka et al (2010). This particularshape captures individuals’ tendency to exaggerate the tiny probabilities of both improbablelarge gains (such as winning a lottery) and improbable large losses (such as encountering aplane crash). In particular, Tversky and Kahneman (1992) propose a specific parametric classof inverse S-shaped weighting functions, which will be used as a baseline example to test theassumptions in our paper.In this paper we study a continuous-time financial portfolio selection model in whichan agent pursues the highest rank-dependent utility (RDU) value. In contrast to classicalcontinuous-time portfolio models such as Merton’s (Merton 1969), a dynamic RDU model isintrinsically time inconsistent ; namely, any “optimal” strategy for today will generally not beoptimal for tomorrow. As a result, there is no notion of a dynamically optimal strategy for atime-inconsistent model because any such a strategy, once devised for this moment, will have to be abandoned immediately (and indeed infinitesimally) at the next moment. The time incon-sistency of the RDU model emanates from the probability weighting which weights the randomoutcomes unevenly according to their probabilities of occurrence. Consider as an example a20-period binomial lattice model with equal probabilities of moving up and down at any givenstate. Standing at t = 0 the probability of reaching the top most state (TMS) at t = 10 isextremely small (2 − ). If the agent has an inverse S-shaped probability weighting then shewill greatly inflate this probability. As time goes by and the agent moves along the latticethe probabilities of this same event – eventually reaching the TMS – keep changing, and so dothe degrees of probability weighting. Indeed, once at t = 9 the probability of finally reachingthe TMS is either 1/2 or 0 – which any reasonably intelligent individual is able to tell – andhence there is no probability weighting at all. We hereby see an inconsistency in the strength of probability weighting over time which is the key reason behind the time inconsistency of anRDU model.Time inconsistency changes fundamentally the way we deal with dynamic optimization ingeneral. Optimization is intimately associated with decision making, and finding an optimalsolution is therefore to advise on the best decisions. Now, there is no optimal solutions undertime inconsistency: so what is the purpose of studying a time-inconsistent problem?In his seminal paper, Strotz (1955) describes three types of agents when facing time in-consistency. Type 1, a “spendthrift” (or a naivet´e as in the more recent literature), does notrecognize the time inconsistency and at any given time seeks an optimal solution for that mo-ment only. As a result, his strategies are always myopic and change all the times. The nexttwo types are aware of time inconsistency but act differently. Type 2 is a “precommitter” whosolves the optimization problem only once at time 0 and then commits to the resulting strat-egy throughout, even though she knows that the original solution may no longer be optimalat later times. Type 3 is a “thrift” (or a sophisticated agent) who is unable to precommitand realizes that her future selves will disobey whatever plans she makes now. Her resolutionis to compromise and choose consistent planning in the sense that she optimizes taking thefuture disobedience as a constraint . The Strotzian approach to time inconsistency is, therefore, descriptive , namely to describe what people actually do – the different reactions and behaviorsin front of the inconsistency, as opposed to being normative , namely to tell people what to do.It is both interesting and challenging to formulate mathematical models for each of the threetypes and solve them. It is interesting because these models are very different from the classicalstochastic control based ones and different from each other, and it is challenging because themost powerful tools for tackling dynamic optimization such as dynamic programming and3artingale analysis are based on time consistency and hence fail for time-inconsistent problems.Recently, there is an upsurge of research interest and effort in the fields of stochastic control andmathematical finance/insurance in studying time-inconsistent models, mostly focusing on threedifferent problems: mean–variance portfolio selection, and those involving non-exponentialdiscounting or probability weighting. Earlier works focused on Type 2, precommitted agents(see, e.g., Li and Ng 2000, Zhou and Li 2000, He and Zhou 2011), and later ones graduallyshifted to Type 3, consistent planners or sophisticated agents (Ekeland and Lazrak 2006, Bjorkand Murgoci 2010, Hu et al 2012, Hu et al 2017, Bjork et al 2014).The Type 3 problem can be mathematically formulated as a game in the following way.The sophisticated agent, anticipating the disagreement between her current and future selves,searches for a dynamic strategy that all the future selves have no incentive to deviate from.The resulting strategy is a (subgame perfect) intra-personal equilibrium that will be carriedthrough. In the continuous-time setting, Ekeland and Lazrak (2006) are the first to givea formal definition of such an equilibrium (albeit for a deterministic Ramsey model withnon-exponential discounting), based on a first-order condition of a “spike variation” of theequilibrium strategy.This paper derives the consistent investment strategies of a Type 3 RDU agent in acontinuous-time market in which the asset prices are described by stochastic differential equa-tions (SDEs). We make several contributions, both methodologically and economically. Firstof all, to our best knowledge, this paper is the first to formulate and attack an RDU invest-ment problem in the continuous-time setting. There are substantial difficulties in approachingthe problem. Most notably, in deriving an equilibrium strategy, one needs to analyze the ef-fect of the aforementioned spike variation on the RDU objective functional. In the absenceof probability weighting, there is a well-developed approach at disposal to do this, based oncalculus of variations and sample-path dependence of SDEs on parameters. With probabilityweighting, however, we need to study the distributional dependence of the SDE solutions onparameters, which is actually a largely unexplored topic to our best knowledge. Starting fromscratch, we carry out a delicate analysis to solve the problem thoroughly. The final solutionrequires the existence of solutions to a highly nonlinear, singular ordinary differential equation The same approach is used to derive the stochastic maximum principle for optimal stochastic controls; see,e.g., Yong and Zhou (1999). revised market wherethe risk premium is properly scaled. The scaling factor depends on the original investmentopportunity set and the agent’s probability weighting function, but not her outcome utilityfunction. This suggests that the additional constraint arising from the consistent planningdue to time inconsistency can be transferred to the market opportunity set. This observationcould be a key leading to identifying market equilibria where all the agents are RDU consistentplanners.When the scaling factor is less than 1, the risk premium is reduced and the agent is morerisk averse than her EUT counterpart. In this case the probability weighting and consistentplanning make the agent to take less risky exposure. We present a necessary and sufficientcondition for this to happen.It should be noted that, in order to derive our main results, we make several assumptions onthe model primitives. Some of them may look quite technical especially those on the weightingfunction. However, we do not impose those assumptions for mathematical convenience; insteadwe make sure that they are mathematically mild and economically reasonable. In particular,all the assumptions on the weighting function are satisfied by the baseline function proposedby Tversky and Kahneman (1992).The rest of the paper is organized as follows. In Section 2, we state our problem and definethe intra-personal equilibrium in the same spirit as in our previous work Hu et al. (2012,2017).In Section 3, we present sufficient conditions under which the equilibrium terminal wealthcan be explicitly derived. In Section 4, we examine these sufficient conditions closely on ourmodel primitives. Section 5 is devoted to a concrete example demonstrating the results ofthe paper. In Section 6, we present an equivalent condition for an effectively reduced risk5remium. Finally, Section 7 concludes. In Appendices, we state related general results ona class of singular ODEs, and verify our assumptions on a family of time-varying Tversky–Kahneman’s weighting functions.
We consider a continuous-time market in a finite time horizon [0 , T ], where there are a risk-free asset and n risky assets being traded frictionlessly with price processes S ( · ) and S i ( · ), i = 1 , · · · , n , respectively. The risk-free interest rate, without loss of generality, is set to be r ( · ) ≡
0, or equivalently, S ( t ) ≡ S (0). The dynamics of S i ( · ), i = 1 , · · · , n , of the risky assetsfollow a multi-dimensional geometric Brownian motion dS i ( t ) = S i ( t ) " µ i ( t ) dt + n X j =1 σ i,j ( t ) dW j ( t ) , i = 1 , · · · , n, where µ i ( · ) and σ i,j ( · ) are all deterministic functions of time t , W ( · ) = ( W ( · ) , · · · , W n ( · )) ⊤ isan n -dimensional standard Brownian motion in a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P )with F t = σ ( W s : s ≤ t ) ∨ N ( P ). Here ⊤ denotes the matrix transpose.Denote µ ( t ) = ( µ ( t ) , · · · , µ n ( t )) ⊤ and σ ( t ) = ( σ i,j ( t )) n × n . We assume that σ ( · ) is invertibleand that the market price of risk is θ ( t ) := σ ( t ) − µ ( t ); so the market is arbitrage free andcomplete.A trading strategy is a self-financing portfolio described by an F t -adapted process π ( · ) =( π ( · ) , · · · , π n ( · )) ⊤ , where π i ( t ) is the dollar amount allocated to asset i at time t . Under sucha portfolio π ( · ), the dynamics of the corresponding wealth process X ( · ) evolve according tothe wealth equation dX ( t ) = π ( t ) ⊤ µ ( t ) dt + π ( t ) ⊤ σ ( t ) dW ( t ) . (2.1)For any t ∈ [0 , T ), we say a trading strategy π ( · ) is admissible on [ t, T ] if E R Tt | π ( s ) ⊤ µ ( s ) | ds < + ∞ and E R Tt | π ( s ) ⊤ σ ( s ) | ds < + ∞ . 6 .2 Rank-dependent utility An agent, with an initial endowment x at time t = 0, pursues the highest possible rank-dependent utility (RDU) of her wealth at t = T by dynamically trading in the market. At anygiven time t with the wealth state X ( t ) = x she takes an admissible portfolio π ( · ) on [ t, T ]leading to the terminal wealth X ( T ). The RDU value of this terminal wealth is J ( X ( T ); t, x ) = Z + ∞ w ( t, P t ( u ( X ( T )) > y )) dy + Z −∞ [ w ( t, P t ( u ( X ( T )) > y )) − dy, where w ( t, · ) is the probability weighting applied at time t , u ( · ) is the (outcome) utility function,and P t := P ( ·|F t ) denotes the conditional probability given F t , which includes the information X ( t ) = x . The agent’s original objective is to maximize J ( X ( T ); t, x ) by choosing a properadmissible investment strategy.Here we allow the weighting function w to depend explicitly on time t . This is not just formathematical generality; the time variation of probability weighting is supported empirically(Dierkes 2013, Cui et al 2020) and argued for on a psychology ground (Cui et al 2020). As discussed in Introduction, probability weighting in general causes time-inconsistency. Thismeans an optimal strategy with respect to J ( X ( T ); 0 , X (0)) is not necessarily still optimal withrespect to J ( X ( T ); t, X ( t )) for a future time t >
0, where X ( · ) is the “optimal” wealth processprojected at t = 0 for the objective J ( X ( T ); 0 , X (0)). In this paper, we study the behaviorsof a sophisticated agent who is aware of the time-inconsistency but lacks commitment, andwho instead seeks a consistent investment among all the different selves t ∈ [0 , T ] by findingintra-personal equilibrium strategies.Precisely, given an admissible trading strategy π ( · ) with the corresponding wealth process X ( · ) starting from X (0) = x , a time t ∈ [0 , T ) and a small number ε ∈ (0 , T − t ), wedefine a slightly perturbed strategy which adds $ k on top of π ( · ) over the small time interval[ t, t + ε ) while keeping π ( · ) unchanged outside of this interval. Here k is an F t -measurablerandom vector . This technique is called a spike variation . It follows from the wealth equation(2.1) that the perturbed terminal wealth is X ( T ) + k ⊤ ∆( t, ε ), where ∆( t, ε ) = R t + εt µ ( s ) ds + R t + ǫt σ ( s ) dW ( s ). 7 efinition 2.1 An admissible strategy π ( · ) with the wealth process X ( · ) starting from X (0) = x is called an equilibrium strategy, if for any t ∈ [0 , T ) and any F t -measurable random vector k , one has lim sup ε ↓ J ( X ( T ) + k ⊤ ∆( t, ε ); t, X ( t )) − J ( X ( T ); t, X ( t )) ε ≤ . (2.2)This definition follows most existing works on time-inconsistent optimal controls in continuoustime; see e.g. Ekeland and Lazrak (2006), Bjork and Murgoci (2010), and Hu et al (2012, 2017).Theoretically, an equilibrium strategy is an “infinitesimally” sub-game perfect equilibriumamong all the selves t ∈ [0 , T ). The objectives of this paper are to find conditions under which there exists an equilibriumstrategy, to derive the terminal wealth under such a strategy, and to draw economic implica-tions and interpretations of the results. Since the market is complete, once the desired terminalwealth profile is identified its replicating portfolio is then the corresponding equilibrium strat-egy for consistent investment.
In this subsection we collect all the assumptions needed on the market parameters as well ason the preference functions. These assumptions are henceforth in force, without necessarilybeing mentioned again in all the subsequent statements of results.
Assumption 2.2
The functions µ ( · ) , σ ( · ) and θ ( · ) are all right continuous and uniformlybounded, and θ ( t ) = 0 ∀ t ∈ [0 , T ] . Definition 2.1 is also in line with the original Strotz’s vision of intrapersonal equilibrium (Strotz 1955),adjusted for the continuous-time setting. More specifically, a Strotzian equilibrium strategy stipulates that forany given self t , all the future selves will commit to the strategy, because any deviation from the strategy willmake the deviating self t worse off. In a continuous-time setting, however, any fixed t alone has no influenceon the terminal wealth because it has a measure of zero. Therefore, one considers instead a small “alliance” ofself t : the interval [ t, t + ε ). Definition 2.1 posits that a “deviation-in-alliance” from equilibrium fares worse ina first-order sense. Throughout this paper, by an “increasing” function we mean a “non-decreasing” function, namely f isincreasing if f ( x ) ≥ f ( y ) whenever x > y . We say f is “strictly increasing” if f ( x ) > f ( y ) whenever x > y .Similar conventions are used for “decreasing” and “strictly decreasing” functions. ssumption 2.3 (i) w ( · , · ) : [0 , T ] × [0 , [0 , is measurable. Moreover, for each t ∈ [0 , T ] , w ( t, · ) : [0 , [0 , is strictly increasing, C , and w ( t,
0) = 0 , w ( t,
1) = 1 ∀ t ∈ [0 , T ] . (ii) u ( · ) : R R is strictly increasing, strictly concave, C , and u ( −∞ ) = −∞ , u ′ ( −∞ ) =+ ∞ , u ′ (+ ∞ ) = 0 . These two are very weak assumptions, representing some “minimum requirements” for themodel primitives.Define I ( x ) := ( u ′ ) − ( x ) , x >
0, and l ( x ) := − ln u ′ ( x ), x ∈ R . Assumption 2.4 (i)
There exists α > such that lim sup x → + ∞ x − α − I ′ ( x ) < + ∞ , or equiva-lently, lim sup x →−∞ − u ′′ ( x ) u ′ ( x ) α < + ∞ . (ii) There exist a > , b > such that l ′ ( x ) ≤ e a | l ( x ) | + b , l ′′ ( x ) ≤ e a | l ( x ) | + b ∀ x ∈ R . (iii) u ′′′ ( x ) ≥ ∀ x ∈ R . (iv) For any t ∈ [0 , T ] , there exist c > and m ∈ ( − , , both possibly depending on t , suchthat w ′ p ( t, p ) ≤ c [ p m + (1 − p ) m ] ∀ p ∈ (0 , . Assumption 2.4-(i) and -(ii) are very mild conditions satisfied by most commonly usedutility functions defined on R (e.g. the exponential utility). Assumption 2.4-(iii) implies the risk prudence of the agent, capturing her tendency to take precautions against future risk.Many common utility functions are prudent. Assumption 2.4-(iv) is to control the level ofprobability weighting on very small and very large probabilities. It is satisfied by some well-known weighting functions, e.g. that of Tversky and Kahneman (1992); see Appendix B.
Assumption 2.5
There exist constants ν > , ζ > , such that Z −∞ u ′ ( x ) − ν dx < + ∞ , Z ∞ u ′ ( x ) ν dx < + ∞ , Z + ∞−∞ u ′ ( x ) e − ζx dx < + ∞ . This is also a mild assumption, which is satisfied by, say, the exponential utility function. Through an experiment with a large number of subjects, Noussair et al (2014) observe that the majorityof individuals’ decisions are consistent with prudence. .5 A crucial function The following real-valued function, generated from the weighting function, will play a centralrole throughout this paper: h ( t, x ) := E (cid:2) w ′ p ( t, N ( ξ )) e xξ (cid:3) > , t ∈ [0 , T ] , x ∈ R , where (and henceforth) w ′ p ( t, p ) := ∂∂p w ( t, p ), ξ is a standard normal random variable with N being its probability distribution function. Similarly, we denote by h ′ x ( t, x ) and h ′′ x ( t, x ) thefirst- and second-order partial derivatives of h in x respectively, and in general h ( n ) x ( t, x ) the n -th order partial derivative of h in x . Lemma 2.6 h ( t, x ) < + ∞ ∀ ( t, x ) ∈ [0 , T ] × R . Proof . Fix ( t, x ) ∈ [0 , T ] × R . It follows from Assumption 2.4-(iv) that w ′ p ( t, p ) ≤ c ( p m + (1 − p ) m ) ∀ p ∈ (0 ,
1) for some c > m ∈ ( − , m := m − ∈ ( − , m ), p := ˆ mm > q := pp − >
1. Applying Young’s inequality a m b ≤ a ˆ m p + b q q ∀ a ≥ , b ≥
0, we obtain E [( N ( ξ )) m e ± xξ ] ≤ E (cid:20) p ( N ( ξ )) ˆ m + 1 q e ± xqξ (cid:21) = 1( ˆ m + 1) p + 1 q e ( qx ) / . So h ( t, x ) ≤ c (cid:0) E [ N ( ξ ) m e xξ ] + E [ N ( − ξ ) m e xξ ] (cid:1) = c ( E [ N ( ξ ) m e xξ ] + E [ N ( ξ ) m e − xξ ]) ≤ c (cid:20)
1( ˆ m + 1) p + 1 q e ( qx ) / (cid:21) < + ∞ . The proof is complete. (cid:3)
The following lemma collects some basic properties of h which will be useful in the sequel. Lemma 2.7
For any t ∈ [0 , T ] , h ( t, · ) has the following properties: (i) h ( t,
0) = 1 . (ii) h ( t, x ) is C ∞ in x ≥ , with h ( n ) x ( t, x ) = E (cid:2) w ′ p ( t, N ( ξ )) ξ n e xξ (cid:3) , x ≥ , ∀ n ≥ . (iii) h ( n ) x ( t, x ) is convex in x ≥ for all even n ≥ , and increasing in x ≥ for all odd n ≥ . (iv) ln h ( t, x ) is convex in x ≥ . roof . (i) We have h ( t,
0) = E [ w ′ p ( t, N ( ξ ))] = Z ∞−∞ w ′ p ( t, N ( x )) dN ( x ) = Z w ′ p ( t, p ) dp = w ( t, − w ( t,
0) = 1 . (ii) For ε = 0, write h ( t, x + ε ) − h ( t, x ) ε = E (cid:20) w ′ p ( t, N ( ξ )) ξ Z e ( x + θε ) ξ dθ (cid:21) . For x ≥
0, we have 0 < Z e ( x + θε ) ξ dθ ≤ e ( x + | ε | ) ξ + e −| ε | ξ . Fix a <
0. For any integer n ≥
0, it follows from h ( t, a ) < + ∞ that there exists a constant C n > | ξ n | ≤ C n ( e aξ/ + e ξ ) . Now, for any ε with | ε | < min( − a/ , | w ′ p ( t, N ( ξ )) ξ Z e ( x + θε ) ξ dθ |≤ w ′ p ( t, N ( ξ )) | ξ | (cid:0) e ( x + | ε | ) ξ + e −| ε | ξ (cid:1) ≤ C w ′ p ( t, N ( ξ )) (cid:0) e ( x + | ε | +1) ξ + e (1 −| ε | ) ξ + e ( x + | ε | + a/ ξ + e ( a/ −| ε | ) ξ (cid:1) < C w ′ p ( t, N ( ξ )) (cid:0) e ( x − a/ ξ + e ( x +1) ξ + e (1+ a/ ξ + e ξ + e ( x + a/ ξ + e xξ + e aξ + e aξ/ (cid:1) , where to deduce the last inequality we have repeatedly used the fact that e y < e y + e y forany y > y and y ∈ [ y , y ]. From the assumption of the lemma, it follows that the last termof the above is a random variable with a finite mean; hence Lebesgue’s dominated convergencetheorem yields h ′ x ( t, x ) = E [ w ′ p ( t, N ( ξ )) ξe xξ ] . Similarly, we can derive the desired expressions of higher-order derivatives.(iii) The result is straightforward by (ii).(iv) It follows from (ii) and the Cauchy–Schwarz inequality that, for x ≥ h ′ x ( t, x ) = E (cid:2) w ′ p ( t, N ( ξ )) ξe xξ (cid:3) = E hq w ′ p ( t, N ( ξ )) ξe xξ/ · q w ′ p ( t, N ( ξ )) e xξ/ i ≤ q E (cid:2) w ′ p ( t, N ( ξ )) ξ e xξ (cid:3) · q E (cid:2) w ′ p ( t, N ( ξ )) e xξ (cid:3) = p h ′′ x ( t, x ) p h ( t, x ) . ∂ ∂x ln h ( t, x ) = h ′′ x ( t, x ) h ( t, x ) − ( h ′ x ( t, x )) h ( t, x ) ≥ , establishing the desired convexity. (cid:3) Our approach to deriving the equilibrium strategies is inspired by an
Ansatz we now make. Ifthe agent was an expected utility maximizer, then her optimal strategy would be to dynamicallyreplicate the terminal wealth I ( κρ ( T )) where ρ ( · ) is the state-price density process defined as ρ ( t ) := exp (cid:18) − Z t | θ ( s ) | ds − Z t θ ( s ) ⊤ dW ( s ) (cid:19) , (3.1)and κ is the Lagrange multiplier for the budget constraint. We conjecture that in the currentRDU setting the terminal wealth from an equilibrium strategy is still of the form I ( κ ¯ ρ ( T ))with a revised state-price density process determined by multiplying the market price of riskfunction θ ( · ) by a scaling function λ ( · ):¯ ρ ( t ) = exp (cid:18) − Z t | λ ( s ) θ ( s ) | ds − Z t λ ( s ) θ ( s ) ⊤ dW ( s ) (cid:19) . (3.2)Since now we have conjectured a specific form of the desired terminal wealth profile X ( T ),we will be able to calculate its RDU value along with that of a slightly perturbed wealthprocess in the spirit of Definition 2.1. Then, the equilibrium condition (2.2) will lead to anequation that can be used to identify λ ( · ) as well as to other conditions.It turns out that the equation to derive λ ( · ) is an ODE, explicitly expressed in the followingform: Λ ′ ( t ) = −| θ ( t ) | (cid:18) h ( t, √ Λ( t )) h ′ x ( t, √ Λ( t )) (cid:19) Λ( t ) , t ∈ [0 , T ) , Λ( T ) = 0 . (3.3)This is a highly nonlinear ODE that is singular at T . The existence of its positive solutionswill be established in Subsection 4.2.The scaling function λ ( · ) in determining (3.2) is then given by λ ( t ) := p − Λ ′ ( t ) / | θ ( t ) | > , t ∈ [0 , T ) , (3.4)where Λ( · ) is a positive solution of (3.3). 12 .2 Terminal wealth of an equilibrium strategy The following result gives a complete solution to our problem by presenting the explicit terminalwealth profile of an equilibrium strategy.
Theorem 3.1
Assume that equation (3.3) admits a solution Λ( · ) ∈ C [0 , T ] ∩ C [0 , T ) with Λ( t ) > ∀ t ∈ [0 , T ) , and that the following inequality holds for any c ∈ R : Z + ∞−∞ w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! (cid:18) g ′′ ( x ) + c − g ( x )Λ( t ) g ′ ( x ) (cid:19) du ( x ) ≥ , a.e.t ∈ [0 , T ) . (3.5) Moreover, assume there is κ > such that the following holds E [ ρ ( T ) I ( κ ¯ ρ ( T ))] = x (3.6) where x > is the initial endowment of the agent at t = 0 . Then the portfolio replicating theterminal wealth X ( T ) := I ( κ ¯ ρ ( T )) (3.7) where ¯ ρ ( T ) is determined through (3.2) – (3.4), is an equilibrium strategy.Proof . Denote E s,t := Z ts λ ( v ) θ ( v ) ⊤ dW ( v ) ∀ ≤ s ≤ t ≤ T, f ( x ) := I ( κe − Λ(0) e − x ) =: g − ( x ) , v ( x ) := u − ( x ) , x ∈ R . It is easy to see that, for any s ∈ [0 , T ) and conditional on F s , E s,t is normal, i.e., E s,T |F s ∼ N (0 , Λ( s )), and (3.7) can be rewritten as X ( T ) = f ( E ,T ).Let π ( · ) be the replicating strategy of X ( T ), which exists by the market completeness.Moreover, the budget constraint (3.6) ensures that π ( · ) is an admissible portfolio starting fromthe initial wealth x . The goal is to prove that π ( · ) is an equilibrium strategy.Fix t ∈ [0 , T ). Consider the perturbed strategy described in Subsection 2.3 with theperturbed final wealth X ( T ) + k ⊤ ∆( t, ε ), where ε ∈ (0 , T − t ). To compute the RDU value ofthis perturbed strategy, we first calculate, for any y ∈ R : P t (cid:0) u ( X ( T ) + k ⊤ ∆( t, ε )) > y (cid:1) = P t (cid:0) X ( T ) > v ( y ) − k ⊤ ∆( t, ε ) (cid:1) = P t (cid:0) E t + ε,T > g ( v ( y ) − k ⊤ ∆( t, ε )) − E ,t + ε (cid:1) = E t (cid:2) P t + ε (cid:0) E t + ε,T > g ( v ( y ) − k ⊤ ∆( t, ε ) (cid:1) − E ,t + ε ) (cid:3) = E t " N E ,t + ε − g ( v ( y ) − k ⊤ ∆( t, ε )) p Λ( t + ε ) ! , P t = P ( ·|F t ) and E t = E [ ·|F t ].Denote m ( s, y ) = v ( y ) − k ⊤ ∆( t, s ) and Y ( s, y ) = E ,t + s − g ( m ( s,y )) √ Λ( t + s ) , s ∈ [0 , ε ), y ∈ R . ApplyingItˆo’s formula and noting that Λ ′ ( t + s ) = − λ ( t + s ) | θ ( t + s ) | , we derive dY ( s ) = " Y ( s )2Λ( t + s ) λ ( t + s ) | θ ( t + s ) | + g ′ ( m ( s )) k ⊤ µ ( t + s ) p Λ( t + s ) − g ′′ ( m ( s )) | k ⊤ σ ( t + s ) | p Λ( t + s ) ds + λ ( t + s ) θ ( t + s ) ⊤ + g ′ ( m ( s )) k ⊤ σ ( t + s ) p Λ( t + s ) dW ( t + s ) , s ∈ [0 , ε ) . Applying Itˆo’s formula again yields dN ( Y ( s )) = N ′ ( Y ( s )) dY ( s ) + 12Λ( t + s ) N ′′ ( Y ( s )) | λ ( t + s ) θ ( t + s ) ⊤ + g ′ ( m ( s )) k ⊤ σ ( t + s ) | ds = N ′ ( Y ( s )) dY ( s ) − t + s ) N ′ ( Y ( s )) Y ( s ) | λ ( t + s ) θ ( t + s ) ⊤ + g ′ ( m ( s )) k ⊤ σ ( t + s ) | ds = A ( s ) ds + B ( s ) dW ( t + s ) , s ∈ [0 , ε ) , where A ( s ) := N ′ ( Y ( s )) √ Λ( t + s ) h g ′ ( m ( s )) k ⊤ µ ( t + s ) − g ′′ ( m ( s ))2 | σ ( t + s ) ⊤ k | − Y ( s ) g ′ ( m ( s )) √ Λ( t + s ) | σ ( t + s ) ⊤ k | − Y ( s ) g ′ ( m ( s )) √ Λ( t + s ) k ⊤ σ ( t + s ) λ ( t + s ) θ ( t + s ) (cid:21) = N ′ ( Y ( s )) √ Λ( t + s ) (cid:20) − (cid:18) g ′′ ( m ( s )) + g ′ ( m ( s )) Y ( s ) √ Λ( t + s ) (cid:19) | σ ( t + s ) ⊤ k | + g ′ ( m ( s )) (cid:18) − Y ( s ) λ ( t + s ) √ Λ( t + s ) (cid:19) θ ( t + s ) ⊤ σ ( t + s ) ⊤ k (cid:21) ,B ( s ) := N ′ ( Y ( s )) √ Λ( t + s ) (cid:2) λ ( t + s ) θ ( t + s ) ⊤ + g ′ ( m ( s )) k ⊤ σ ( t + s ) (cid:3) . (3.8)Integrating from s = 0 to s = δ , where δ ∈ (0 , ε ], and then taking conditional expectations onthe above, we obtain E t [ N ( Y ( δ ))] = N ( Y (0)) + Z δ E t [ A ( s )] ds, (3.9)where we have used the fact that R · B ( s ) dW ( t + s ) is a martingale on [0 , ε ] for sufficientlysmall ε > E t Z δ B ( s ) dW ( t + s ) = 0 , δ ∈ (0 , ε ] . (3.10) To save space, we will omit to write out the dependence of m and Y in y from this point of the proof,except when it is important to spell out this dependence. However, the reader is urged to bear in mind thisdependence while reading the proof.
14 proof of this martingality will be delayed to Subsection 3.3. Now, we havelim sup ǫ ↓ w (cid:0) t, P t ( u ( X ( T ) + k ⊤ ∆( t, ε )) > y ) (cid:1) − w ( t, P t ( u ( X ( T )) > y )) ε = lim sup ε ↓ w ( t, E t [ N ( Y ( ε ))]) − w ( t, N ( Y (0))) ε = lim sup ε ↓ w (cid:0) t, N ( Y (0)) + R ε E t [ A ( s )] ds (cid:1) − w ( t, N ( Y (0))) ε = w ′ p ( t, N ( Y (0))) · lim sup ε ↓ ε Z ε E t [ A ( s )] ds = w ′ p ( t, N ( Y (0))) A (0) . Hence (from this point on we will write back the variable y ) lim sup ε ↓ J ( X ( T )+ k ⊤ ∆( t,ε ); t,X ( t )) − J ( X ( T ); t,X ( t )) ε = lim sup ε ↓ R + ∞−∞ w ( t, E t [ N ( Y ( ε,y ))]) − w ( t,N ( Y (0 ,y ))) ε dy ≤ R + ∞−∞ lim sup ε ↓ w ( t, E t [ N ( Y ( ε,y ))]) − w ( t,N ( Y (0 ,y ))) ε dy = R + ∞−∞ w ′ p ( t, N ( Y (0 , y ))) A (0 , y ) dy = −| σ ( t ) ⊤ k | √ Λ( t ) R + ∞−∞ w ′ p ( t, N ( Y (0 , y ))) N ′ ( Y (0 , y )) (cid:18) g ′′ ( m (0 , y )) + g ′ ( m (0 , y )) Y (0 ,y ) √ Λ( t ) (cid:19) dy + θ ( t ) ⊤ σ ( t ) ⊤ k √ Λ( t ) R + ∞−∞ w ′ p ( t, N ( Y (0 , y ))) N ′ ( Y (0 , y )) g ′ ( m (0 , y )) (cid:18) − Y (0 ,y ) √ Λ( t ) λ ( t ) (cid:19) dy. (3.11) In the above, the first inequalitylim sup ε ↓ R + ∞−∞ w ( t, E t [ N ( Y ( ε,y ))]) − w ( t,N ( Y (0 ,y ))) ε dy ≤ R + ∞−∞ lim sup ε ↓ w ( t, E t [ N ( Y ( ε,y ))]) − w ( t,N ( Y (0 ,y ))) ε dy (3.12)will be proved in Subsection 3.4.By Definition 2.2, the underlying strategy is an equilibrium if the right hand side of (3.11)is non-positive for any k ∈ R n . However, the right hand side, being quadratic in σ ( t ) ⊤ k while σ ( t ) is invertible, is non-positive for any k if and only if (noting θ ( t ) = 0) Z + ∞−∞ w ′ p ( t, N ( Y (0 , y ))) N ′ ( Y (0 , y )) g ′ ( m (0 , y )) − Y (0 , y ) p Λ( t ) λ ( t ) ! dy = 0 , (3.13)and Z + ∞−∞ w ′ p ( t, N ( Y (0 , y ))) N ′ ( Y (0 , y )) g ′′ ( m (0 , y )) + g ′ ( m (0 , y )) Y (0 , y ) p Λ( t ) ! dy ≥ . (3.14) Recall that t is fixed; so any F t -measurable random vector k is almost surely deterministic conditional on F t . m (0 , y ) = v ( y ) and Y (0 , y ) = E ,t − g ( v ( y )) √ Λ( t ) . By changing variables x = v ( y ) and z = E ,t − g ( x ) √ Λ( t ) , the equation (3.13) becomes0 = Z + ∞−∞ w ′ p t, N E ,t − g ( x ) p Λ( t ) !! N ′ E ,t − g ( x ) p Λ( t ) ! g ′ ( x ) (cid:18) − E ,t − g ( x )Λ( t ) λ ( t ) (cid:19) u ′ ( x ) dx = − Z + ∞−∞ w ′ p t, N E ,t − g ( x ) p Λ( t ) !! N ′ E ,t − g ( x ) p Λ( t ) ! (cid:18) − E ,t − g ( x )Λ( t ) λ ( t ) (cid:19) du ′ ( x )= − Z + ∞−∞ w ′ p ( t, N ( z )) N ′ ( z ) − z p Λ( t ) λ ( t ) ! de √ Λ( t ) z −E ,t = − e −E ,t Z + ∞−∞ w ′ p ( t, N ( z )) N ′ ( z )( p Λ( t ) − zλ ( t )) e √ Λ( t ) z dz = − e −E ,t E h w ′ p ( t, N ( ξ )) (cid:16)p Λ( t ) − ξλ ( t ) (cid:17) e √ Λ( t ) ξ i , where ξ ∼ N (0 , p Λ( t ) E h w ′ p ( t, N ( ξ )) e √ Λ( t ) ξ i = λ ( t ) E h w ′ p ( t, N ( ξ )) ξe √ Λ( t ) ξ i , or λ ( t ) = h ( t, p Λ( t )) h ′ ( t, p Λ( t )) p Λ( t ) , which holds true by the facts that Λ solves equation (3.3) and Λ ′ ( t ) = − λ ( t ) | θ ( t ) | .Similarly, we can rewrite the inequality (3.14) as Z + ∞−∞ w ′ p t, N E ,t − g ( x ) p Λ( t ) !! N ′ E ,t − g ( x ) p Λ( t ) ! (cid:18) g ′′ ( x ) + g ′ ( x ) E ,t − g ( x )Λ( t ) (cid:19) du ( x ) ≥ , (3.15)which is satisfied under (3.5). (cid:3) In the above proof, there are two technical results, the martingale condition (3.10) and theinequality (3.12), left unproved. We provide proofs in the next two subsections.
Proposition 3.2
The martingale condition (3.10) holds for sufficiently small ε > .Proof . As before the variable y is suppressed. It suffices to prove that B ( s ) is locally squareintegrable at s = 0+. Note that λ ( · ) , θ ( · ) , σ ( · ) are all bounded, and √ Λ( t + s ) is locally bounded16t s = 0+. It thus follows from the expression of B ( · ) that we only need to estimate a boundof N ′ ( Y ( s )) g ′ ( m ( s )).Because lim sup y → + ∞ y − α − I ′ ( y ) < + ∞ (Assumption 2.4-(i)) and − I ′ ( y ) ≡ − u ′′ ( I ( y )) is increas-ing in y (Assumption 2.4-(iii)), we have that there exists K > − I ′ ( y ) ≤ K ( y α + 1) ∀ y > . (3.16)Recalling that m ( s ) = f ( E ,t + s − p Λ( t + s ) Y ( s )) and f ′ ( x ) = − ˜ κI ′ (˜ κe − x ) e − x where ˜ κ := κe − Λ(0) , we have g ′ ( m ( s )) = g ′ ( f ( E ,t + s − p Λ( t + s ) Y ( s )))= 1 f ′ ( E ,t + s − p Λ( t + s ) Y ( s ))= 1 − ˜ κI ′ (˜ κe √ Λ( t + s ) Y ( s ) −E ,t + s ) e E ,t + s − √ Λ( t + s ) Y ( s ) ≤ K ˜ κ (cid:16) ˜ κ α e ( α − √ Λ( t + s ) Y ( s ) − ( α − E ,t + s + e E ,t + s − √ Λ( t + s ) Y ( s ) (cid:17) , where the last inequality is due to (3.16). However, N ′ ( x ) e γx is bounded in x ∈ R for anygiven γ ∈ R ; hence 0 < N ′ ( Y ( s )) g ′ ( m ( s )) ≤ c e − ( α − E ,t + s + c e E ,t + s . The right hand side above is locally square integrable as a process in s . The proof is complete. (cid:3) We now prove the inequality (3.12). In view of Fatou’s lemma, it suffices to show that the inte-grand on its left hand side is dominated by an integrable function. Throughout this subsection,we keep t ∈ [0 , T ) fixed. Lemma 3.3
For any constant γ ∈ (0 , and ε ∈ (0 , T − t ) , there exists a constant c ( γ ) > such that | A ( δ, y ) | N ′ ( Y ( δ, y )) − γ = | A ( δ, y ) | N ′ ( − Y ( δ, y )) − γ ≤ c ( γ ) e γ t + δ ) E ,t + δ ∀ δ ∈ (0 , ε ] , y ∈ R . Proof . As before, we drop the y variable to save space in this proof. Since g ( x ) = l ( x ) + ln κ − Λ(0), Assumption 2.4-(ii) is satisfied with l ( · ) replaced by g ( · ) (and the constants a and b | A ( δ ) | N ′ ( Y ( δ )) − γ ≤ c ( γ ) N ′ ( Y ( δ )) γ ( e a | g ( m ( δ )) | + b + | Y ( δ ) | e a | g ( m ( δ )) | +2 b )= c ( γ ) √ π e − γ Y ( δ ) ( e a | g ( m ( δ )) | + b + | Y ( δ ) | e a | g ( m ( δ )) | +2 b ) ≤ c ( γ ) e − γ Y ( δ ) e a | g ( m ( δ )) | +2 b = c ( γ ) e − γ t + δ ) ( E ,t + δ − g ( m ( δ ))) e a | g ( m ( δ )) | +2 b , where c ( γ ) > c ( γ ) > x + y ) ≥ x − y , we deduce | A ( δ ) | N ′ ( Y ( δ )) − γ ≤ c ( γ ) e − γ t + δ ) ( E ,t + δ − g ( m ( δ ))) e a | g ( m ( δ )) | +2 b ≤ c ( γ ) e γ t + δ ) E ,t + δ e − γ t + δ ) g ( m ( t + δ )) +2 a | g ( m ( t + δ )) | +2 b ≤ c ( γ ) e n ( γ ) e γ t + δ ) E ,t + δ , (3.17)where n ( γ ) := max x ∈ R {− γ t + δ x + 2 a | x | + 2 b } < + ∞ . This completes the proof. (cid:3) Lemma 3.4
For any γ ∈ (0 , , there exists ε ∈ (0 , T − t ) and a constant M ( γ ) < + ∞ suchthat E t | A ( δ, y ) | [ E t N ( Y ( δ, y ))] γ ≤ M ( γ ) ∀ δ ∈ (0 , ǫ ] , y ∈ R . Proof . Again we omit to write out y . Denote m = 1 /γ and n = mm − . Then Cauchy–Schwarz’sinequality yields E t | A ( δ ) | [ E t N ( Y ( δ ))] γ ≡ E t | A ( δ ) | [ E t N ( Y ( δ ))] /m ≤ (cid:0) E t (cid:2) | A ( δ ) | n N ( Y ( δ )) − n/m (cid:3)(cid:1) /n . Write | A ( δ ) | n N ( Y ( δ )) − n/m = (cid:18) N ′ ( Y ( δ )) N ( Y ( δ )) (cid:19) n/m (cid:18) | A ( δ ) | N ′ ( Y ( δ )) /m (cid:19) n = (cid:18) N ′ ( Y ( δ )) N ( Y ( δ )) (cid:19) n/m (cid:18) | A ( δ ) | N ′ ( Y ( δ )) ( γ +1) / (cid:19) n N ′ ( Y ( δ )) n (1 − γ ) / . Noting N ′ ( x ) N ( x ) ≤ C ( | x | + 1) ∀ x ∈ R for some C >
0, we conclude that the last term aboveconverges to 0 faster than the first term going to + ∞ when Y ( δ ) goes to + ∞ ; hence we can18nd a bound M ( γ ) > | A ( δ ) | n N ( Y ( δ )) − n/m ≤ M ( γ ) (cid:18) | A ( δ ) | N ′ ( Y ( δ )) ( γ +1) / (cid:19) n ≤ M ( γ ) c ((1 − γ ) / n e t + δ ) E ,t + δ , where the last inequality is due to Lemma 3.3. When ε > M ( γ ) < ∞ such that E t [ e t + δ ) E ,t + δ ] < M ( γ ) for any δ ∈ (0 , ǫ ]. This leads to the desiredinequality. (cid:3) Proposition 3.5
We have the following conclusions: (i)
For any γ > , there are a sufficiently small ε ∈ (0 , T − t ) and a function H ( · ; γ ) with R y ( γ ) −∞ H ( y ; γ ) dy < + ∞ for some y ( γ ) < , such that when δ ∈ (0 , ε ) , we have ( E t | A ( δ ) | ) γ ≤ H ( y ; γ ) ∀ y ∈ ( −∞ , y ( γ )] . (ii) For any γ > , there are a sufficiently small ε ∈ (0 , T − t ) and a function H ( · ; γ ) with R + ∞ ¯ y ( γ ) H ( y ; γ ) dy < + ∞ for some ¯ y ( γ ) > , such that when δ ∈ (0 , ε ) , we have ( E t | A ( δ ) | ) γ ≤ H ( y ; γ ) ∀ y ∈ [¯ y ( γ ) , + ∞ ) . Proof . We prove only (i), while (ii) being similar. Applying γ = 1 to the second inequality of(3.17), we get | A ( δ ) | ≤ c (1) e t + δ ) E ,t + δ e − t + δ ) g ( m ( δ )) +2 a | g ( m ( δ )) | +2 b . Define n := max x ∈ R {− t + δ ) x + 2 a | x | + 2 b } < + ∞ . Then | A ( δ ) | ≤ c (1) e n e t + δ ) E ,t + δ e − t + δ ) g ( m ( δ )) . So E t | A ( δ ) | ≤ c (1) e n E t [ e t + δ ) E ,t + δ e − t + δ ) g ( m ( δ )) ] ≤ c (1) e n q E t e t + δ ) E ,t + δ q E t e − t + δ ) g ( m ( δ )) . Set n := c (1) e n E t [ e t + δ E ,t + δ ] which is a finite constant when ε (and hence δ ) issufficiently small, and ξ := k ⊤ ∆( t, δ ) ∼ N (0 , η ) conditional on F t , where η ≡ η ( δ ) =19ar( ξ ) ≤ | k | R t + δt | σ ( s ) | ds . Then( E t | A ( δ ) | ) ≤ n E t [ e − t + δ ) g ( m ( δ )) ]= n E t [ e − t + δ ) ln u ′ ( v ( y ) − ξ ) ]= n √ π Z + ∞−∞ e − t + δ ) ln u ′ ( v ( y )+ ηz ) e − z / dz = n √ π Z + ∞−∞ e − t + δ ) ln u ′ ( η ˜ z ) e − (˜ z − v ( y ) /η ) / d ˜ z = n √ π "Z v ( y ) / (2 η ) −∞ + Z v ( y ) / (2 η ) + Z + ∞ e − t + δ ) ln u ′ ( η ˜ z ) e − (˜ z − v ( y ) /η ) / d ˜ z. We now find an integrable bound (as a function of y ) for each of the three integrals in theabove.For the first integral, take y < v ( y ) < u ′ ( v ( y ) / > v ( y ) < u ′ ( v ( y ) / ≥ u ′ ( v ( y ) / > ∀ y ≤ y . Thus Z v ( y ) / (2 η ) −∞ e − t + δ ) ln u ′ ( η ˜ z ) e − (˜ z − v ( y ) /η ) / d ˜ z < √ πe − t + δ ) ln u ′ ( v ( y ) / ≤ c ( a ) u ′ ( v ( y ) / − a , y ≤ y , for any a > c ( a ) < + ∞ . For the second integral, we have Z v ( y ) / (2 η ) e − t + δ ) ln u ′ ( η ˜ z ) e − (˜ z − v ( y ) /η ) / d ˜ z ≤ e − v ( y ) / (8 η ) | v ( y ) / (2 η ) | ≤ c e − v ( y ) / (9 η ) for some constant c > Z + ∞ e − t + δ ) ln u ′ ( η ˜ z ) e − (˜ z − v ( y ) /η ) / d ˜ z = Z + ∞ e − t + δ ) ln u ′ ( η ˜ z ) e − ˜ z / e ˜ zv ( y ) /η e − v ( y ) / (2 η ) d ˜ z< e − v ( y ) / (2 η ) Z + ∞ e − ˜ z / d ˜ z = r π e − v ( y ) / (2 η ) . Combining the three integrals, we conclude that, for any a > , a >
0, when ε ∈ (0 , T − t )is small enough and y < v ( y ) sufficiently negative, there exists constant c > δ ∈ (0 , ε ),( E t | A ( δ ) | ) ≤ c [ u ′ ( v ( y )) − a + e − a v ( y ) ] := H ( y ; γ ) /γ ∀ y ≤ y . This is based on the fact (which can be easily shown) that given α >
0, for any a > c ( a ) > e − α ln x ≤ c ( a ) x − a ∀ x > γ >
0, we can take suitable a and a (which may depend on γ ) such that,in view of Assumption 2.5, R y ( γ ) −∞ H ( y ) dy < + ∞ , where y ( γ ) is sufficiently negative. (cid:3) Theorem 3.6
There exists an integrable function H ( · ) such that when ε > is sufficientlysmall, it holds that (cid:12)(cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ε, y ))]) − w ( t, N ( Y (0 , y ))) ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ H ( y ) ∀ y ∈ R . Proof . Fix ε >
0. For any y ∈ R , by the mean-value theorem, there exists δ ∈ [0 , ǫ ] (whichdepends on t and y ) such that (cid:12)(cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ε, y ))]) − w ( t, N ( Y (0 , y ))) ε (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) w ′ p ( t, E t [ N ( Y ( δ, y )]) E t [ A ( δ )] (cid:12)(cid:12) ≤ w ′ p ( t, E t [ N ( Y ( δ, y )]) E t [ | A ( δ ) | ] , where we have used the fact that, by virtue of (3.9), ddδ E t [ N ( Y ( δ, y ))] = E t [ A ( δ )] . By Assump-tion 3.1-(iv), we have (cid:12)(cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ε, y ))]) − w ( t, N ( Y (0 , y ))) ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ c E t [ | A ( δ ) | ]( E t [ N ( Y ( δ, y )]) − m + c E t [ | A δ | ](1 − E t [ N ( Y ( δ, y )]) − m for some c > m ∈ ( − , E t [ N ( Y ( δ, y )] = P t (cid:0) u ( X ( T ) + k ⊤ ∆( t, δ )) > y (cid:1) . Choose y and y such that E t [ N ( Y ( δ, y )] < / E t [ N ( Y ( − δ, y )] < / δ ∈ (0 , ε ].Using notations in Lemma 3.4 and Proposition 3.5, denote y := y ((1 + m ) / ∧ y , ¯ y := ¯ y ((1 + m ) / ∨ y , and ˆ H ( y ) := 2 cH ( y ; (1 + m ) / M ((2 m ) / ( m − (1 − m ) / , ˆ H ( y ) := 2 cH ( y ; (1 + m ) / M ((2 m ) / ( m − (1 − m ) / . Then:(i) For any y < y , we deduce (cid:12)(cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ε, y ))]) − w ( t, N ( Y (0 , y ))) ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ c E t [ | A ( δ ) | ]( E t [ N ( Y ( δ, y )]) − m ≤ c ( E t [ | A ( δ ) | ]) (1+ m ) / ( E t [ | A ( δ ) | ]) (1 − m ) / ( E t [ N ( Y ( δ, y )]) − m ≤ cH ( y ; (1 + m ) / M ((2 m ) / ( m − (1 − m ) / = ˆ H ( y ) . y > ¯ y , we have (cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ε,y ))]) − w ( t,N ( Y (0 ,y ))) ε (cid:12)(cid:12)(cid:12) ≤ ˆ H ( y ).(iii) Finally, (cid:12)(cid:12)(cid:12) w ( t, E t [ N ( Y ( ǫ,y ))]) − w ( t,N ( Y (0 ,y ))) ǫ (cid:12)(cid:12)(cid:12) ≤ cM ( − m ) when y ∈ [ y, ¯ y ].Then H ( · ), where H ( y ) = ˆ H ( y ) y The following two statements are equivalent: (i) There exists κ > such that (3.6) holds. (ii) There exists ¯ κ > such that E [ ¯ ρ ( T ) I (¯ κ ¯ ρ ( T ))] = x . (4.1) Proof . Let Q and ¯ Q be respectively the equivalent martingale measures corresponding to ρ ( T )and ¯ ρ ( T ); namely, d Q d P = ρ ( T ) , d ¯ Q d P = ¯ ρ ( T ) . By Girsanov’s theorem, ˜ W ( · ) and ¯ W ( · ) are respectively Brownian motions under Q and ¯ Q ,where ˜ W ( t ) := W ( t ) + Z t θ ( s ) ds, ¯ W ( t ) := W ( t ) + Z t λ ( s ) θ ( s ) ds, t ∈ [0 , T ] . E [ ρ ( T ) I ( κ ¯ ρ ( T ))] = E Q h I ( κe − R T | λ ( s ) θ ( s ) | ds − R T λ ( s ) θ ( s ) ⊤ dW ( s ) ) i = E Q h I ( κe − R T | λ ( s ) θ ( s ) | ds + R T λ ( s ) | θ ( s ) | ds − R T λ ( s ) θ ( s ) ⊤ d ˜ W ( s ) ) i = E h I ( κe − R T | λ ( s ) θ ( s ) | ds + R T λ ( s ) | θ ( s ) | ds − R T λ ( s ) θ ( s ) ⊤ dW ( s ) ) i = E ¯ Q h I ( κe − R T | λ ( s ) θ ( s ) | ds + R T λ ( s ) | θ ( s ) | ds − R T λ ( s ) θ ( s ) ⊤ d ¯ W ( s ) ) i = E h ¯ ρ ( T ) I ( κe − R T | λ ( s ) θ ( s ) | ds + R T λ ( s ) | θ ( s ) | ds − R T λ ( s ) θ ( s ) ⊤ d ¯ W ( s ) ) i = E h ¯ ρ ( T ) I (cid:16) κe R T λ ( s )(1 − λ ( s )) | θ ( s ) | ds ¯ ρ ( T ) (cid:17)i . This establishes the desired equivalence with ¯ κ = κe R T λ ( s )(1 − λ ( s )) | θ ( s ) | ds . (cid:3) So the assumption in Theorem 3.1 regarding the existence of a positive constant κ satisfying(3.6) boils down to the familiar condition (4.1). The latter condition is standard in the classicalMerton problem in which the pricing kernel is ¯ ρ ( T ) or, equivalently, the market price of riskprocess is λ ( · ) θ ( · ). Note that because the probability weighting function w has been embeddedinto λ ( · ), the existence of a positive ¯ κ satisfying (4.1) becomes a condition on the utilityfunction u only, which is satisfied by, say, the exponential utility. More important, with Theorem 4.1, Theorem 3.1 shows that the investment behavior of thesophisticated RDU agent is indistinguishable from an EUT maximizer in a market where themarket price of risk is revised from θ ( · ) to λ ( · ) θ ( · ). This finding may have important economicimplications especially in the study of market equilibria. The main result of this paper depends crucially on the existence of a positive solution to theODE (3.3). Note that this equation is highly nonlinear, and singular at t = T in that thedenominator of the right hand side of the equation is 0 at T . In this subsection we provideconditions under which (3.3) admits positive solutions, by applying a general existence resultfor a class of ODEs with singular initial/terminal values (see Appendix A). Indeed, if this condition fails, than it is usually an indication that the original problem is not well-posedand/or an optimal solution is not attainable; see Jin et al (2008) for a detailed analysis on this constraint. y ( t ) = Λ( T − t ), (3.3) is equivalent to y ′ ( t ) = | θ ( T − t ) | (cid:18) h ( T − t, √ y ( t )) h ′ x ( T − t, √ y ( t )) (cid:19) y ( t ) , t ∈ (0 , T ] ,y (0) = 0 . (4.2)In the rest of this subsection we study equation (4.2) instead of (3.3). The key idea is to firstestablish the local existence in the right neighborhood of t = 0, and then extend it globally tothe whole time interval [0 , T ].We introduce the following assumption on the function h ( · , · ) (which depends directly onthe probability weighting function) and on the market represented by θ ( · ): Assumption 4.2 (i) h ′ x ( t, ≥ and h ′′′ x ( t, ≥ ∀ t ∈ [0 , T ] . (ii) lim sup t ↑ T h ′ x ( t, √ | θ ( t ) | ( T − t ) < and lim inf t ↑ T | θ ( t ) | > . (iii) sup t ∈ [0 ,T ] h ( t, < + ∞ and lim sup t ↑ T h ′′ x ( t, < + ∞ . (iv) inf t ∈ [0 ,T ] h ′′ x ( t, > . Assumption 4.2-(iii) and -(iv) are mild. Assumption 4.2-(i) can be relaxed with a moresubtle analysis than the one to be given below; although we will not pursue in that direction.The first part of Assumption 4.2-(ii) is the most important of all, which regulates how theprobability weighting function w ( t, · ) should behave, given the market, when t is sufficientlyclose to the terminal time T . Mathematically, this terminal behavior of weighting functions istranslated into the singularity of the ODE (3.3) at T , which is why Assumption 4.2 is neededfor a proof of the existence of (3.3). Luckily, we will show in Appendix B that all the partsof Assumption 4.2 are satisfied by a family of time-varying Tversky–Kahnamen’s weightingfunctions.We first strengthen Lemma 2.7 under Assumption 4.2-(i). Lemma 4.3 Under Assumption 4.2-(i) in addition to the same assumption of Lemma 2.7, forany t ∈ [0 , T ] , h ( t, · ) has the following properties: (i) h ( t, x ) and h ′′ ( t, x ) are both increasing in x ≥ , and h ′ x ( t, x ) is convex in x ≥ . We believe that this is a distinctive feature of the continuous-time setting. In the discrete-time case, thereis no infinitesimal issue of the weighting functions close to the terminal time. xh ′′ x ( t, ≤ h ′ x ( t, x ) ≤ h ′ x ( t, 0) + xh ′′ x ( t, x ) ∀ x ≥ .Proof . (i) It follows from Lemma 2.7-(iii) that h ′ x ( t, x ) and h (3) x ( t, x ) are both increasing in x ≥ 0. Assumption 4.2-(i) then leads to the desired results readily.(ii) Applying the Taylor expansion, we have for any x > 0, there exists ζ ∈ [0 , x ] such that h ′ x ( t, x ) = h ′ x ( t, 0) + xh ′′ x ( t, 0) + 12 x h (3) x ( t, ζ ) ≥ xh ′′ x ( t, x ) . On the other hand, the convexity of h ′ x ( t, x ) in x ≥ h ′ x ( t, x ) ≤ h ′ x ( t, 0) + xh ′′ x ( t, x ) . The proof is complete. (cid:3) Lemma 4.4 Under Assumption 4.2, there exist k > greater than any given number and δ > such that | θ ( T − t ) | (cid:18) √ th ( T − t, √ k t ) h ′ x ( T − t, √ k t ) (cid:19) < ∀ t ∈ (0 , δ ] . (4.3) Moreover, there exist k > less than any given positive number and δ > such that | θ ( T − t ) | (cid:18) √ th ( T − t, √ k t ) h ′ x ( T − t, √ k t ) (cid:19) > ∀ t ∈ (0 , δ ] . (4.4) Proof . By Lemma 4.3-(i), h ( T − t, x ) is increasing when x ≥ 0; hencelim sup t ↓ h ( T − t, √ kt ) ≤ lim sup t ↓ h ( T − t, < + ∞ , (4.5)where the finiteness is due to Assumption 4.2-(iii). On the other hand, it follows from Lemma4.3-(ii) that, for any k > h ′ x ( T − t, √ kt ) √ t ≥ √ kh ′′ x ( T − t, ≥ √ k inf s ∈ [0 ,T ] h ′′ x ( s, > , (4.6)where the last inequality is due to Assumption 4.2-(iv). Combining (4.5) and (4.6) and notingthe boundedness of θ ( · ), we conclude that there is k = k greater than any given number suchthat (4.3) is satisfied.Next, by Lemma 4.3-(ii), we deduce that for sufficiently small k > t ↓ h ′ x ( T − t, √ kt ) p | θ ( T − t ) | t ≤ lim sup t ↓ ( h ′ x ( T − t, p | θ ( T − t ) | t + √ k h ′′ x ( T − t, √ kt ) p | θ ( T − t ) | ) ≤ lim sup t ↓ h ′ x ( T − t, p | θ ( T − t ) | t + √ k lim sup t ↓ h ′′ x ( T − t, p | θ ( T − t ) | < , h ( T − t, √ kt ) ≥ h ( T − t, 0) = 1, we conclude that there is k = k less than any given positive number so that(4.4) is satisfied. (cid:3) The following establishes the local existence of the ODE (4.2). Proposition 4.5 Under Assumption 4.2, equation (4.2) admits a solution y ( · ) ∈ C [0 , δ ] ∩ C (0 , δ ] on a time interval [0 , δ ] for some δ > , with y ( t ) > ∀ t ∈ (0 , δ ] .Proof . Fix k > < k < k as in Lemma 4.4, and take δ = δ ∧ δ . Then it is easyto check that β ( t ) = k t and α ( t ) = k t satisfy all the requirements in Theorem A.2 on (0 , δ ];hence the result. (cid:3) Next we extend the local solution y ( · ) obtained in Proposition 4.5 to the whole time interval[0 , T ]. Denote y ( δ ) = x > 0. Without loss of generality, we assume x < δ closer to 0 to reduce x ). Proposition 4.6 Under Assumption 4.2-(i) and (iv), for any δ > and < x < , theequation y ′ ( t ) = | θ ( T − t ) | (cid:18) h ( T − t, √ y ( t )) h ′ x ( T − t, √ y ( t )) (cid:19) y ( t ) , t ∈ ( δ, T ] ,y ( δ ) = x (4.7) has a solution y ( · ) ∈ C [ δ, T ] with y ( t ) > ∀ t ∈ [ δ, T ] .Proof . Since ln h ( x, t ) is convex in x ≥ h ′ x ( t, x ) h ( t, x ) ≥ h ′ x ( t, √ x ) h ( t, √ x ) ≥ √ x h ′′ x ( t, h ( t, √ x ) ≥ √ x h ′′ x ( t, h ( t, ≥ c > ∀ t ∈ [0 , T ] , x ≥ √ x , where the second inequality is due to Lemma 4.3-(ii) and c (which may depend on x ) is aconstant arising from Assumption 4.2-(iii) and -(iv). Hence for any y ≥ x , we have | θ ( T − t ) | (cid:18) h ( T − t, √ y ) h ′ x ( T − t, √ y ) (cid:19) y ≤ c y, where c > c and the bound of | θ ( · ) | .Denote c = x e c T < + ∞ , and a truncation function r ( y ) = ( y ∨ x ) ∧ c for y ≥ y ′ ( t ) = f ( t, y ( t )) , t ∈ ( δ, T ] ,y ( δ ) = x (4.8)26here f ( t, y ) := | θ ( T − t ) | h ( T − t, p r ( y )) h ′ x ( T − t, p r ( y )) ! r ( y ) . It is easy to show that f ( · , · ) satisfies the conditions in Theorem A.3 on ( δ, T ]; hence (4.8)admits a solution y ( · ). Moreover, since y ′ ( t ) ≥ y ( t ) ≥ x ∀ t ∈ [ δ, T ].Take t := inf { t ∈ ( δ, T ] : y ( t ) ≥ c } ∧ T . If t < T , then y ( t ) = c and y ( t ) < c ∀ t Theorem 4.7 Under Assumption 4.2, the equation (3.3) admits a solution Λ( · ) ∈ C [0 , T ] ∩ C [0 , T ) satisfying Λ( t ) > ∀ t ∈ [0 , T ) . We now provide conditions on the model primitives under which the inequality (3.5) holds. Weassume that the ODE (3.3) admits a positive solution Λ( · ) ∈ C [0 , T ] ∩ C [0 , T ), and introducethe following additional assumption. Assumption 4.8 For a.e. t ∈ [0 , T ) , w ( t, · ) is either convex or inverse S-shaped. A convex weighting function captures risk aversion in terms of exaggerating the smallprobability of very “bad” events while downplaying very “good” events; see Yaari (1987).On the other hand, as discussed in Introduction, an inverse S-shaped probability weightingfunction is more interesting as it reflects the tendency of inflating the small probabilities of both tails which are consistent with the conclusions of many experimental and empirical works. Lemma 4.9 Under Assumptions 4.2 and 4.8, we have lim | y |→ + ∞ w ′ p ( t, N ( y )) N ′ ( y ) I ′ ( e √ Λ( t ) y − c ) = 0 ∀ c ∈ R , ∀ t ∈ [0 , T ) . (4.9) Proof . It follows from (3.16) that0 ≤ w ′ p ( t, N ( y )) − I ′ ( e √ Λ( t ) y − c ) ≤ Kw ′ p ( t, N ( y ))( e α √ Λ( t ) y − αc + 1) . Z + ∞−∞ w ′ p ( t, N ( y ))( e α √ Λ( t ) y − αc + 1) N ′ ( y ) dy = e − αc h ( t, α p Λ( t )) + 1 < + ∞ , implying lim | y |→ + ∞ w ′ p ( t, N ( y ))( e α √ Λ( t ) y − αc + 1) N ′ ( y ) = 0 . This completes the proof. (cid:3) We now analyze the integral in (3.5) by decomposing it into M + M , where M = Z + ∞−∞ w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! c − g ( x )Λ( t ) g ′ ( x ) du ( x ) , and M = Z + ∞−∞ w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! g ′′ ( x ) du ( x )= Z + ∞−∞ w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′ ( x ) dg ′ ( x ) . Applying integration by parts and noting that N ′′ ( x ) = − xN ′ ( x ), we can further decompose M into − ( M + M + M ) where M = Z + ∞−∞ g ′ ( x ) w ′′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) !! u ′ ( x ) − g ′ ( x ) p Λ( t ) dxM = Z + ∞−∞ g ′ ( x ) w ′ p t, N c − g ( x ) p Λ( t ) !! N ′′ c − g ( x ) p Λ( t ) ! u ′ ( x ) − g ′ ( x ) p Λ( t ) dx = − Z + ∞−∞ g ′ ( x ) w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′ ( x ) c − g ( x ) p Λ( t ) − g ′ ( x ) p Λ( t ) dx = Z + ∞−∞ g ′ ( x ) w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′ ( x ) c − g ( x ) p Λ( t ) dx ≡ M ,M = Z + ∞−∞ g ′ ( x ) w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′′ ( x ) dx, assuming that for any c ∈ R , w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′ ( x ) g ′ ( x ) = 0 when | x | → + ∞ . (4.10)28ince g ′ ( x ) = − u ′′ ( x ) u ′ ( x ) , (4.10) can be written as w ′ p t, N c − g ( x ) p Λ( t ) !! N ′ c − g ( x ) p Λ( t ) ! u ′′ ( x ) = 0 when | x | → ∞ , which is equivalent to (4.9).Now, denoting y = c − g ( x ) √ Λ( t ) , we have M + M = − ( M + M )= − Z + ∞−∞ g ′ ( x ) N ′ ( y ) (cid:18) w ′′ p ( t, N ( y )) N ′ ( y ) u ′ ( x ) d y d x + w ′ p ( t, N ( y ))) u ′′ ( x ) (cid:19) dx = − Z + ∞−∞ g ′ ( x ) N ′ ( y ) d (cid:2) w ′ p ( t, N ( y )) u ′ ( x ) (cid:3) = − Z −∞ + ∞ g ′ ( f ( c − p Λ( t ) y )) N ′ ( y ) d h w ′ p ( t, N ( y )) u ′ ( f ( c − p Λ( t ) y )) i = Z + ∞−∞ " N ′ ( y ) f ′ ( c − p Λ( t ) y ) d h w ′ p ( t, N ( y )) u ′ ( f ( c − p Λ( t ) y )) i . Hence, inequality (3.5) is equivalent to Z + ∞−∞ N ′ ( y ) f ′ ( c − p Λ( t ) y ) d h w ′ p ( t, N ( y )) u ′ ( f ( c − p Λ( t ) y )) i ≥ c ∈ R , a.e. t ∈ [0 , T ).An obvious sufficient condition for (4.11) is that w ′ p ( t, N ( y )) u ′ ( f ( c − p Λ( t ) y )) ≡ w ′ p ( t, N ( y )) e √ Λ( t ) y − c is increasing in y for any c ∈ R , which holds automatically if w ( t, · ) is convex. Theorem 4.10 Under Assumptions 4.2 and 4.8, the inequality (3.5) holds.Proof . We have shown above that (3.5) holds at t when w ( t, · ) is convex. Let us now focus onthe case when w ( t, · ) is inverse S-shaped.First of all, it follows from Theorem 4.7 that the ODE (3.3) admits a positive solutionΛ( · ) ∈ C [0 , T ] ∩ C [0 , T ).Fix t ∈ [0 , T ). Noting f ′ ( x ) = − ˜ κI ′ (˜ κe − x ) e − x where ˜ κ = κe − Λ(0) > 0, we can rewrite29ondition (4.11) as0 ≤ Z + ∞−∞ N ′ ( y ) f ′ ( c − p Λ( t ) y ) d h w ′ p ( t, N ( y )) u ′ ( f ( c − p Λ( t ) y )) i = Z + ∞−∞ N ′ ( y ) − ˜ κI ′ (˜ κe √ Λ( t ) y − c ) e √ Λ( t ) y − c d h w ′ p ( t, N ( y )) e √ Λ( t ) y − c i = Z + ∞−∞ N ′ ( y ) − ˜ κI ′ (˜ κe √ Λ( t ) y − c ) (cid:20) w ′ p ( t, N ( y )) p Λ( t ) + ∂w ′ p ( t, N ( y )) ∂y (cid:21) dy = p Λ( t ) M + M , where M = R + ∞−∞ w ′ p ( t,N ( y )) N ′ ( y ) − ˜ κI ′ (˜ κe √ Λ( t ) y − c ) dy > 0, and M = Z + ∞−∞ ∂w ′ p ( t,N ( y )) ∂y − ˜ κI ′ (˜ κe √ Λ( t ) y − c ) N ′ ( y ) dy = E [ F ( ξ ) G ( ξ )]with F ( y ) = ∂w ′ p ( t,N ( y )) ∂y and G ( y ) = − ˜ κI ′ (˜ κe √ Λ( t ) y − c ) > w ( t, · ) is inverse S-shaped, there exists q ∈ (0 , 1) such that w ′′ p ( t, p ) ≤ , q ), w ′′ p ( t, p ) ≥ q, w ′′ p ( t, q ) = 0. Furthermore, for any b > Z b ∂w ′ p ( t, N ( y )) ∂y N ′ ( y ) dy = Z b N ′ ( y ) d [ w ′ p ( t, N ( y ))]= ( w ′ p ( t, N ( y )) N ′ ( y )) | y = by =0 − Z ba w ′ p ( t, N ( y )) dN ′ ( y )= w ′ p ( t, N ( b )) N ′ ( b ) − √ π w ′ p ( t, 12 ) + Z b w ′ p ( t, N ( y )) ydN ( y ) . Since w ′′ p ( t, p ) ≥ p , R b ∂w ′ p ( t,N ( y )) ∂y N ′ ( y ) dy is increasing in b for suf-ficiently large b . Hence lim b ↑ + ∞ R b ∂w ′ p ( t,N ( y )) ∂y N ′ ( y ) dy exists. However, + ∞ > h ′ x ( t, ≡ E [ w ′ p ( t, N ( ξ )) ξ ], we conclude that lim b ↑ + ∞ R b w ′ t ( N ( y )) ydN ( y ) exists and is finite. This im-plies that a := lim b ↑ + ∞ w ′ p ( t, N ( b )) N ′ ( b ) exists. It then follows from the fact that + ∞ >h ( t, ≡ E [ w ′ p ( t, N ( ξ ))] that a = 0, i.e., Z + ∞ ∂w ′ p ( t, N ( y )) ∂y N ′ ( y ) dy = − √ π w ′ p (cid:18) t, (cid:19) + Z + ∞ w ′ p ( t, N ( y )) ydN ( y ) . Similarly, we can show that Z −∞ ∂w ′ p ( t, N ( y )) ∂y N ′ ( y ) dy = 1 √ π w ′ p (cid:18) t, (cid:19) + Z −∞ w ′ p ( t, N ( y )) ydN ( y ) . 30s a result, E [ F ( ξ )] = Z + ∞−∞ ∂w ′ p ( t, N ( y )) ∂y N ′ ( y ) dy = Z + ∞−∞ w ′ p ( t, N ( y )) ydN ( y ) = h ′ x ( t, ≥ . Denote c = N − ( q ). Then F ( ξ ) is negative on ( −∞ , c ), positive on ( c, + ∞ ), and F ( c ) = 0.Since u ′′ ( x ) is increasing in x , so is I ′ ( x ) = 1 /u ′′ ( I ( x )); hence G ( · ) is increasing. Then we have F ( ξ )( G ( ξ ) − G ( c )) ≥ 0. Consequently,0 ≤ E [ F ( ξ )( G ( ξ ) − G ( c ))] = E [ F ( ξ ) G ( ξ )] − G ( c ) E [ F ( ξ )] = E [ F ( ξ ) G ( ξ )] − G ( c ) h ′ x ( t, , which implies E [ F ( ξ ) G ( ξ )] ≥ G ( c ) h ′ x ( t, ≥ . The proof is complete. (cid:3) In this section, we give a concrete example to demonstrate our results.Define F ( x ) := N ( x/ √ , and w ( t, p ) := F ( N − ( p )) , where N is the probability distribution function of a standard normal. So in this example ourweighting function is time-invariant, and we will drop t and write w ( p ) = w ( t, p ) throughoutthis section.It is clear that w (0) = 0 , w (1) = 1, w ′ ( p ) = F ′ ( N − ( p )) N ′ ( N − ( p )) > 0. So w is a probability weightingfunction satisfying Assumption 2.3-(i). Moreover, we have w ( N ( x )) = F ( x ) = N ( x/ √ ,w ′ ( N ( x )) = F ′ ( x ) N ′ ( x ) = √ N ′ ( x/ √ N ′ ( x ) = 1 √ e x / ,h ( t, x ) ≡ E [ w ′ ( N ( ξ )) e xξ ] = 1 √ Z R √ π e xz e − z / dz = e x . m ∈ ( − , − / p ↓ w ′ ( p ) p m = lim x →−∞ w ′ ( N ( x )) N ( x ) m = lim x →−∞ √ e x / N ( x ) m = 1 √ lim x →−∞ e x / (4 m ) N ( x ) ! m = 1 √ lim x →−∞ xe x / (4 m ) / mN ′ ( x ) ! m = 1 √ (cid:18) m lim x →−∞ xe x ( m + ) (cid:19) m = 0 . Also, we have lim p ↑ w ′ ( p )(1 − p ) m = lim q ↓ w ′ (1 − q ) q m = lim q ↓ w ′ ( q ) q m = 0 . So Assumption 2.4-(iv) holds.Since h ( t, x ) = e x , it is straightforward to verify that Assumption 4.2 is satisfied. Moreover, w ′ ( p ) = e ( N − ( p )) / ; so w is concave on [0 , / 2] and convex on [1 / , w in this paper.With the explicit form of h , the ODE (3.3) becomesΛ ′ ( t ) = − | θ ( t ) | , t ∈ [0 , T ) , Λ( T ) = 0 , whose solution is Λ( t ) = R Tt | θ ( s ) | ds . Thus, λ ( t ) = p − Λ ′ ( t ) / | θ ( t ) | = 1 / . As discussed at the end of Section 4.1, λ ( t ) = 1 / π ∗ when the exponentialutility function is u ( x ) = 1 − e − αx , x ∈ R , for some α > 0, which clearly satisfies all theassumptions on the utility function in this paper.32n this case, u ′ ( x ) = αe − αx and hence I ( x ) = ln α − ln xα , x > 0. It is easy to check that thereexists κ > X ∗ ( T ) = I ( κ ¯ ρ ( T )) = c + 1 α Z T λ ( t ) θ ( t ) ⊤ d ˜ W ( t ) , where ˜ W ( t ) = W ( t ) + R t θ ( s ) ds is a standard Brownian motion under the risk-neutral measure, Q , of the original market, and c is a constant dependent of the market parameters.By the pricing theory, the replicating wealth process X ∗ ( · ) of X ∗ ( T ) is a Q -martingale(recall that the risk-free rate has been assumed to be 0); hence X ∗ ( t ) = c + 1 α Z t λ ( s ) θ ( s ) ⊤ d ˜ W ( s ) , t ∈ [0 , T ] . Matching the above with the wealth equation dX ∗ ( t ) = π ∗ ( t ) ⊤ σ ( t ) d ˜ W ( t ), we obtain the equi-librium portfolio π ∗ ( t ) = 1 α λ ( t )( σ ( t ) ⊤ ) − θ ( t ) . Recall that the optimal portfolio of an EUT agent with the same exponential utility is π EUT ( t ) = α ( σ ( t ) ⊤ ) − θ ( t ) . Hence the risky investment of the RDU agent at t is that of the EUT agentmultiplied by λ ( t ). Note that this result does not depend on the spcific form of the weightingfunction so long as λ ( · ) exists. In the special case when w ( t, p ) = F ( N − ( p )), λ ( t ) ≡ / 2; sothe risk exposure is reduced by half. We have seen that the sophisticated RDU agent behaves as if the risk premium is factored by λ ( · ). In the example with the specific probability weighting function presented in Section 5, λ ( t ) = 1 / 2; so the risk premium is reduced and the agent acts more cautiously than her EUTcounterpart. In this section, we answer the general question of when there is a reduction inrisk premium or, equivalently, when λ ( t ) < · ) exists. It follows from (3.3) that λ ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p Λ( t ) h ( t, p Λ( t ) ) h ′ x ( t, p Λ( t ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > , t ∈ [0 , T ) . x ∈ R + , define a probability measure Q x by d Q x d P = e − x + xξ , under which ξ − x ∼ N (0 , h ( t, x ) = e x E [ w ′ p ( t, N ( ξ )) e − x + xξ ]= e x E Q x [ w ′ p ( t, N ( ξ − x + x ))]= e x E [ w ′ p ( t, N ( ξ + x ))]= e x H ( t, x ) , where H ( t, x ) := E [ w ′ p ( t, N ( ξ + x ))] ≥ 0. The following characterizes the condition λ ( t ) < Theorem 6.1 Assume that Λ( · ) exists. Then, for any t ∈ [0 , T ) , λ ( t ) < if and only if H ′ x ( t, p Λ( t )) > .Proof . We have xh ( t, x ) h ′ x ( t, x ) = xe x H ( t, x ) e x H ′ x ( t, x ) + xe x H ( t, x ) = xH ( t, x ) H ′ x ( t, x ) + xH ( t, x ) . Noting λ ( t ) > ∀ t ∈ [0 , T ), we conclude that, for any t ∈ [0 , T ), λ ( t ) = xh ( t,x ) h ′ x ( t,x ) | x = √ Λ( t ) < H ′ x ( t, x ) | x = √ Λ( t ) ≡ H ′ x ( t, p Λ( t )) > (cid:3) Corollary 6.2 For any t ∈ [0 , T ) , λ ( t ) < if H ( t, x ) is strictly increasing in x ∈ R + . Inparticular, if w ( t, p ) is strictly convex in p , then λ ( t ) < .Proof . This is obvious. (cid:3) In the example in Section 5, H ( t, x ) = e − x / h ( t, x ) = e x / , which is strictly increasing in x ∈ R + ; hence Corollary 6.2 applies. On the other hand, as explained earlier, a strict convexityof w ( t, p ) in p underlines strict overweighting of the left tail (i.e. the bad events) and strictunderweighting of the right tail (i.e. the good events); so it enhances the level of risk aversionleading to a smaller risk premium and less risky exposure.Even if w is more general including being inverse S-shaped, it is still possible that λ ( t ) < 1, as demonstrated by the example in Section 5. In this case, the equivalent condition H ′ x ( t, p Λ( t )) > Concluding Remarks A continuous-time RDU portfolio selection problem is inherently time inconsistent. A sophis-ticated agent, realizing that in the future she herself might disagree to her current planning,resorts to consistent investment by implementing intra-personal equilibrium strategies fromwhich she will have no incentive to deviate at any point in time. We have solved the openproblem of deducing such equilibrium strategies, by developing an approach that to our bestknowledge is new to the literature. The main technical thrust of our approach is to expressthe first-order derivative of the small deviation from an equilibrium as a quadratic functionof the deviating amount. The definition of the equilibrium requires the quadratic function tohave a constant sign whatever the amount might be. This leads to an equality (which in turnleads to an ODE) and an inequality, the two constituting the main sufficient conditions forderiving explicitly the final wealth profile and, hence, the resulting strategy courtesy of themarket completeness.With an intra-personal equilibrium strategy, the agent, at any given time, in effect solves a constrained dynamically optimal RDU model in which the constraint is to honor all her futurestrategies. One may think this would lead to an extremely complicated terminal wealth profile.Our result, however, shows that the terminal wealth is surprisingly simple – it resembles that ofan optimal Merton portfolio, except that the investment opportunity set needs to be modifiedproperly. In other words, the RDU agent behaves as if she was an EUT agent, only that she isin a fictions market where she blends her probability weighting function into the market priceof risk. This observation may in turn shed lights on finding intertemporal market equilibriafor markets where all the agents are EUT and/or RDU consistent planners.We derived our equilibrium strategy based on an Ansatz; as such, our results do not ruleout the possibility of having other equilibria beyond our Ansatz. In general, uniqueness ofintra-personal equilibrium for time-inconsistent problems remains a very challenging researchquestion. It is particularly the case for the RDU model, or so we believe. To our best knowledge, Hu et al (2017) is the only paper that addresses the uniqueness in continuous time. ppendicesA Existence of Solutions to a Class of ODEs In this appendix we present some general existing results on a class of ODEs, taken fromAgarwal and O’Regan (2004). Consider the following ODE y ′ ( t ) = f ( t, y ( t )) , t ∈ (0 , T ] ,y (0) = 0 ∈ R , (A.1)where f may not be defined at t = 0.Denote by AC [0 , s ] the set of absolutely continuous functions on [0 , s ] where s > 0. Thefollowing assumption is introduced in Agarwal and O’Regan (2004). Assumption A.1 There exists t ∈ (0 , T ] such that1. There is β ∈ AC [0 , t ] ∩ C (0 , t ] with β (0) ≥ such that β ′ ( t ) ≥ f ( t, β ( t )) , t ∈ (0 , t ] . 2. There is α ∈ AC [0 , t ] ∩ C (0 , t ] with α ( t ) ≤ β ( t ) ∀ t ∈ [0 , t ] and α (0) ≤ such that α ′ ( t ) ≤ f ( t, α ( t )) , t ∈ (0 , t ] . 3. The function f ∗ ( t, y ) = f ( t, β ( t )) + g ( β ( t ) − y ) y ≥ β ( t ) ,f ( t, y ) α ( t ) < y < β ( t ) ,f ( t, α ( t )) + g ( α ( t ) − y ) y ≤ α ( t ) , is, in the region (0 , t ] × R , continuous in y for any t and measurable in t for any y ,where g ( x ) = x | x |≤ + sign(x) | x | > is the radial retraction. The following two propositions, both drawn from Agarwal and O’Regan (2004), concernthe local and global existence of the ODE (A.1) respectively. In [1], f ∗ is assumed to be (jointly) continuous; but from the proofs therein, we can easily weaken it tothis current version. roposition A.2 (Agarwal and O’Regan 2004, Theorem 3.1) Under Assumption A.1,the ODE (A.1) admits a solution y ∈ AC [0 , t ] satisfying α ( t ) ≤ y ( t ) ≤ β ( t ) for t ∈ [0 , t ] . Proposition A.3 (Agarwal and O’Regan 2004, Theorem 1.4) Given t ∈ (0 , T ] , assumethat f ( t, y ) is continuous in y for any t ∈ [ t , T ] , measurable in t for any y ∈ R , and thereexists g ( · ) ∈ L [ t , T ] such that | f ( t, y ) | ≤ g ( t ) ∀ ( t, y ) ∈ [ t , T ] × R . Then the equation y ′ ( t ) = f ( t, y ( t )) , t ∈ ( t , T ] ,y ( t ) = a ∈ R , (A.2) admits a solution y ∈ AC [ t , T ] . B Tversky–Kahnamen’s Probability Weighting Functions In this appendix we verify that a class of time–varying Tversky–Kahnamen (TK) probabilityweighting functions satisfy all the technical assumptions required in the paper.First of all, the original TK weighting function, introduced in Tversky and Kahnamen(1992), is w T K ( p ; δ ) := p δ [ p δ + (1 − p ) δ ] /δ , p ∈ [0 , , (B.1)where δ ∈ (0 , 1] is a parameter. This is an inverse S-shaped function, with w ′ T K ( p ; δ ) > p is close to both 0 and 1. Moreover, a smaller δ implies a stronger degree of probabilityweighting. When δ = 1, there is no probability weighting.We now vary the parameter δ over time to generate a family of time-dependent TK func-tions. Given a measurable function δ : [0 , T ] (0 , 1] with 0 < δ ( t ) < t ∈ [0 , T ) and δ ( T ) = 1, define ˜ w ( t, p ) := w T K ( p ; δ ( t )) , ( t, p ) ∈ [0 , T ] × [0 , . (B.2)The purpose of this appendix is to show that ˜ w satisfies all the assumptions in the paper,under proper conditions on δ ( · ).Clearly, ˜ w satisfies Assumption 2.3-(i). Proposition B.1 ˜ w satisfies Assumption 2.4-(iv).Proof . As t ∈ [0 , T ) is fixed in Assumption 2.4-(iv), from the construction of ˜ w it suffices toprove the conclusion for w T K ( · ; δ ) with δ ∈ (0 , 1) fixed.37enote a function α ( p ; δ ) := [ p δ + (1 − p ) δ ] − δ , p ∈ [0 , , (B.3)with parameter δ ∈ (0 , ≤ p δ + (1 − p ) δ < 2; hence2 − δ < α ( p ; δ ) ≤ . (B.4)Moreover, α ( p ; δ ) = α (1 − p ; δ ), w T K ( p ; δ ) = p δ α ( p ; δ ). Hence α ′ ( p ; δ ) = − δ [ p δ + (1 − p ) δ ] − δ − δ [ p δ − − (1 − p ) δ − ]= − α ( p ; δ ) δ [ p δ − − (1 − p ) δ − ] ,w ′ KT ( p ; δ ) = δp δ − α ( p ; δ ) + p δ α ′ ( p ; δ )= δα ( p ; δ ) p δ − + α ( p ; δ ) δ p δ (1 − p ) δ − − α ( p ; δ ) δ p δ − < p δ − + (1 − p ) δ − , where the last inequality is by (B.4). This completes the proof. (cid:3) Before we move to the next assumption, for any probability weighting function w : [0 , [0 , h ( x ; w ) := E [ w ′ ( N ( ξ )) e xξ ], x ∈ R . It then followsfrom Lemma 2.7-(ii) that, for any odd number n ≥ h ( n ) (0; w ) = E [ w ′ ( N ( ξ ); δ ) ξ n ] (B.5)= Z −∞ w ′ ( N ( y )) y n N ′ ( y ) dy + Z + ∞ w ′ ( N ( y )) y n N ′ ( y ) dy = Z ∞ w ′ ( N ( − y ))( − y ) n N ′ ( − y ) d ( − y ) + Z + ∞ w ′ ( N ( y )) y n N ′ ( y ) dy = − Z + ∞ w ′ (1 − N ( y )) y n N ′ ( y ) dy + Z + ∞ w ′ ( N ( y )) y n N ′ ( y ) dy = Z + ∞ [ w ′ ( N ( y )) − w ′ (1 − N ( y ))] y n N ′ ( y ) dy, (B.6)and h ′′ (0; w ) = Z −∞ w ′ ( N ( y )) y N ′ ( y ) dy + Z + ∞ w ′ ( N ( y )) y N ′ ( y ) dy = Z ∞ w ′ ( N ( − y ))( − y ) N ′ ( − y ) d ( − y ) + Z + ∞ w ′ ( N ( y )) y N ′ ( y ) dy = Z + ∞ w ′ (1 − N ( y )) y N ′ ( y ) dy + Z + ∞ w ′ ( N ( y )) y N ′ ( y ) dy = Z + ∞ [ w ′ ( N ( y )) + w ′ (1 − N ( y ))] y N ′ ( y ) dy. (B.7)38 roposition B.2 ˜ w satisfies Assumption 4.2-(i).Proof . With a slight abuse of notation, define h ( x ; δ ) := E [ w ′ KT ( N ( ξ ); δ ) e xξ ], x ∈ R , where δ ∈ (0 , t ∈ [0 , T ] is fixed in Assumption 4.2-(i), we can drop t and need only toshow that h ′ (0; δ ) ≥ h ′′′ (0; δ ) ≥ w ′ T K (1 − p ; δ ) = δ (1 − p ) δ − α ( p ; δ ) − (1 − p ) δ α ′ ( p ; δ ) , we have w ′ T K ( p ; δ ) − w ′ T K (1 − p ; δ ) = δp δ − α ( p ; δ ) + p δ α ′ ( p ; δ ) − (cid:2) δ (1 − p ) δ − α ( p ; δ ) − (1 − p ) δ α ′ ( p ; δ ) (cid:3) = δα ( p ; δ )[ p δ − − (1 − p ) δ − ] + α ′ ( p ; δ )[ p δ + (1 − p ) δ ]= δα ( p ; δ )[ p δ − − (1 − p ) δ − ] + α ′ ( p ; δ ) α ( p ; δ ) − δ = δα ( p ; δ )[ p δ − − (1 − p ) δ − ] + α ( p ; δ )[(1 − p ) δ − − p δ − ]= ( δ − α ( p ; δ )[ p δ − − (1 − p ) δ − ] > ∀ p ∈ (1 / , . (B.8)Now Assumption 4.2-(i) holds by virtue of (B.6) where we take w = w P K ( · ; δ ). (cid:3) Proposition B.3 If inf t ∈ [0 ,T ] δ ( t ) > , then ˜ w satisfies Assumption 4.2-(iv).Proof . We have w ′ T K ( p ; δ ) + w ′ T K (1 − p ; δ ) = δp δ − α ( p ; δ ) + p δ α ′ ( p ; δ ) + (cid:2) δ (1 − p ) δ − α ( p ; δ ) − (1 − p ) δ α ′ ( p ; δ ) (cid:3) = δα ( p ; δ )[ p δ − + (1 − p ) δ − ] + α ′ ( p ; δ )[ p δ − (1 − p ) δ ] > δα ( p ; δ )[ p δ − + (1 − p ) δ − ] > − δ δ ∀ p ∈ (1 / , . It hence follows from (B.7) that h ′′ x ( t, > Z + ∞ − δ ( t ) δ ( t ) y dN ( y ) = 2 − δ ( t ) δ ( t ) / ≥ inf t ∈ [0 ,T ] [2 − δ ( t ) δ ( t )] / > , owing to the condition that inf t ∈ [0 ,T ] δ ( t ) > (cid:3) For Assumption 4.2-(iii), we need the following lemma. Lemma B.4 For any ǫ ∈ (0 , , we have E [ N ( ξ ) − ǫ e xξ ] < + ∞ ∀ x ∈ R .Proof . The statement is symmetric for x ≤ x ≥ 0; hence it suffices to prove for the casewhen x < x = 0 is trivial). 39ince E [ N ( ξ ) − ǫ e xξ ξ ≥− ] ≤ e x E [ N ( ξ ) − ǫ ] = e x − ǫ , we only need to focus on E [ N ( ξ ) − ǫ e xξ ξ< − ].By the fact that for any γ > 0, lim y →−∞ N ( y ) e γy = lim y →−∞ N ′ ( y ) − γe − γy = 0, we deduce that thereexists a constant K > N ( y ) (1 − ǫ ) / e xy ≤ K ∀ y < − 1. Thus E [ N ( ξ ) − ǫ e xξ ξ< − ] ≤ K E [ N ( ξ ) − (1+ ǫ ) / ] < + ∞ . The proof is complete. (cid:3) Proposition B.5 If inf t ∈ [0 ,T ] δ ( t ) > , then ˜ w satisfies Assumption 4.2-(iii).Proof . Take ǫ := 1 ∧ inf t ∈ [0 ,T ] δ ( t ) > 0. For any t ∈ [0 , T ], it follows from the bound of w ′ T K ( p ; δ )(see the proof of Proposition B.1) that h ( t, 2) = E [ w ′ T K ( N ( ξ ); δ ( t )) e ξ ] ≤ E [ N ( ξ ) δ ( t ) − e ξ ] + E [ N ( − ξ ) δ ( t ) − e ξ ]= E [ N ( ξ ) δ ( t ) − e ξ ] + E [ N ( ξ ) δ ( t ) − e − ξ ] ≤ E [ N ( ξ ) ǫ − e ξ ] + E [ N ( ξ ) ǫ − e − ξ ] . According to Lemma B.4, we know h ( t, ≤ K for some constant K independent of t . However, h ( t, x ) is increasing in x ; so sup t ∈ [0 ,T ] h ( t, ≤ sup t ∈ [0 ,T ] h ( t, < K .Next, by the fact that ξ < e ξ + e − ξ we have h ′′ x ( t, ≤ h ( t, 2) + h ( t, < K + 1. (cid:3) Finally, we check Assumption 4.2-(ii). We first need two lemmas. Lemma B.6 There exists ¯ δ ∈ (0 , such that sup δ ∈ [¯ δ, Z + ∞ ln[ N ( y ) δ + N ( − y ) δ ] dy < + ∞ . Proof . Fix δ ∈ (0 , . Observe thatlim y → + ∞ ln[ N ( y ) δ + N ( − y ) δ ] y − = lim y → + ∞ δN ′ ( y ) y − − N ( y ) δ − − N ( − y ) δ − N ( y ) δ + N ( − y ) δ ≤ lim y → + ∞ δN ′ ( y ) y − N ( − y ) δ − N ( y ) δ + N ( − y ) δ ≤ lim y → + ∞ N ′ ( y ) N ( − y ) − δ y − δ 2= lim y → + ∞ δ N ′ ( y ) − δ/ N ( − y ) − δ N ′ ( y ) δ/ y − = 0 . K > δ ), such thatln[ N ( y ) δ + N ( − y ) δ ] < Ky − ∀ y > . As a result, Z + ∞ ln[ N ( y ) δ + N ( − y ) δ ] dy ≤ Z ln( N ( y ) δ + N ( − y ) δ ) dy + K Z T y − dy < + ∞ . Note that the integrand on the left hand side is decreasing in δ ; hence the above finiteness isuniform when δ is sufficiently close to 1. (cid:3) Recall we have defined h ( x ; δ ) = E [ w ′ T K ( N ( ξ ); δ ) e xξ ], x ∈ R , δ ∈ (0 , Lemma B.7 We have h ′ (0; δ ) = 1 − δδ Z + ∞ ln[ N ( y ) δ + N ( − y ) δ ] dy + o (cid:18) − δδ (cid:19) when δ is sufficiently close to 1.Proof . By (B.6) and (B.8), we have h ′ (0; δ ) = Z + ∞ ( δ − α ( N ( y ); δ )[ N ( y ) δ − − N ( − y ) δ − ] ydN ( y )= δ − δ Z + ∞ α ( N ( y ); δ ) yd [ N ( y ) δ + N ( − y ) δ ]= δ − δ δδ − Z + ∞ yd [ N ( y ) δ + N ( − y ) δ ] − /δ = Z + ∞ { − [ N ( y ) δ + N ( − y ) δ ] − /δ } dy, where we have used the fact that lim y → + ∞ { − [ N ( y ) δ + N ( − y ) δ ] − /δ } y = 0.Applying the general Taylor expansion x ǫ = 1+ ǫ ln x + o ( | ǫ | )(ln x ) for x > | ǫ | , we deduce Z + ∞ { − [ N ( y ) δ + N ( − y ) δ ] − /δ } dy = 1 − δδ Z + ∞ ln[ N ( y ) δ + N ( − y ) δ ] dy + o (cid:18) − δδ (cid:19) , where we have used the finiteness R + ∞ { ln[ N ( y ) δ + N ( − y ) δ ] } dy < + ∞ , which follows fromthe inequalities 0 < ln[ N ( y ) δ + N ( − y ) δ ] < ln 2 and Lemma B.6. (cid:3) Proposition B.8 If lim sup t ↑ T − δ ( t ) δ ( t ) √ T − t = 0 and lim inf t ↑ T | θ ( t ) | > , then ˜ w satisfies Assumption 4.2-(ii). roof . This follows immediately from Lemmas B.6 and B.7. 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