Non-concave expected utility optimization with uncertain time horizon: an application to participating life insurance contracts
NNon-concave expected utility optimization with uncertain timehorizon: an application to participating life insurance contracts
Christian Dehm ∗ Thai Nguyen † Mitja Stadje ‡ Abstract
We examine an expected utility maximization problem with an uncertain time horizon, a clas-sical example being a life insurance contract due at the time of death. Life insurance contractsusually have an option-like form leading to a non-concave optimization problem. We considergeneral utility functions and give necessary and sufficient optimality conditions, deriving a compu-tationally tractable algorithm. A numerical study is done to illustrate our findings. Our analysissuggests that the possible occurrence of a premature stopping leads to a reduced performance ofthe optimal portfolio compared to a setting without time-horizon uncertainty.
A classical expected utility maximization approach is the following Merton problemmax π ∈ Π(0 ,x ) E [ U ( P πT )] , where the investment horizon T > P πT is the terminal portfolio generated by anadmissible strategy π ∈ Π(0 , x ) starting with an initial investment x , and U is a concave utilityfunction. Such a utility maximization problem in a continuous-time setting dates back to Merton [31].The general solution for this problem is well-known, see e.g. Biagini [6] for a broad discussion. Merton’spioneering work has been extended in several directions e.g. by assuming more general structuresof preferences, by incorporating additional (possibly untradable) randomness to the underlying riskprocesses, or by including a risk constraints to the optimization problem.In this work, we consider an expected utility maximization problem with random time horizonhaving a participating life insurance contract in mind, which concludes in the case that the policy-holder dies before T . Many of the world’s largest financial institutions in terms of revenues are lifeinsurance companies, and these contracts have been extensively used in European and non-Europeanlife insurance markets; see for instance [4, 5, 13, 24, 2, 23, 32, 39]. Typically, to buy a participating in-surance policy, the policyholder pays a lump sum premium x upfront and the capital saved is investedin a self-financing way, subject to annual interest, where the insurance company offers a (minimal)guarantee. In positive economic developments, the policyholder receives a surplus, while in case of bad ∗ Institute of Insurance Science and Institute of Financial Mathematics, University of Ulm, Ulm, Germany,[email protected]. † ´Ecole d’Actuariat, Universit´e Laval, Qu´ebec city, Canada, and School of Economic Mathematics-Statistics, Universityof Economics Ho Chi Minh city, Vietnam, [email protected]. ‡ Institute of Insurance Science and Institute of Financial Mathematics, University of Ulm, Ulm, Germany,[email protected]
Keywords and phrases. non-concave optimization, uncertain horizon, expected utility, asset allocation, partipating in-surance contracts.
AMS Classification. a r X i v : . [ q -f i n . P M ] M a y conomic developments, the insurance company carries the loss. The risk of possible losses thereforeis completely covered by the insurer. Hence, a participating insurance contract may be regarded as anoption-type financial instrument, leading to a non-concave utility function; see e.g. Chen et al. [16],Lin e al. [30], Nguyen and Stadje [33]. The literature on non-concave optimization with certain timehorizon is vast, see for instance Aumann and Perles [1], Carpenter [15], Ross [38], Reichlin [35], Dai etal. [18], Larsen [29], Carassus and Pham [14], Dong and Zheng [22], Rieger [37], Basak and Makarov[3], Chen et al. [16], Nguyen and Stadje [33] and Bichuch and Sturm [8].Classically, an insurance policy covers a stochastic risk and ends with the death of the insuredperson. In this case, the time point of the payment to the insured and therefore the maturity of theinvestment of the insurance company is random. In the literature which considers uncertain investmenthorizons, independence of time-horizon uncertainty from market developments is usually assumed. Werefer to the early work of Yaari [40], who looks at the investment problem of an individual with anuncertain time of death in a simplified case with purely deterministic investment opportunities. Thelatter work is extended to discrete-time settings with multiple risky assets by Hakansson [25, 26]. Inthe work of Merton [31], the case of an investor retiring at an uncertain date is also addressed asa special case, where the time horizon uncertainty is reflected by the first jump of an independentPoisson process with constant intensity. Richard [36] solves in closed-form an optimal portfolio choiceproblem with an uncertain time of death and the presence of life insurance. Blanchet et al. [11] andBouchard and Pham [12] consider concave utility maximization with general stopping times.Note that a participating life insurance contract typically ends either when the policyholder dies orin case of survival at a fixed pre-specified maturity. We take this feature into account in our analysisof the optimal investment problem for non-concave target functions with a random time horizon. Ourwork therefore combines two streams of literature: non-concave portfolio optimization and portfoliooptimization under time horizon uncertainty. More precisely, we extend the results in Reichlin [35] tothe setting with a random investment horizon and complement the results in Blanchet et al. [11] andBouchard and Pham [12] by allowing non-concave utility functions. The independence assumptionenables us to establish a hedging representation for the stopped portfolio process in the enlarged fil-tration. We follow the general approach of concavificiation techniques as described in [37, 35] to dealwith non-concavity. To illustrate our finding, we carry out an numerical analysis which is challengingbecause of the multiplier being a random process that depends on the random time. Our numericalresults confirm the intuition that an uncertain time horizon risk leads to a reduced performance (interms of certainty equivalent) of the optimal portfolio compared to a setting with a certain time-horizon.The remainder of the paper is organized as follows: First, we describe a specific complete financialmarket setting and introduce the uncertain investment time in Section 2. We show a representationtheorem for progressive enlarged filtrations in Section 3. Section 4 introduces a concavification tech-nique for general utility functions, which allows to prove an optimality and existence result for thenon-concave expected utility optimization with uncertain horizon. Section 5 gives characterizationsof the optimal solution in terms of Lagrange multipliers adapted to the financial market. In Section6, we consider the case of power utility and derive optimal investment solutions and strategies. InSection 7, we perform a numerical study to compare concave and non-concave optimization with andwithout time-horizon uncertainty. Finally, we summarize our main results in Section 8. Let [0 , T ] with 0 < T < ∞ be the maximal time span of the economy.2 .1 The Financial market For the market setup, the n -dimensional random process W is the driver of the stock prizes S modelledas geometric Brownian motion, i.e., dS it S it = µ it dt + n (cid:88) j =1 σ i,jt dW jt , i = 1 , . . . , n, where the superscript i denotes the i -th entry of the corresponding vector or ( i, j ) the entry in the i -th row and j -th colomn of a matrix and we use the subscript t to denote the time index t . We usethe notation µ = ( µ i ) ≤ i ≤ n and σ = ( σ i,j ) ≤ i,j ≤ n for the corresponding vector or matrix, respectively.Additionally to these risky assets, we consider a bond B , given by dB t = B t r t dt , where r denotes the(deterministic) interest rate. The information in the market is captured by the augmented filtration F = ( F t ) t ≥ generated by the Brownian motion, satisfying the usual conditions and F is trivial. Weassume that the coefficients µ , r ≥ σ is bounded,deterministic, invertible with bounded inverse σ − .In this arbitrage-free financial market, there exists a unique equivalent martingale measure Q withRadon-Nikodym density M as the solution of dM t = − M t θ t dW t , where θ t := σ − t ( µ t − r t ). Furtherwe define H t := exp (cid:16) − (cid:82) t r s ds (cid:17) M t . By Itˆo’s formula dH t = − H t r t dt − H t θ t dW t , (2.1)and H t = exp (cid:18) − (cid:90) t ( r s + 12 θ Ts θ s ) ds − (cid:90) t θ s dW s (cid:19) . (2.2)We consider the economy in the usual frictionless setting, where stocks and bond are infinitely divisibleand there are no market frictions, no transaction costs etc. Additional to the financial market setting, we consider a random time-horizon τ , where τ is a positivediscrete random variable independent of F . In particular, τ is not an F -stopping time. The presenceof time horizon randomness induces additional uncertainty into the economy. Since τ has a discretedistribution, we can take time points 0 ≤ T < T < · · · < T n < T and probabilities 0 < p i < (cid:80) ni =1 p i < P ( τ = T i ) = p i , ≤ i ≤ n and P ( τ = T ) = 1 − n (cid:88) i =1 p i > . If P ( τ = T ) = 0, we simply redefine T to be equal to T n .Let F τ = ( F τt ) ≤ t ≤ T with F τt being the σ -algebra generated by ( τ ≤ s ) ≤ s ≤ t . We consider G = F ∨ F τ to be the progressive enlargement of the filtration F with random time τ , i.e., G t = F t ∨F τt for all t ≥ G is the smallest (right-continuous) filtration containing F such that τ is a G -stoppingtime (see e.g. Bielecki and Rutkowski, [9], Chapter 5 or Pham, [34]).By Bielecki and Rutkowski, [9] we know that for any G t -measurable random variable Y , thereexists an F t -measurable random variable ˜ Y such that τ>t Y = τ>t ˜ Y . This fact is noted as ”thefiltration G coincides with F before τ ” in Jeanblanc and Le Cam [27] (see also Dellacherie and Meyer[21]). 3 .3 Admissible strategy We consider an investor putting the amount π is in each risky asset at time 0 ≤ s ≤ T . By consideringa self-financing portfolio, the amount P s − (cid:80) ni =1 π is is invested in the bond. We use the notation( P t,π,xs , t ≤ s ≤ T ) for the wealth process at time s , developed from an initial capital P πt := P t,π,xt = x at time t under a self-financing strategy π where π i denotes the amount invested in asset i . Then P t,π,xs evolves according to the stochastic differential equation dP t,π,xs = P t,π,xs r s ds + π s [( µ s − r s ) ds + σ s dW s ] . (2.3)We call a portfolio ( π t , ≤ t ≤ T ) admissible , if π is progressively measurable w.r.t. G , locallysquare-integrable, i.e., (cid:80) ni =1 (cid:82) T ( π is ) ds < ∞ a.s., the associated wealth process is non-negative and( H s P t,π,xs ) s is a square-integrable martingale. By Girsanov’s theorem as long as π is locally square-integrable ( H s P t,π,xs ) s is always a local martingale. For the set of admissible portfolios with initialcapital x at time t , we use the notationΠ( t, x ) := (cid:110) π s : t ≤ s ≤ T, P t,π,xt = x, π is admissible and P t,π,xs ≥ (cid:111) . (2.4) We suppose that the payoff of the participating life insurance contract has the form α ( x − B ) + + K where K is the guarantee, x is the terminal value of the portfolio due to the returns in the financialmarkets, α is the participation rate and B is the threshold from which the participations rate kicksin, see [16, 33, 2, 4, 5, 13, 23, 32, 39]. If u is a concave utility function we overall get a non-concaveutility function U ( x ) := u ( α ( x − B ) + + K ) . Hence, in the sequel we consider a general not necessarily concave utility function U : (0 , ∞ ) → R with U ( ∞ ) := lim x →∞ U ( x ) >
0, which is non-constant, increasing, upper-semicontinuous with thegrowth condition lim x →∞ U ( x ) x = 0 . (2.5)We set U ( x ) = −∞ for x < U (0) := lim x (cid:38) U ( x ), U ( ∞ ) :=lim x →∞ U ( x ). We note that we do not assume that U is concave, continuous or strictly increas-ing. The assumption U ( ∞ ) > U (0) > −∞ , since adding constants does notchange preferences. In a concave setting, Equation (2.5) is equivalent to U (cid:48) ( ∞ ) = 0, which is part ofthe Inada condition.We note that Equation (2.5) and the assumption U ( ∞ ) > U : R → R ∪ {−∞} that dominates U , i.e., ¯ U ≥ U . The concave envelope U c of U is the smallestconcave function U c : R → R ∪ {−∞} that dominates U , i.e., U c ( x ) ≥ U ( x ) for all x ∈ R . Formally, U c := min { f : f concave, f ( x ) ≥ U ( x ) for all x ∈ R } . Lemma 2.1 (Reichlin [35])
The concave envelope U c of U is finite, continuous on (0 , ∞ ) and sat-isfies Equation (2.5) . The set { U < U c } := { x ∈ (0 , ∞ ) : U ( x ) < U c ( x ) } is open and its (countable)connected components are bounded (open) intervals. Moreover, U c is locally affine on the set { U < U c } ,in the sense that it is locally affine on each of the above intervals. .5 Optimization problem In our complete financial market setup, we consider the problem V τ ( x, U ) = sup π ∈ Π(0 ,x ) E [ U ( P πτ ∧ T )] , (2.6)for suitable portfolios (see Equation (2.4)) inΠ( t, x ) = (cid:110) π s : t ≤ s ≤ T, P t,π,xt = x, π is admissible (cid:111) . From now on, we omit the superscript π in case of no ambiguity. We define ˜ P t := P t exp (cid:16) − (cid:82) t r s ds (cid:17) as the discounted portfolio process and C ( x ) := (cid:40) P τ ∧ T : P τ ∧ T is a non-negative G τ ∧ T -measurable random variable with H τ ∧ T P τ ∧ T square-integrable. (2.7) Definition 2.2
We call P ∗ τ ∧ T optimal , if E [ U ( P ∗ τ ∧ T )] = V ( x, U ) . In this case, P ∗ τ ∧ T is a maximizer of V ( x, U ) . In this section we discuss how hedging is possible in the enlarged filtration G (see also Dellacherie andMeyer [21]). Theorem 3.1
Let T be a fixed positive time and let F t be the augmented filtration generated by the d -dimensional Brownian motion W ( t ) ( ≤ t ≤ T ). Let M be a square-integrabel G -adapted martingale.Then there exists ( ξ u ) ∈ L ( d P × ds, ( F u ) ≤ u ≤ T ) such that M τ ∧ t = M + (cid:82) τ ∧ t ξ u dW u , ≤ t < T . A proof can be found in the appendix.
Corollary 3.2
1. For P τ ∧ T ∈ C ( x ) , we can find π ∈ Π(0 , x ) such that ˜ P τ ∧ T = ˜ P πτ ∧ T =: x + (cid:82) τ ∧ T σ s π s dW Q s . Furthermore t → P ( π ) t and t → ˜ P ( π ) t can be extended as stochastic integrals to [0 , T ] as F -adapted processes.2. Let π ∈ Π(0 , x ) , then ˜ P πτ ∧ T := x + (cid:82) τ ∧ T σ s π s dW Q s ∈ C ( x ) . Remark 3.3
Using this result, we conclude that V τ ( x, U ) = sup π ∈ Π(0 ,x ) E [ U ( P πτ ∧ T )] = sup P ∈ C ( x ) E [ U ( P )] . Proof.
We prove the two parts of the corollary separately.1. Let P ∈ C ( x ). Then P is non-negative. We consider the process H τ ∧ t P τ ∧ t := E Q [ H τ ∧ T P |G τ ∧ t ],which is by assumption a square-integrable martingale. In particular, it holds that E Q (cid:104) ˜ P τ ∧ T (cid:105) ≤ x . By Theorem 3.1, we can find an F -adapted (and square-integrable) process ( ξ t ) ≤ t ≤ T suchthat H τ ∧ t P τ ∧ t = P + (cid:90) τ ∧ t ξ s dW s , a.s. P t and ˜ P t might only be defined up to T ∧ τ . However, we can define for0 ≤ t ≤ T : H t P t = P + (cid:90) t ξ u dW u . We extend then also ˜ P t = e − rt P t for 0 ≤ t ≤ T . Itˆo’s Lemma and Girsanov’s theorem give˜ P τ ∧ t = P + (cid:90) τ ∧ t π s σ s dW Q s . with π t := ξ t σ − t H t + P t θ t σ − t for 0 ≤ t ≤ T . Note that ( π t ) ≤ t ≤ T is F -adapted and locallysquare-integrable.2. Conversely, let π ∈ Π(0 , x ) and denote the corresponding wealth process by P . Setting ˜ P t = e − rt P t we note that ˜ P τ ∧ T = x + (cid:82) τ ∧ T π s σ s dW Q s . We observe that this process is a non-negativelocal martingale under Q and thus a Q -supermartingale, i.e., E Q (cid:104) ˜ P τ ∧ T (cid:105) ≤ x . Clearly, by thedefinition of Π(0 , x ), we also have that H τ ∧ T P τ ∧ T is square-integrable. (cid:50) Remark 3.4
We note that due to the independence of H t and τ E Q (cid:104) ˜ P τ ∧ T (cid:105) = E [ H τ ∧ T P τ ∧ T ] = E (cid:34) n (cid:88) i =1 p i H T i P T i + (cid:32) − n (cid:88) i =1 p i (cid:33) H T P T (cid:35) ≤ x. Here, we recall important concepts, which are helpful in proving our results in the next section. InKramkov and Schachermayer [28], the concept of asymptotic elasticity (AE) is introduced and it isessentially shown that an upper bound of AE ( U ) is necessary and sufficient to ensure the existenceof an optimizer. This condition can be formulated in terms of the Fenchel-Legendre transform of U defined by J ( y ) := sup x> { U ( x ) − xy } (see Deelstra et al. [19]). In the following, we assume E [ J ( λH τ ∧ T )] < ∞ for all λ > AE ( J ) := lim sup y → sup q ∈ ∂J ( y ) | q | yJ ( y ) . By Deelstra et al. [19], we know the following connection between AE ( J ) and AE ( U ): If thefunction U is strictly concave, we have J (cid:48) = − ( U (cid:48) ) − , and the conditions U (cid:48) ( ∞ ) = 0 and J (cid:48) (0) = −∞ are equivalent. The conditions AE ( U ) < AE ( J ) < ∞ are equivalent under the condition U (cid:48) ( ∞ ) = 0.To exclude trivial cases, it is necessary to assume that V τ ( x, U ) < ∞ for some x >
0. However,even in the case of a concave utility, this is not sufficient to guarantee the existence of a maximizer (seeChapter 5 in Kramkov and Schachermayer [28] or Chapter 5 in Biagini and Guasoni [7]). By a slightmodification of Lemma 3.2 in Reichlin [35], one may see that under the assumption that AE ( J ) < ∞ , AE ( U ) < V τ ( x, U ) < ∞ for some x >
0. Later in Theorem 4.4, we will actually showthat V τ ( x, U ) = V τ ( x, U c ). Here, we see that under the assumption that AE ( J ) < ∞ and AE ( U ) < .2 Concavification with uncertain time horizon In this section, we generalize Theorem 5.1 and Proposition 5.3 of Reichlin [35] to our setting of arandom investment horizon.
Theorem 4.1
For any F t -measurable P t with E [ H t P t ] ≤ x for fixed ≤ t ≤ T with P ( τ = t ) > ,there exists an F t -measurable P ∗ t with E [ H t P ∗ t ] ≤ x such that { P ∗ t ∈ { U < U c }} = ∅ . In other words, P ∗ t ∈ { U = U c } P − a.s for ≤ t ≤ T with P ( τ = t ) > . Moreover, it holds that E [ U ( P ∗ t )] = E [ U c ( P ∗ t )] = E [ U c ( P t )] ≥ E [ U ( P t )] . In particular, for any P τ ∧ T ∈ C ( x ) , there exists P ∗ τ ∧ T ∈ C ( x ) with P ∗ τ ∧ T ∈ { U = U c } P − a.s and wehave E [ U ( P ∗ τ ∧ T )] = E [ U c ( P ∗ τ ∧ T )] = E [ U c ( P τ ∧ T )] ≥ E [ U ( P τ ∧ T )] . Proof.
One can show analogously to Reichlin [35] that there exists an F t -measurable P ∗ t takingvalues in U = U c with E [ H t P ∗ t ] ≤ E [ H t P t ] ≤ x and E [ U c ( P ∗ t )] = E [ U c ( P t )] , for 0 ≤ t ≤ T. (4.1)In particular, E Q (cid:104) ˜ P ∗ τ ∧ T (cid:105) = n (cid:88) j =1 p j E (cid:104) H T j P ∗ T j (cid:105) ≤ x. Now P τ ∧ T ∈ C ( x ) by Corollary 3.2 corresponds to a process ( P t ) ≤ t ≤ T stopped at time τ ∧ T . Hence, E [ U c ( P ∗ τ ∧ T )] = n (cid:88) j =1 (cid:88) i E (cid:104) U c ( P ∗ T j ) (cid:105) p j + − n (cid:88) j =1 p j (cid:88) i E [ U c ( P ∗ T )]= n (cid:88) j =1 (cid:88) i E (cid:2) U c ( P T j ) (cid:3) p j + − n (cid:88) j =1 p j (cid:88) i E [ U c ( P T )] = E [ U c ( P τ ∧ T )] . (4.2) (cid:50) Lemma 4.2
In the setting of the lemma above, it holds for a portfolio process ( P t ) ≤ t ≤ T that:If P t ∈ { U < U c } for some t ∈ { T , . . . , T n , T } with p t (cid:54) = 0 , then E [ U ( P τ ∧ T )] < E (cid:104) U ( P (1) τ ∧ T ) (cid:105) for some P (1) τ ∧ T ∈ C ( x ) . Proof.
Let P t ∈ { U < U c } . By the Lemma above, we can find ( P ∗ t ) with the property P ∗ t ∈ { U = U c } or equivalently E [ U ( P ∗ t )] = E [ U c ( P ∗ t )]. Now, due to P t ∈ { U < U c } and p t (cid:54) = 0, we obtain byEquation (4.2) E [ U ( P ∗ τ ∧ T )] = E [ U c ( P ∗ τ ∧ T )] ≥ E [ U c ( P τ ∧ T )] > E [ U ( P τ ∧ T )] , which is the asserted inequality. Hence, we can choose P (1) τ ∧ T := P ∗ τ ∧ T . (cid:50) emark 4.3 From Lemma 4.2, we see that the optimal solution has the property that P opt t ∈ { U = U c } for all t ∈ { T , . . . , T n , T } with p t (cid:54) = 0 . This means that the optimal solution does not take values inthe non-concave part of the utility function at the (possible) times to stop (if this time has non-zeroprobability). Theorem 4.4
It holds that V τ ( x, U ) = V τ ( x, U c ) for all x > . (4.3) Every maximizer of the problem V τ ( x, U ) also maximizes the concavified problem V τ ( x, U c ) . The prob-lem V τ ( x, U ) admits a maximizer if and only if the concavified problem V τ ( x, U c ) admits a maximizer.Proof. By definition, U ≤ U c . Therefore the inequality V τ ( x, U ) ≤ V τ ( x, U c ) is obvious. For theother inequality, we choose a maximizing sequence ( P n ) n ∈ N withlim n →∞ E [ U c ( P n )] = sup π ∈ Π(0 ,x ) E [ U c ( P π )] = V τ ( x, U c ) . Now, using Theorem 4.1, we can find ( P ∗ n ) n ∈ N such that E [ U c ( P n )] = E [ U c ( P ∗ n )] = E [ U ( P ∗ n )] ≤ sup P ∈ C ( x ) E [ U ( P )] = V τ ( x, U ) , which shows the other inequality.For the second part, we first take P τ ∧ T to be a maximizer of V τ ( x, U ). Then we have V τ ( x, U ) = E [ U ( P τ ∧ T )] ≤ E [ U c ( P τ ∧ T )] ≤ V τ ( x, U c ) = V τ ( x, U ) . Conversely, assume that P τ ∧ T is the maximizer of V τ ( x, U c ). By Theorem 4.1, we can find P ∗ τ ∧ T suchthat E [ U ( P ∗ τ ∧ T )] = E [ U c ( P ∗ τ ∧ T )] = E [ U c ( P τ ∧ T )]. Thus, we obtain V τ ( x, U ) = V τ ( x, U c ) = E [ U c ( P τ ∧ T )] = E [ U c ( P ∗ τ ∧ T )] = E [ U ( P ∗ τ ∧ T )] . (cid:50) The aim of this chapter is to generalize results with time horizon uncertainty (see for instance Theorem2 in Blanchet et al. [11]) to the case of non-concave optimization. We consider a special choice ofa non-concave objective function, e.g., a payoff function which has an option-type form. We define u : R → R for given K >
B > u ( x ) = (cid:40) U ( α ( x − B ) + + K ) for x ≥ , −∞ else , (5.1)where U is concave, strictly increasing and at least twice differentiable. Recall that such a payoffstructure arises for participating contracts with guarantees, where α is the participation rate, K is theguarantee and B is the threshold for the participation, see [16, 33, 2, 23, 32, 39, 22]. However, thisfunction u could also be understood as a managerial compensation with a fixed payment K > x with strike B , see Carpenter [15]. The concave envelope ˜ u : R → R is thengiven by ˜ u ( x ) := −∞ for x < ,u (0) + u (cid:48) (ˆ x ( B )) x for 0 ≤ x ≤ ˆ x ( B ) ,u ( x ) for x > ˆ x ( B ) , (5.2)8here ˆ x ( B ) := min { x > u ( x ) = ˜ u ( x ) } . Then ˜ u dominates u with equality for x = 0 and x ≥ ˆ x ( B )(for fixed B > m ∈ R is in the subdifferential of ˜ u (also denoted by ˜ u (cid:48) ), if it holdsthat for every x, y ≥ u ( y ) − ˜ u ( x ) ≤ m ( y − x ) . (5.3)The function ˜ u is not differentiable. Nevertheless, the subdifferential ˜ u (cid:48) may be identified with theset-valued function ˜ u (cid:48) ( x ) := [ u (cid:48) (ˆ x ( B )) , ∞ ) for x = 0 , { u (cid:48) (ˆ x ( B )) } for 0 < x ≤ ˆ x ( B ) , { u (cid:48) ( x ) } for x > ˆ x ( B ) . (5.4)Now, we are able to define the function i : (0 , ∞ ) → [0 , ∞ ) by i ( y ) := (cid:20) α (cid:16) I (cid:16) yα (cid:17) − K (cid:17) + B (cid:21) { y ≤ u (cid:48) (ˆ x ( B )) } . (5.5)We note that i is the generalized inverse function of ˜ u (cid:48) in the sense that y ∈ ˜ u (cid:48) ( i ( y )) for all B > . (5.6)The objective function is given by J ( x ) = sup π ∈ Π(0 ,x ) E [ u ( P πτ ∧ T )] , (5.7)under the budget constraint E [ H τ ∧ T P τ ∧ T ] ≤ x , which is a non-concave optimization problem. Theorem 5.1
We consider i ( x ) as the inverse of ˜ u ( x ) (in the sense of Equation (5.6) ). If there existsa deterministic function ν with i ( ν ) = x such that the process ( H t i ( ν t H t )) t ≥ is a martingale, thenthe wealth process P (cid:63) defined by P (cid:63)t = i ( ν t H t ) is optimal.Proof. The proof is similar as in Blanchet et al. [11]. (cid:50)
The following optimality and existence result is the main theorem of this section generalizingresults by Blanchet et al. [11] to non-concave settings. We need the following assumption.
Assumption 1
We assume that E (cid:2) max i | u ( P (cid:63)T i ) | (cid:3) < ∞ and E (cid:2) ˜ u (cid:48)− (cid:0) (1 − δ ) P (cid:63)T i (cid:1) P (cid:63)T i (cid:3) < ∞ for some δ > for i = 1 , . . . , n , and E (cid:20) max i ( P ∗ T i H T i ) (cid:90) T H s π s σ s ds (cid:21) < ∞ . Let ( ν T j ) j be a sequence of F T j -measurable random variables. Condition 1
We state the following properties: ( i ) the process ( H t P t , t ≥ is a square-integrable martingale adapted to F and P T j = i ( ν T j H T j ) for j = { , . . . , n + 1 } with T n +1 := T . ( ii ) the random variable (cid:80) ni =1 p i ν T i + (1 − (cid:80) ni =1 p i ) ν T is constant. Theorem 5.2
Under Assumption 1 it holds that: . If the problem (5.7) admits a solution P (cid:63) , then P (cid:63)t = i ( ν t H t ) for t ∈ { T , . . . , T n , T } , for an F -adapted process ν which satisfies ( i ) and ( ii ) in Condition 1.2. If there exists an F -adapted process ν such that ( i ) and ( ii ) in Condition 1 hold, then P t = i ( ν t H t ) is an optimal solution of problem (5.7) ( t ∈ { T , . . . , T n , T } ).Proof. We note P is optimal for ˜ u as well as for u by Theorem 4.4.1. Let t ∈ { T , . . . , T n , T } . Choose ν t = H − t ˜ u (cid:48) ( P (cid:63)t ) where at 0 we identify u (cid:48) with its right-handside derivative. Then it holds P (cid:63)t = i ( ν t H t ) for t ∈ { T , . . . , T n , T } by Lemma 4.2. Moreover,( H t i ( ν t H t )) t ≥ is an F -measurable square-integrable martingale since it coincides with the wealthprocess HP (cid:63) for t ∈ { T , . . . , T n , T } .Now, let Y be another wealth process not a.s. equal to P (cid:63) , i.e., a non-negative process suchthat ( H t Y t , t ≥
0) is a square-integrable martingale with the same initial wealth x . We definefor 0 ≤ ε ≤ χ by Φ( ε ) := E [ χ ( ε, ω )] and χ ( ε, ω ) := n (cid:88) i =1 ˜ u (cid:0) εP (cid:63)T i + (1 − ε ) Y T i (cid:1) p i + (cid:32) − n (cid:88) i =1 p i (cid:33) ˜ u ( εP (cid:63)T + (1 − ε ) Y T ) . We note that χ is differentiable w.r.t. ε where we identify the derivatives at the boundary pointswith the left-hand or right-hand sides derivatives, respectively. Moreover, it holds (for some k > E [ | χ ( ε, ω ) | ] ≤ E (cid:20) max t ∈ T ,...,T n ,T | ˜ u ( P ∗ t ) | (cid:21) ≤ E (cid:20) max t ∈ T ,...,T n ,T k | u ( P (cid:63)t ) | (cid:21) < ∞ . It is worth noting that for ε > − δ we have (1 − δ ) P (cid:63)T i < εP (cid:63)T i + (1 − ε ) Y T i . We calculate | χ (cid:48) ( ε, ω ) | ≤ n (cid:88) i =1 p i (cid:12)(cid:12) ˜ u (cid:48) (cid:0) (1 − δ ) P (cid:63)T i (cid:1)(cid:12)(cid:12) P (cid:63)T i + (cid:32) − n (cid:88) i =1 p i (cid:33) (cid:12)(cid:12) ˜ u (cid:48) ((1 − δ ) P (cid:63)T ) (cid:12)(cid:12) P (cid:63)T ≤ max t ∈ T ,...,T n ,T (cid:12)(cid:12) ˜ u (cid:48) ((1 − δ ) P (cid:63)t ) (cid:12)(cid:12) P (cid:63)t . Under Assumption 1 we obtain with Dominated ConvergenceΦ (cid:48) ( ε ) = E (cid:20) n (cid:88) i =1 ˜ u (cid:48) (cid:0) εP (cid:63)T i + (1 − ε ) Y T i (cid:1) (cid:0) P (cid:63)T i − Y T i (cid:1) p i + (cid:32) − n (cid:88) i =1 p i (cid:33) ˜ u (cid:48) ( εP (cid:63)T + (1 − ε ) Y T ) ( P (cid:63)T − Y T ) (cid:21) . We know that the function Φ attains its maximum at ε = 1, since P (cid:63)t is the optimal solution (asshown in Theorem 5.1). Hence, 0 ≤ Φ (cid:48) (1) . Thus, 0 ≤ E (cid:20) n (cid:88) i =1 ˜ u (cid:48) (cid:0) P (cid:63)T i (cid:1) H − T i p i (cid:0) H T i P (cid:63)T i − H T i Y T i (cid:1) + (cid:32) − n (cid:88) i =1 p i (cid:33) ˜ u (cid:48) ( P (cid:63)T ) H − T ( H T P (cid:63)T − H T Y T ) (cid:21) .
10e consider C t := ( P (cid:63)t − Y t ) H t and D t := (cid:82) t ˜ u (cid:48) ( P (cid:63)s ) H − s µ ( ds ) with µ ( ds ) = p i s = T i . Now,integration by parts yields (cid:90) T C t dD t = [ C t D t ] Tt =0 − (cid:90) T D t dC t − (cid:104) C, D (cid:105) T = C T D T − (cid:90) T D t dC t . We note that the last integral has expectation zero, since ( C t ) ≤ t ≤ T is a square-integrable mar-tingale. This means E (cid:34) n (cid:88) i =1 ˜ u (cid:48) (cid:0) P (cid:63)T i (cid:1) H − T i p i (cid:0) H T i P (cid:63)T i − H T i Y T i (cid:1)(cid:35) = E (cid:34) ( P (cid:63)T − Y T ) H T n (cid:88) i =1 ˜ u (cid:48) (cid:0) P (cid:63)T i (cid:1) H − T i p i (cid:35) . In total, we have0 ≤ E (cid:34) H T ( P (cid:63)T − Y T ) (cid:32) n (cid:88) i =1 ˜ u (cid:48) (cid:0) P (cid:63)T i (cid:1) H − T i p i + (cid:32) − n (cid:88) i =1 p i (cid:33) H − T ˜ u (cid:48) ( P (cid:63)T ) (cid:33)(cid:35) . (5.8)We note that this equality holds for any admissible Y T . Define Z := n (cid:88) i =1 ˜ u (cid:48) (cid:0) P (cid:63)T i (cid:1) H − T i p i + (cid:32) − n (cid:88) i =1 p i (cid:33) H − T ˜ u (cid:48) ( P (cid:63)T ) . Then Equation (5.8) means that E (cid:104) H T ˆ Y T Z (cid:105) ≥ Y T ≤ P ∗ T such that E (cid:104) H T ˆ Y T (cid:105) = 0and ˆ Y T H T is square-integrable. In other words, E (cid:104) ˆ˜ Y T Z (cid:105) ≥ Y T being F T -measurable such that E (cid:104) ˆ˜ Y T (cid:105) = 0 and ˆ˜ Y ≤ P ∗ T H T . Multiplying ˆ˜ Y T with a positive constant,we see that E (cid:104) ˜ Y Z (cid:105) ≥ Y such that E [ ˜ Y ] = 0 , ˜ Y ≤ P ∗ T H T ≤ δ for a δ >
0, and ˜ Y boundedfrom above else. (5.9) implies E [ ˜ Y ˜ Z ] ≥ Z := Z − E [ Z ] . (5.10)Let us now show that from (5.10) it follows that Z defined above is constant. If ˜ Z ≥ Z ≤ E [ ˜ Z ] = 0 . So assume that ˜ Z takes positive and negativevalues with strictly positive probability. Suppose then first that P { P ∗ T H T > δ, ˜ Z ≤ } > δ >
0. Define ˜ Y = − a ˜ Z> + b ˜ Z ≤ ,P ∗ T H T >δ with a, b >
0, chosen such that E [ ˜ Y ] = 0 which is possible as by assumption P [ P ∗ T H T > δ, ˜ Z ≤ >
0. Then ˜ Y ≤ P ∗ T H T ≤ δ and ˜ Y is bounded from above. Hence, denoting by ˜ Z + and ˜ Z − the positive andnegative parts of ˜ Z , respectively, (5.10) entails0 ≤ E [ ˜ Y ˜ Z ] = − a E [ ˜ Z + ] − b E [ ˜ Z − P ∗ T H T >δ ]Thus, ˜ Z + = 0 a.s. Since E [ ˜ Z ] = 0 this implies that ˜ Z = 0 a.s. From the definition of ˜ Z we canconclude then that Z is constant.Next suppose that for every δ > P [ P ∗ T H T > δ, ˜ Z ≤
0] = 0, i.e., up to a zero-set { ˜ Z ≤ } ⊂{ P ∗ T H T ≤ δ } . By the budget constraint we must have then for a δ > P [ { ˜ Z > } ∩ { P ∗ T H T > δ } ] > . (5.11)11onsider the continuous function f ( c ) = E (cid:104) − c< ˜ Z< ∞ + < ˜ Z ≤ c,P ∗ T H T >δ (cid:105) =: E [ − B + A ] , with c ≥
0. Note that if B is a zero set, then ˜ Z ≤ c a.s. and it follows then from (5.11) that A can not be a zero set. Hence, we always have that P ( A ) + P ( B ) > f (0) < c → + ∞ f ( c ) >
0. Hence, there exists ¯ c > f (¯ c ) = 0. Let¯ Y := A − B . Then E [ ¯ Y ] = 0 and ¯ Y is bounded from above and satisfies¯ Y ≤ P ∗ T H T ≤ δ. Thus, by (5.10) 0 ≤ E [ ¯ Y ˜ Z ] = E [ ˜ Z A ] − E [ ˜ Z B ] < Z ( ω ) < ˜ Z ( ω (cid:48) ) for every ω ∈ A and ω (cid:48) ∈ B and A and B have the same measure. Equation (5.12) is a contradiction. Hence, by the first part of the proof˜ Z = 0 a.s. and therefore Z = ˜ Z − E [ ˜ Z ] is constant.To obtain the representation of Z as stated in condition ( ii ), we recall our definition ν t = H − t ˜ u (cid:48) ( P (cid:63)t ) (for t ∈ { T , . . . , T n , T } ) from the very beginning of this proof.2. Let ν t be an adapted process such that conditions ( i ) and ( ii ) in Condition 1 hold. Let P (cid:63)t = i ( ν t H t ) for t ∈ { T , . . . , T n , T } , which is a wealth process by condition ( i ). Furthermore, let Y be another wealth process not a.s. equal to P (cid:63) . Then we consider as in the proof of Theorem5.1 E [ u ( Y τ ∧ T ) − u ( P (cid:63)τ ∧ T )] ≤ E [˜ u ( Y τ ∧ T ) − ˜ u ( P (cid:63)τ ∧ T )] ≤ E (cid:34) n (cid:88) i =1 p i ν T i ( Y T i − P (cid:63)T i ) H T i (cid:35) + (cid:32) − n (cid:88) i =1 p i (cid:33) E [( Y T − P (cid:63)T ) ν T H T ] , where we used in the last equation that by definition ν T i H T i is an element of the subgradient of˜ u ( P ∗ T i ). Similar to the first part of this proof E (cid:34) n (cid:88) i =1 p i ν T i ( Y T i − P (cid:63)T i ) H T i (cid:35) = E (cid:34) ( Y T − P (cid:63)T ) H T n (cid:88) i =1 p i ν T i (cid:35) by integration by parts. Overall, we have E (cid:34) n (cid:88) i =1 p i ν T i ( Y T i − P (cid:63)T i ) H T i (cid:35) + (cid:32) − n (cid:88) i =1 p i (cid:33) E [( Y T − P (cid:63)T ) ν T H T ]= E (cid:34) H T ( Y T − P (cid:63)T ) (cid:32) n (cid:88) i =1 p i ν T i + (cid:32) − n (cid:88) i =1 p i (cid:33) H − T ν T H T (cid:33)(cid:35) = 0 , where the last factor is constant by condition ( ii ). We obtain the last equality since H T ( Y T − P (cid:63)T )is a martingale. Therefore P (cid:63)t = i ( ν t H t ) is optimal for t ∈ { T , . . . , T n , T } . (cid:50) orollary 5.3 In Condition 1, ( ii ) is equivalent to: ( iii ) The random variable (cid:80) ni = k p i ν T i + (1 − (cid:80) ni = k p i ) ν T is F T k -measurable for every k = 1 , . . . , n − .Proof. Clearly ( iii ) ⇒ ( ii ). Let us now show the other direction. From the proof of Theorem 5.2 wecan actually see that there exists a function, say f , such that ν t = f ( t, P t , H t ) a.s. Now on τ > T k the corollary can be seen as in the proof of Theorem 5.2 by considering conditional expectationsinstead of classical expectations, using that P T k ∈ { U = U c } , that τ is independent of S and thatproblem (2.6) is time consistent. Hence, A = (cid:80) ni = k +1 p i ν T i + (1 − (cid:80) ni = k +1 p i ) ν T can be written as A = g (( H s ) ≤ s ≤ T k ) on τ > T k for a suitable function g . Since H is independent of τ we must havethen that A = g (( H s ) ≤ s ≤ T k ) on τ ≤ T k as well. In particular A is F T k -measurable. (cid:50) Corollary 5.4
The optimal solution P opt admits the representation P opt s = i ( ν s H s ) for s ∈ { T , . . . , T n , T } . Remark 5.5
This result generalizes the representation P (cid:63)τ ∧ T = i ( ν τ ∧ T H τ ∧ T ) of the optimal solutionto a non-concave setup (generalizing also parts of Bouchard and Pham [12]). Moreover, taking thespecial choice of τ = T P − a.s. we obtain the non-concave optimization discussed in Carpenter [15]. Remark 5.6
The theorem does not hold if T = 0 . For instance, if x is too small at time , it is notin the image of ˜ u (cid:48) and the representation P opt = i ( ν H ) does not hold. We will provide a representation of the optimal solution in the form P t = I ( ν t H t ) in the concave caseand in the non-concave case ( t ∈ { T , . . . , T n , T } ), where we are able to calculate the correspondingfunction ν t . For this, we have to do some preparations. Lemma 6.1
Let q ∈ R . It holds (for ≤ t ≤ T ) that E (cid:2) H qT |F t (cid:3) = E (cid:20)(cid:18) H T H t (cid:19) q (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) H qt = f ( q, t, T ) H qt , (6.1) with f given by f ( q, t, T ) := exp (cid:16) − q (cid:82) Tt ( r s + θ s ) ds + q (cid:82) Tt θ s ds (cid:17) . We note that the above defined function f has the property: f ( q, , T ) f ( q, t , T ) = f ( q, , t ) or equivalently f ( q, , T ) = f ( q, , t ) · f ( q, t , T ) , since the parameters are the boundaries of the integrals. The next result provides a slight generaliza-tion: Lemma 6.2
Let q ∈ R , ≤ t < T and let ν T be F t -measurable. Then it holds that E (cid:2) H qT ν T H T ≤ u (cid:48) (ˆ x ( B )) |F t (cid:3) = E (cid:20)(cid:18) H T H t (cid:19) q ν T H T ≤ u (cid:48) (ˆ x ( B )) (cid:12)(cid:12) F t (cid:21) H qt = g ( q, t, T ) H qt , (6.2) with g given by g ( q, t, T ) := f ( q, t, T ) · Φ ( d ) , where Φ denotes the cumulative distribution functionof the standard normal distribution, d ( q, t, T, W t ) = − a ( q,T ) − ( W t − ( T − t ) qθ ) √ T − t where a ( q, T ) is chosen suchthat ν T H T ≤ u (cid:48) (ˆ x ( B )) if and only if W T ≥ a ( q, T ) , meaning that a ( q, T ) = − log( u (cid:48) (ˆ x ( B ))) − log( ν T )+ q ( r + θ ) Tqθ . To allow Equation (6.2) to hold for 0 ≤ t ≤ T , we define g ( q, T, T ) := ν T H T ≤ u (cid:48) (ˆ x ( B )) .13 .2 Concave optimization with power utility We start with a Merton-type problem: V τ ( x, U ) = sup P τ ∧ T ∈ C ( x ) E [ U ( P τ ∧ T )] , where U is a strictly concave utility function and C ( x ) as in Equation (2.7) is given by C ( x ) := (cid:40) P τ ∧ T : P τ ∧ T is a non-negative G τ ∧ T -measurable random variable with H τ ∧ T P τ ∧ T square-integrable.We consider for 0 ≤ τ < T a discrete random variable, i.e., there are times T := 0 < T < T < · · ·
B > u ( x ) = (cid:40) U ( α ( x − B ) + + K ) for x ≥ , −∞ else . The objective function is given by J ( x ) = sup π ∈ Π(0 ,x ) E [ u ( P πτ ∧ T )] , under the budget constraint E [ H τ ∧ T P τ ∧ T ] ≤ x , which is a non-concave optimization problem. Asabove, we take power utility for U and the same discrete distribution of τ . We know that for t ∈{ T , . . . , T n , T } we can find a function ν such that P t = i ( ν t H t ) by Corollary 5.4, where the function i is given by Equation (5.5). Remark 6.3
At time T n the optimization problem has a known time horizon. Hence, we can calculatethe optimal solution P (cid:63)T , depending on the (random) wealth x T n := P T n (compare to Carpenter [15]).Then ν T is F T n -measurable and depending on x T n . For t > T n − , we can invest capital x T n − := P T n − in the time horizon T n − < t ≤ T . We know by Theorem 5.2 (and Corollary 5.3) that the randomvariable p n ν T n + (1 − p n ) ν T is F T n -measurable. This implies that ν T n is F T n -measurable but typicallynot F T n − -measurable. The same argument yields that every ν T i is F T i -measurable ( ≤ i ≤ n ), buttypically not F T i − -measurable. In particular, this implies that ν T is not deterministic, but F T -measurable. We note that the case n = 1 is included using the abbreviation T := 0 . The budget constraint yields E [ H τ ∧ T P τ ∧ T ] = n (cid:88) i =1 p i E [ H T i P T i ] + (cid:32) − n (cid:88) i =1 p i (cid:33) E [ H T P T ] = x, (6.9)which is an equation depending on (the random variables) ν T , . . . , ν T n , ν T . Moreover, we know for0 ≤ s ≤ T by the budget constraint H τ ∧ s P τ ∧ s = E [ H τ ∧ T P τ ∧ T |F s ∧ τ ] . We distinguish the cases:1. s = T : We have H τ ∧ s P τ ∧ s = n (cid:88) i =1 H T i P T i τ = T i + H T P T τ = T = E [ H τ ∧ T P τ ∧ T |F s ∧ τ ] = n (cid:88) i =1 H T i P T i τ = T i + E [ H T P T |F s ∧ τ ] τ = T , which is a true statement. Note that ν T can only be calculated implicitly using Equation (6.9).2. s = T n < T : We calculate H τ ∧ s P τ ∧ s = H T P T τ = T + · · · + H T n P T n τ = T n + H T n P T n τ>T n = H T P T τ = T + · · · + H T n P T n τ = T n + E [ H T P T τ>T n |F T n ] .
17e define for T i ∈ { T , . . . , T n , T } P s,T i := E (cid:20) H T i H s α − γ − γ ( H T i ν T i ) − γ ν Ti H Ti ≤ u (cid:48) (ˆ x ( B )) (cid:12)(cid:12) F s (cid:21) − (cid:18) Kα − B (cid:19) g (1 , s, T i ) . (6.10)The above equation means for τ > T n (noting that ν T is F T n -measurable) P T n = P T n ,T = α − γ − γ g (cid:18) γ − γ , T n , T (cid:19) H − γ T n ν − γ T n − (cid:18) Kα − B (cid:19) g (1 , T n , T ) . Comparing the coefficients of dP T n with the portfolio equation for the Brownian part yields π T n = θ T n γσ T n P T n + 1 σ T n (cid:18) Kα − B (cid:19) (cid:18) γ g (1 , T n , T ) θ T n − f (1 , T n , T )Φ (cid:48) ( d (1 , T n , T, W T n )) 1 √ T − T n (cid:19) + 1 σ T n ν − γ T α − γ − γ H − γ T n f (cid:18) γ − γ , T n , T (cid:19) Φ (cid:48) (cid:18) d (cid:18) γ − γ , T n , T, W T n (cid:19)(cid:19) √ T − T n , with d ( q, s, T, W s ) = − a ( q,T ) − ( W s − ( T − s ) qθ ) √ T − s as in Lemma 6.2.We know that the representation P t = i ( ν t H t ) for t ∈ { T , . . . , T n , T } holds, i.e., P T n = (cid:20) α − γ − γ ( ν T n H T n ) − γ − (cid:18) α K − B (cid:19)(cid:21) ν Tn H Tn ≤ u (cid:48) (ˆ x ( B )) , where the indicator is equal to 1 a.s. if P T n is non-zero. In the case that P T n = 0, we choose ν T n = ∞ .In the case that P T n >
0, Corollary 5.4 implies that P T n ≥ ˆ x ( B ), which means that ν T n ≤ u (cid:48) (ˆ x ( B )) H Tn .We have then ν − γ T n = − α γ − γ (cid:0) Kα − B (cid:1) (1 − g (1 , T n , T )) H − γ T n (cid:16) − g (cid:16) γ − γ , T n , T (cid:17)(cid:17) . (6.11)3. s = T j (for some fixed 1 ≤ j < n ): We note that conditional on F s ∧ τ , we know whether apossible stop occurred at some time points T , . . . , T j − or not. The martingale property yields E [ H τ ∧ T P τ ∧ T |F s ∧ τ ] = H τ ∧ s P τ ∧ s = j (cid:88) i =1 H T i P T i τ = T i + H s P s τ>T j . By the representation P t = i ( ν s H s ) for t ∈ { T j +1 , . . . , T n , T } , this is (on τ > T j = s ) H s P s = n (cid:88) i = j +1 p i E [ H T i P T i |F s ] + − n (cid:88) i = j +1 p i E [ H T P T |F s ] . With Equation (6.10), the above equation can be written as P s = n (cid:88) i = j +1 p i P s,T i + − n (cid:88) i = j +1 p i P s,T .
18. 0 < s < T : We calculate using Lemma 6.2 H s P s = n (cid:88) i =1 E [ H T i P T i |F s ] p i + (cid:32) − n (cid:88) i =1 p i (cid:33) E [ H T P T |F s ] . Hence, using Equation (6.10) P s = n (cid:88) i =1 p i P s,T i + (cid:32) − n (cid:88) i =1 p i (cid:33) P s,T . T j < s < T j +1 (for some fixed 1 ≤ j < n ): We know that the information whether stoppingoccurred at time points T , . . . , T j is included F s ∧ τ . Further note that it is not possible to stopin s due to our discrete framework. We calculate H τ ∧ s P τ ∧ s = j (cid:88) i =1 H T i P T i τ = T i + H s P s τ>T j = E [ H τ ∧ T P τ ∧ T |F s ∧ τ ]= j (cid:88) i =1 H T i P T i τ = T i + n (cid:88) i = j +1 p i E [ H T i P T i |F s ] + − n (cid:88) i = j +1 p i E [ H T P T |F s ] τ>T j . For τ > T j this is H s P s = n (cid:88) i = j +1 p i E [ H T i P T i |F s ] + − n (cid:88) i = j +1 p i E [ H T P T |F s ] . The portfolio process is given on ( τ > T j ) P s = n (cid:88) i = j +1 p i P s,T i + − n (cid:88) i = j +1 p i P s,T . T n < s < T : This case is included in the case T j < s < T j +1 with the convention (cid:80) ni = n +1 x i = 0.The portfolio process for τ > T n is given by (noting that ν T is F T n -measurable) P s = P s,T = ν − γ T α − γ − γ g (cid:18) γ − γ , s, T (cid:19) H − γ s − (cid:18) Kα − B (cid:19) g (1 , s, T ) . As above, we deduce that the optimal strategy has the form (for τ > T n ) π s = θ s γσ s P s + 1 σ s (cid:18) Kα − B (cid:19) (cid:18) γ g (1 , s, T ) θ s − f (1 , s, T )Φ (cid:48) ( d (1 , s, T, W s )) 1 √ T − s (cid:19) + 1 σ s ν − γ T α − γ − γ H − γ s f (cid:18) γ − γ , s, T (cid:19) Φ (cid:48) (cid:18) d (cid:18) γ − γ , s, T, W s (cid:19)(cid:19) √ T − s . Overall, we find the optimal portfolio process for τ > sP s = n (cid:88) i = j +1 p i P s,T i + − n (cid:88) i = j +1 p i P s,T for T j ≤ s < T j +1 (0 ≤ j ≤ n ) , T := 0 and T n +1 := T . The representation P t = i ( ν t H t ) is shown for t ∈{ T , . . . , T n , T } in Corollary 5.4. In the above derivation, we can calculate ν s explicitly for τ > s = T n as ν − γ s = − α γ − γ ( Kα − B ) (1 − g (1 ,T n ,T )) H − γTn (cid:16) − g (cid:16) γ − γ ,T n ,T (cid:17)(cid:17) in the case that P T n >
0. On the other hand, we choose ν T n = ∞ for P T n = 0. Remark 6.4
We note that we can obtain the Merton solution for power utility as a special case bysetting p i = 0 for all ≤ i ≤ n and noting that ν T is then deterministic. Further we set B = K = 0 and α = 1 . In this setup u coincides with ˜ u and therefore ˆ x = 0 . This gives a = ∞ , so Φ( a ) = 1 . Inthis case, f (cid:16) γ − γ , t , t (cid:17) = g (cid:16) γ − γ , t , t (cid:17) for any t , t ∈ [0 , T ] . Remark 6.5
The case that P ( τ = T ) = 1 has been considered in Carpenter [15]. We consider a classical Black-Scholes market with a risky asset S and a bond B . The drift µ , volatility σ and riskless interest rate r are assumed to be constant. Let θ := µ − rσ be the market price of risk(Sharp ratio). Moreover, we take time points 0 < T < T and a probability p < P ( τ = T ) = p and P ( τ = T ) = 1 − p (cid:54) = 0 . For power utility, we consider a non-concave optimization with uncertain investment horizon J ( x ) = sup π ∈ Π(0 ,x ) E [ u ( P πτ ∧ T )] , with u given by Equation (5.1) for given K >
B > E [ τ ] = ˜ T .Unless stated otherwise, for the non-concave optimization problem with certain time horizon ˜ T [15]we use the following values (as in Chen, Nguyen and Stadje [17]) on the right part of the followingtable: µ r σ γ ˜ T x p T T α B K P s = i ( ν s H s ) for s ∈ { T , T } . As statedin Remark 6.3, ν T is F T -measurable and depending on the wealth level at time T . Idea of finding ν T and ν T : Working with the extended wealth process by Theorem 5.2 we canchoose the random variables ν T and ν T and C ∈ R such that it holds that E [ H T P T |F T ] = H T P T , (7.1) pν T + (1 − p ) ν T = C, (7.2) E [ H T P T |F ] = P = x, (7.3)where Equation (7.3) can only be solved numerically.20e note that P T = 0 implies that P T = 0. In this case choose ν T = ∞ (which of course is greaterthan u (cid:48) (ˆ x ( B )) H T ). Note that ν T can be chosen F T -measurable as in Carpenter [15] simply throughensuring that (7.1) holds. For P T >
0, Equation (7.1) is equivalent to E [ H T P T |F T ] = α − γ − γ ν − γ T g (cid:18) γ − γ , T , T (cid:19) H γ − γ T − (cid:18) α K − B (cid:19) g (1 , T , T ) H T = H T P T = α − γ − γ ν − γ T H γ − γ T − H T (cid:18) Kα − B (cid:19) , with g ( q, t, T ) := f ( q, t, T ) · Φ ( d ( q, t, T, W t )), where Φ denotes the cumulative distribution functionof the standard normal distribution, d ( q, t, T, W t ) = − a ( q,T ) − ( W t − ( T − t ) qθ ) √ T − t where a ( q, T ) is chosen as a ( q, T ) = − log( u (cid:48) (ˆ x ( B ))) − log( ν T )+ q ( r + θ ) Tqθ (see Lemma 6.2 above). Putting this together with Equation(7.2) gives α − γ − γ H − γ T (cid:32)(cid:18) Cp − − pp ν T (cid:19) − γ − ν − γ T g (cid:18) γ − γ , T , T (cid:19)(cid:33) + (cid:18) Kα − B (cid:19) ( g (1 , T , T ) −
1) = 0 . (7.4)We note that the above term is in fact an implicit formula for ν T since g also depends on ν T . (a) Stopped portfolio variance (b) Certainty equivalent Figure 1: Impact of premature stopping on the portfolio variance and mean with p = 1 / Simulation strategy:
First we simulate n = 10 realizations of H (to get H T ). Then we choosea C ∈ R arbitrarily. For every H and C , we solve Equation (7.4) to find ν T . Then we find ν T byEquation (7.2). Finally, we can estimate the expectation in Equation (7.3) by taking the average overall H T P T . If the deviation is too large, we readjust the choice of C accordingly, using that ν T and ν T are monotone increasing in C . We note that our simulation idea is only based on the values at T and T . We therefore do not need to simulate the whole path of ( W s , H s ).21 iscussion : Having determined the multiplier values ν T and ν T , we can calculate the values of theoptimal stopped portfolio P τ ∧ T . To compare the performance of the optimal wealth with uncertaintime horizon with that without time horizon uncertainty, we compare the respective certanty equiv-alents in terms of expected utility. The left panel of Figure 1 depicts the variance of the stoppedportfolio. We can observe that given that E [ τ ] = ˜ T = 10, the larger the variance of τ , the higherthe variance of the stopped portfolio of the uncertain time horizon problem. These values are alwayshigher than those of the problem with certain maturity. Intuitively, to deal with the uncertain timehorizon risk, the agent might have to be more conservative in making investment decisions to beprepared for a possible premature stopping. This explains the result reported in the right panel ofFigure 1 that time horizon uncertainty overall leads to a reduced performance (in terms of certaintyequivalent) of the optimal portfolio compared to the setting with certain time-horizon.The effect of the premature stopping probability p is illustrated in Figure 2 for T = 8 , T = 12fixed. A smaller (larger) value of p generates an optimal stopped portfolio that is closer to theterminal wealth without time-horizon uncertainty with higher (smaller) time horizon, which explainsthe monotone effect of the premature stopping probability in terms of certainty equivalent. (a) Variance (b) Certainty equivalent Figure 2: Effect of the premature stopping probability with T = 8 , T = 12 fixed. We analyze a non-concave optimization with uncertain investment horizon for general non-concaveutility functions and prove that every maximizer also maximizes the concavified problem extendingthe results of Reichlin [35] to a framework with random time horizon. For a discrete distributionof the maturity τ , we prove existence and optimality characterizations. Moreover, we calculate theoptimal solution in the non-concave setting for power utility, confirming and generalizing results fromBlanchet et al. [11] to non-concave utility functions. Finally, we illustrate our findings numericallyfor the non-concave optimization problem with random maturity, comparing them with results witha fixed investment time horizon. Our results suggest that the occurrence of a possible prematurestopping leads to a reduced performance of the optimal solution.22 Proof of Theorem 3.1
We assume w.l.o.g. M = 0. Since G coincides with F before τ and the martingale representationtheorem for all t ∈ [0 , T ], there exists ¯ M t ∈ L ( d P , F t ) and ( ξ tu ) ∈ L ( d P × du, ( F u ) u ) with M t τ>t = τ>t ¯ M t = τ>t (cid:90) t ξ tu dW u . (A.1)Fix i ∈ { , . . . , n } and choose s, t ∈ [0 , T ] with T i ≤ t ≤ s < T i +1 . In the sequel, ¯ M always refersto a random variable measurable with respect to F . Let ¯ M s,t be F s -measurable with τ>s ¯ M s,t = τ>s ( M s − M t ). Noting that { τ > s } ⊆ { τ > t } and using Equation (A.1), we obtain on τ > s ¯ M s,t = ( M s − M t )= (cid:90) s ξ su dW u − (cid:90) t ξ tu dW u = (cid:90) t (cid:0) ξ su − ξ tu (cid:1) dW u + (cid:90) st ξ su dW u . We note that E (cid:2)(cid:82) st ξ su dW u |F t (cid:3) = 0. Using extensively that τ is G t -measurable, we calculate E (cid:2) ¯ M s,t |F t (cid:3) τ>s = E (cid:2) ¯ M s,t |G t (cid:3) τ>s = E (cid:2) ¯ M s,t (1 − τ ≤ T i ) |G t (cid:3) = E (cid:2) ¯ M s,t τ>s |G t (cid:3) = E [( M s − M t ) (1 − τ ≤ T i ) |G t ]= E [( M s − M t ) |G t ] − E [( M s − M t ) |G t ] τ ≤ T i = 0 , where the first equality holds since ¯ M s,t is F s -measurable and therefore independent of τ .We conclude that τ>s (cid:82) t (cid:0) ξ su − ξ tu (cid:1) dW u = 0. Thus,0 = E (cid:34) τ>s (cid:18)(cid:90) t (cid:0) ξ su − ξ tu (cid:1) dW u (cid:19) (cid:35) = P ( τ > s ) E (cid:20)(cid:90) t (cid:0) ξ su − ξ tu (cid:1) du (cid:21) This means that ξ su = ξ tu for 0 ≤ u ≤ t ≤ s < T i +1 . By the continuity of M , we obtain for generalfixed s < T t (cid:37) s E (cid:104) ( M s τ>s − M t τ>s ) (cid:105) = lim t (cid:37) s E (cid:104) ( M s τ>s − M t τ>s τ>t ) (cid:105) = P ( τ > s ) lim t (cid:37) s (cid:18) E (cid:20)(cid:90) t (cid:0) ξ su − ξ tu (cid:1) du (cid:21) + E (cid:20)(cid:90) st ( ξ su ) du (cid:21)(cid:19) ≥ . This implies that lim t (cid:37) s ξ tu = ξ su in L ( P × du ). Overall, we have ξ su = ξ T i u for u ≤ s, for T i ≤ s ≤ T i +1 < T, ≤ i ≤ n − , and ξ su = ξ T n u for u ≤ s, for T n ≤ s < T.
23n particular, ξ su and ξ s (cid:48) u coincide for 0 ≤ u ≤ s ∧ s (cid:48) < T and we define consistently ξ u := ξ su for0 ≤ u ≤ s < T . Finally, we calculate for T i ≤ t < T i +1 M τ ∧ t = i (cid:88) j =1 M T j τ = T j + M t τ>t = i (cid:88) j =1 lim s (cid:48) (cid:37) T j M s (cid:48) τ>s (cid:48) τ = T j + M t τ>t = i (cid:88) j =1 lim s (cid:48) (cid:37) T j (cid:90) s (cid:48) ξ u dW u τ = T j + τ>t (cid:90) t ξ u dW u = i (cid:88) j =1 (cid:90) T j ξ u dW u τ = T j + τ>t (cid:90) t ξ u dW u = (cid:90) t ∧ τ ξ u dW u . (cid:50) B Proofs of Chapter 6.1
Proof of Lemma 6.1 . The first equality holds since H t is F t -measurable. Therefore, we calculate: E (cid:2) H qT |F t (cid:3) = E (cid:20) exp (cid:18) − q (cid:90) t ( r s + 12 | θ s | ) ds − q (cid:90) Tt ( r s + 12 | θ s | ) ds − q (cid:90) t θ s dW s − q (cid:90) Tt θ s dW s (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = H qt exp (cid:18) − q (cid:90) Tt ( r s + 12 | θ s | ) ds + q (cid:90) Tt | θ s | ds (cid:19) = H qt f ( q, t, T ) . By the last line, we obtain the representation of the function f as f ( q, t, T ) =exp (cid:16) − q (cid:82) Tt ( r s + | θ s | ) ds + q (cid:82) Tt | θ s | ds (cid:17) (cid:50) Proof of Lemma 6.2.
We define the random variable (since ν T is F t -measurable) a ( q, T ) := − log ( u (cid:48) (ˆ x ( B ))) − log ( ν T ) + q (cid:0) r + θ (cid:1) Tqθ .
Then a is chosen such that ν T H T ≤ u (cid:48) (ˆ x ( B )) ⇐⇒ W T ≥ a ( q, T ) . Now, we define ψ ( y ) := exp (cid:18) − q (cid:18) r + 12 θ (cid:19) T − qθy (cid:19) y ≥ a . We consider (for t < T ) E [ ψ ( W T ) |F t ] = E [ ψ ( W T − W t + W t ) |F t ] = Ψ( W t ) , where Ψ is given byΨ( x ) = E [ ψ ( w + x )] = 1 (cid:112) π ( T − t ) (cid:90) ∞−∞ ψ ( w + x ) exp (cid:18) − w T − t ) (cid:19) dw = exp (cid:0) − q (cid:0) r + θ (cid:1) T (cid:1)(cid:112) π ( T − t ) (cid:90) ∞−∞ exp ( − qθy ) exp (cid:18) − ( y − x ) T − t ) (cid:19) y ≥ a dy, y = w + x in the last step. Further we note that for c := ( T − t ) qθ and denoting Φ asthe standard normal cdf we haveΨ( x ) = exp (cid:18) − q (cid:18) r + 12 θ (cid:19) T − qθx + 12 ( T − t ) q θ (cid:19) Φ (cid:18) − a ( q, T ) − ( x − c ) √ T − t (cid:19) . Finally we calculate (noting that a is F t -measurable)Ψ( W t ) = exp (cid:18) − q (cid:18) r + 12 θ (cid:19) T − qθW t + 12 ( T − t ) q θ (cid:19) Φ (cid:18) − a ( q, T ) − ( W t − c ) √ T − t (cid:19) = H qt exp (cid:18) − q (cid:18) r + 12 θ (cid:19) ( T − t ) + 12 ( T − t ) q θ (cid:19) Φ (cid:18) − a ( q, T ) − ( W t − c ) √ T − t (cid:19) . We define d ( q, t, T, W t ) := − a ( q,T ) − ( W t − ( T − t ) qθ ) √ T − t and g ( q, t, T ) := exp (cid:18) − q (cid:18) r + 12 θ (cid:19) ( T − t ) + 12 ( T − t ) q θ (cid:19) Φ (cid:18) − a ( q, T ) − ( W t − ( T − t ) qθ ) √ T − t (cid:19) . Then the statement follows by the definition of f ( q, t, T ) (in Lemma 6.1). (cid:50) References [1]
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