A Variational Analysis Approach to Solving the Merton Problem
aa r X i v : . [ q -f i n . P M ] M a r A Variational Analysis Approach to Solving the MertonProblem
Ali Al-Aradi, Sebastian Jaimungal
Department of Statistical Sciences, University of Toronto
Abstract
We address the Merton problem of maximizing the expected utility of terminal wealth using tech-niques from variational analysis. Under a general continuous semimartingale market model withstochastic parameters, we obtain a characterization of the optimal portfolio for general utility func-tions in terms of a forward-backward stochastic differential equation (FBSDE) and derive solutionsfor a number of well-known utility functions. Our results complement a previous study conducted inFerland and Watier (2008) on optimal strategies in markets driven by Brownian noise with randomdrift and volatility parameters.
Keywords:
Merton problem; Portfolio selection; Stochastic control; Convex analysis; Variationalanalysis.
1. Introduction
The Merton problem is among the most well-known and well-studied problems in mathematicalfinance. Introduced in the seminal works of Merton (1969) and Merton (1971), the problem isone of dynamic asset allocation and consumption in which an investor chooses to allocate theirwealth between a risk-free asset and a risky asset with the goal of maximizing expected utility ofconsumption and terminal wealth. Although the two initial papers consider a number of variationsto the problem such as stochastic additions to wealth other than capital gains (e.g. wages), makingthe risk-free asset defaultable and using alternatives to geometric Brownian motion for model-ing asset price behavior, the papers still managed to spawn numerous extensions in other direc-tions. For example, the incorporation of transaction costs in Magill and Constantinides (1976) andDavis and Norman (1990), uncertain investment horizon in Blanchet-Scalliet et al. (2008), taxesto capital gains in Tahar et al. (2010) and illiquid assets in Ang et al. (2014). A number of worksalso consider variations of the Merton problem with partial information, e.g. B¨auerle and Rieder ✩ SJ and AA would like to acknowledge the support of the Natural Sciences and Engineering Research Council ofCanada (NSERC). For SJ, under the funding reference numbers RGPIN-2018-05705 and RGPAS-2018-522715.
Email addresses: [email protected] (Ali Al-Aradi), [email protected] (Sebastian Jaimungal)
2. Model Setup
Let (Ω , F , F , P ) be a filtered probability space, where F = {F t } t ≥ is the natural filtrationgenerated by all processes in the model. We assume that the market consists of n risky assets andone risk-free asset which are defined as follows: Definition 1.
The stock price process for risky asset i , P i = ( P it ) t ≥ for all i ∈ N := { , . . . , n } ,is a positive semimartingale satisfying the stochastic differential equation (SDE): dP it = α it P it dt + P it dM it , P i = p i , (2.1)2 here α i = ( α it ) t ≥ is an F -adapted process representing the asset’s instantaneous rate of return and M i = ( M it ) t ≥ is an F -martingale with M i = 0 representing the asset’s noise component .The risk-free asset price process , P = ( P t ) t ≥ is a positive semimartingale satisfying the SDE dP t = r t P t dt , P = p , (2.2) where r = ( r t ) t ≥ is an F -adapted process representing the risk-free rate. Next, we specify the assumptions made on the various market model processes. To this end, wefirst define the spaces L pT ( R n ) = (cid:26) f : Ω × [0 , T ] → R n s.t. E (cid:20)Z T ( k f t k p ) p dt (cid:21) < ∞ (cid:27) , < p < ∞ , (2.3)and L ∞ ,MT ( R n ) = ( f : Ω × [0 , T ] → R n s.t. sup t ∈ [0 ,T ] k f t k ∞ ≤ M, P − a.s. ) , (2.4) where k x k p := ( P ni =1 | x i | p ) /p and k x k ∞ := max i ∈ N | x i | for x ∈ R n denote the p -norm and ∞ -norm on R n , respectively. Furthermore, we will make use of the shorthand notation k x k := k x k to denotethe usual Euclidean norm. Assumption 1.
The risk-free rate and rate of return processes are continuous and bounded, i.e. r ∈ L ∞ ,MT ( R ) , α ∈ L ∞ ,MT ( R n ) , where α t = ( α t , ..., α nt ) ⊺ . Additionally, the martingale noise pro-cesses are assumed to be continuous with finite second moments, i.e. E ( k M t k ) < ∞ for all t ≥ . We also assume that the quadratic co-variation processes associated with the noise componentsatisfy
Assumption 2.
Let Σ be the matrix whose ij -th element is the quadratic covariation processbetween M i and M j , Σ ijt := h M i , M j i t . We assume that, for each x ∈ R n , there exists ε > and C < ∞ such that ε k x k ≤ x ⊺ Σ t x ≤ C k x k , ∀ t ≥ . (2.5)This is an extension of the usual non-degeneracy and bounded variance conditions. Note thatsince M is continuous, Σ is a continuous process as well.Next, we define portfolio processes which will constitute the investor’s control in the optimizationproblem. 3 efinition 2. A portfolio is an F -predictable, vector-valued process π = ( π t ) t ≥ , with π ∈ L T ( R n ) where π t = ( π t , ..., π nt ) ⊺ such that for all t ≥ , π it represents the proportion of wealthinvested in risky asset i and π t = 1 − π t − · · · − π nt is the proportion invested in the risk-free asset.We denote the set of all portfolios by A = n π : Ω × [0 , T ] → R n s.t. π ∈ L T ( R n ) , F -predictable o . (2.6)Given the model dynamics and portfolio assumptions, the portfolio value process X π =( X π t ) t ≥ associated with an arbitrary portfolio π satisfies the SDE dX π t = X π t (cid:0) r t + π ⊺ t θ t (cid:1) dt + X π t π ⊺ t d M t , X π = x > , (2.7)where θ t = α t − r t is the vector of excess returns, is a vector of ones and x is the investor’s initialwealth. It will also be convenient at times to work with the logarithm of wealth, which satisfiesthe SDE d log X π t = γ π t dt + π ⊺ t d M t , (2.8)where γ π t = r t + π ⊺ t θ t − π ⊺ t Σ t π t is the portfolio growth rate .Due to some technical requirements that will come into play when solving the optimizationproblem, we initially restrict ourselves to strategies where wealth is transferred to the risk-free assetfor the remainder of the investment horizon once a certain level of wealth is reached. To formalizethis restriction, we first define the K -stopped version of π as the portfolio π K = ( π K,t ) t ≥ givenby π K,t = ( π t , if | X π t | ≤ K , if | X π t | > K We also define the stopping time associated with reaching the wealth threshold, namely τ π K = inf (cid:8) t ≥ | X π t | > K (cid:9) , (2.9)along with the associated indicator process π K = ( π K t ) t ≥ defined as π K t = { τ π K > t } . (2.10)Finally, we define the set of constrained portfolios for a given wealth threshold. Definition 3.
The set of admissible portfolios for the K -constrained problem consists ofthose strategies that are stopped once the wealth threshold K is reached, denoted A K = (cid:8) π K : π ∈ A (cid:9) . (2.11)4 . Stochastic Control Problem The stochastic control problem we consider is the Merton problem without consumption. Morespecifically, the investor’s objective is to determine the portfolio process π ∈ A that maximizestheir expected utility of terminal wealth at the end of their investment horizon T . In mathematicalterms our stochastic control problem is to find the optimal portfolio π ∗ which, if the supremumis attained in the set of admissible strategies, achievessup π ∈A H ( π ) , (3.1)where H : A → R is the performance criteria of an admissible portfolio π ∈ A given by H ( π ) := E [ U ( X π T )] . (3.2)We approach solving (3.1) by solving a sequence of nested constrained problems where thesearch space is reduced to A K , namely sup π K ∈A K H K ( π K ) , (3.3)where H K : A K → R is the performance criteria of a K -stopped portfolio π K ∈ A K given by H K ( π K ) := E [ U ( X π K T )] . (3.4)In the expressions above, U is a von Neumann-Morgenstern utility function which reflects theinvestor’s preferences and satisfies Assumption 3.
The investor’s utility function U is three times continuously differentiable, in-creasing and strictly concave, i.e. U (1) ( x ) > and U (2) ( x ) < for all x > , where U ( k ) is the k thderivative of U . For convenience we will define the utility process Z π = ( Z π t ) t ≥ as Z π t := U ( X π t ) . (3.5)We proceed with solving the optimal control problem in four parts: (i) we establish the existenceand uniqueness of a global optimizer for the stochastic control problems (3.1) and (3.3); (ii) wecompute the Gˆateaux derivative associated with the functional H K ; (iii) we find an element in theadmissible set A K which makes the derivative vanish and relate it to the solution of a FBSDE; (iv)we take the limit as K tends to infinity to obtain an expression for π ∗ .For the remainder of the paper we address the constrained problem (3.3) with a fixed K un-less explicitly stated and we will omit the subscripts from the control processes for notationalconvenience. 5 .1. Existence and Uniqueness of a Global Maximum To show the existence and uniqueness of a global optimizer for (3.3), we use the strict concavityof a related control problem that uses the dollar amount process as the investor’s control anddemonstrate a one-to-one correspondence between the control processes of the two problems. Ananalogous argument can be used to prove a similar result for (3.1), which we omit.
Proposition 1.
The stochastic control problem (3.3) has a unique global maximizer.
Proof.
We define an auxiliary control problem with the same performance criteria as (3.4) butwhere the control process is a vector of dollar amounts , rather than proportions of wealth, investedin each asset and denote the new control process by e π = ( e π t ) t ≥ where e π t = ( e π t , ..., e π nt ) ⊺ . Inparticular, we are interested in sup e π ∈A ∗ K J ( e π ) , (3.6)where J : A ∗ K → R is given by J ( e π ) := E (cid:2) U (cid:0) X e π T (cid:1)(cid:3) and A ∗ K is the set of admissible portfolios expressed in terms of dollar amounts. The controlprocesses in the two optimization problems are related via π it = e π it e π t + · · · + e π nt for i ∈ N , or through the wealth process as follows X π t π t = e π t . Given a fixed initial wealth and using the fact that portfolios are self-financing, there exists a one-to-one mapping between A K and A ∗ K . Additionally, the numerical value of the two functionals H and J are equal when taking two controls that map to one another. This implies that if theauxiliary control problem has a unique global maximizer then so does the control problem (3.3).To show that the auxiliary control problem has a unique solution we show that the functional J is strictly concave in the control e π and that the search space A ∗ K is a convex set. This is done intwo separate lemmas. Lemma 1.
The functional J is strictly concave in the dollar amount process e π . Proof.
The wealth process controlled via the dollar amounts follows the dynamics dX e π t = (cid:0) r t X e π t + e π ⊺ t θ t (cid:1) dt + e π ⊺ t d M t , X e π = x > . (3.7)6his process can be linearized by defining κ ,t X e π t where κ s,t = e − R ts r u du . This process satisfies d (cid:0) κ ,t X e π t (cid:1) = κ ,t dX e π t − r t κ ,t X e π t dt = κ ,t e π ⊺ t θ t dt + κ ,t e π ⊺ t d M t . Integrating we obtain an expression for X e π T X e π T = x κ T, + Z T κ T,t e π ⊺ t θ t dt + Z T κ T,t e π ⊺ t d M t . Since this expression is linear in e π , for any e π , e ω ∈ A ∗ K and c ∈ [0 ,
1] we have X c e π +(1 − c ) e ω T = cX e π T + (1 − c ) X e ω T , and by the strict concavity of U it follows that U (cid:16) X c e π +(1 − c ) e ω T (cid:17) > c U ( X e π T ) + (1 − c ) U ( X e ω T ) . Taking expectations establishes the strict concavity of the functional J . Lemma 2.
The search space A ∗ K for the auxiliary control problem (3.6) is a convex set. Proof.
Fix c ∈ [0 ,
1] and e π , e π ∈ A ∗ K and consider the convex combination e ω = c e π + (1 − c ) e π .The goal is to show that this is an admissible dollar amount process. Clearly, e ω is F -predictableand corresponds to a portfolio process in A K , so all that remains is to show that it is stopped once X e ω hits the wealth threshold K .Recall from the proof of the previous lemma that X e ω = cX e π + (1 − c ) X e π . Moreover, if asubportfolio e π i is not stopped then the absolute value of its associated wealth is necessarily less thanthe threshold K . With this in mind, there are three cases to consider: neither of the subportfoliosis stopped, exactly one subportfolio is stopped or both subportfolios are stopped. In the first twocases it can be easily verified that | X e ω | ≤ K and in the last case | X e ω | = K and the portfolio e ω isstopped since both e π and e π are stopped. Therefore, e ω ∈ A ∗ K and hence A ∗ K is a convex set.Since the functional J is strictly concave and the set A ∗ K is convex, the auxiliary control problem(3.6) has a unique global maximizer and therefore so does the constrained control problem (3.3).This completes the proof. In this section we derive an expression for the Gˆateaux derivative of the functional H K in anumber of incremental steps, starting with the following lemma:7 emma 3. Fix ǫ > and two portfolio processes π , ω ∈ A K and define the following processes: F π , ( k ) t := U ( k ) ( X π t )( X π t ) k for k = 1 , , , (3.8a) I ω t := Z t ω ⊺ u ( θ u − Σ u π u ) du + Z t ω ⊺ u d M u , (3.8b) g π t := F π , (1) t θ t + F π , (2) t Σ t π t , (3.8c) h π t := (cid:16) F π , (1) t + F π , (2) t (cid:17) ( r t + π ⊺ t θ t ) + (cid:16) F π , (2) t + F π , (3) t (cid:17) π ⊺ t Σ t π t . (3.8d) Then we have H K ( π + ǫ ω ) = E [ Z π T ∧ τ π + ǫ ω ] + ǫ E "Z T ∧ τ π + ǫ ω (cid:16) ω ⊺ t g π t + I ω t h π t (cid:17) dt + o ( ǫ ) . (3.9) Proof.
First, using (2.7) and the definition of F π , ( k ) t , we obtain the dynamics of Z π t by applyingItˆo’s lemma which gives dZ π t = (cid:16) F π , (1) t (cid:0) r t + π ⊺ t θ t (cid:1) + F π , (2) t π ⊺ t Σ t π t (cid:17) dt + F π , (1) t π ⊺ t d M t . (3.10) We are interested in the dynamics of the perturbed utility process Z π + ǫ ω , i.e. the utility processinduced by the control π + ǫ ω . It is important to note that in order for π + ǫ ω to be an admissibleportfolio it must be stopped in the usual manner once its associated wealth process reaches thethreshold K . To arrive at the SDE satisfied by this process, we first perturb the growth rate γ π ,then the wealth process X π and the auxiliary processes F π , (1) and F π , (2) , and finally Z π . In eachstep, we write the perturbed process as a sum of the unperturbed process and an adjustment termthat is linear in ǫ plus higher order terms. The perturbed growth rate is γ π + ǫ ω t = r t + ( π t + ǫ ω t ) ⊺ θ t − ( π t + ǫ ω t ) ⊺ Σ t ( π t + ǫ ω t )= r t + π ⊺ t θ t + ǫ ω ⊺ t θ t − h π ⊺ t Σ t π t + 2 ǫ ω ⊺ t Σ t π t + o ( ǫ ) i = γ π t + ǫ ω ⊺ t ( θ t − Σ t π t ) + o ( ǫ ) . Next, we derive the perturbed wealth process using the perturbed growth rate process obtainedabove. Substituting, we have d log X π + ǫ ω t = γ π + ǫ ω t dt + ( π t + ǫ ω t ) ⊺ d M t = (cid:16) γ π t + ǫ ω ⊺ t ( θ t − Σ t π t ) + o ( ǫ ) (cid:17) dt + π ⊺ t d M t + ǫ ω ⊺ t d M t = d log X π t + ǫ n ω ⊺ t ( θ t − Σ t π t ) dt + ω ⊺ t d M t o + o ( ǫ ) . Rearranging the equation above and integrating, while noting that X π + ǫ ω = X π = x , yieldslog (cid:18) X π + ǫ ω t X π t (cid:19) − log (cid:18) X π + ǫ ω X π (cid:19) = ǫ (cid:26)Z t ω ⊺ u ( θ u − Σ u π u ) du + Z t ω ⊺ u d M u (cid:27) + o ( ǫ ) . X π + ǫ ω t = X π t exp (cid:0) ǫI ω t + o ( ǫ ) (cid:1) = ⇒ X π + ǫ ω t = X π t (1 + ǫI ω t + o ( ǫ )) . (3.11)Prior to considering the auxiliary processes F π , (1) and F π , (2) , we need to write the derivatives of U (cid:0) X π + ǫ ω t (cid:1) in terms of the unperturbed wealth process. To do so, we use the expression in (3.11)to write U ( k ) (cid:0) X π + ǫ ω t (cid:1) = U ( k ) (cid:16) X π t + ǫX π t I ω t + o ( ǫ ) (cid:17) . Since U is sufficiently differentiable we can write this expression as a Taylor series around X π t ,namely U ( k ) (cid:16) X π t + ǫX π t I ω t + o ( ǫ ) (cid:17) = U ( k ) ( X π t ) + ǫU ( k +1) ( X π t ) X π t I ω t + o ( ǫ ) . (3.12)Using (3.11) and (3.12), the perturbed auxiliary processes F π + ǫ ω , ( k ) for k = 1 , F π + ǫ ω , ( k ) t = U ( k ) ( X π + ǫ ω t )( X π + ǫ ω t ) k = (cid:16) U ( k ) ( X π t ) + ǫU ( k +1) ( X π t ) X π t I ω t + o ( ǫ ) (cid:17)(cid:16) X π t + ǫX π t I ω t + o ( ǫ ) (cid:17) k = (cid:16) U ( k ) ( X π t ) + ǫU ( k +1) ( X π t ) X π t I ω t + o ( ǫ ) (cid:17)(cid:16) ( X π t ) k + ǫk ( X π t ) k I ω t + o ( ǫ ) (cid:17) = F π , ( k ) t + ǫI ω t (cid:0) kU ( k ) ( X π t )( X π t ) k + U ( k +1) ( X π t )( X π t ) k +1 (cid:1) + o ( ǫ )= F π , ( k ) t + ǫI ω t (cid:16) kF π , ( k ) t + F π , ( k +1) t (cid:17) + o ( ǫ ) . (3.13)Now we have all the components to find the dynamics of the perturbed utility process Z π + ǫ ω .Starting from (3.10) and using (3.11) and (3.13) dZ π + ǫ ω t = (cid:16) F π + ǫ ω , (1) t (cid:0) r t + ( π t + ǫ ω t ) ⊺ θ t (cid:1) + F π + ǫ ω , (2) t ( π t + ǫ ω t ) ⊺ Σ t ( π t + ǫ ω t ) (cid:17) dt + F π t + ǫ ω , (1) t ( π t + ǫ ω t ) ⊺ d M t = (cid:18) h F π , (1) t + ǫI ω t (cid:16) F π , (1) t + F π , (2) t (cid:17)i (cid:0) r t + π ⊺ t θ t + ǫ ω ⊺ t θ t (cid:1) + h F π , (2) t + ǫI ω t (cid:16) F π , (2) t + F π , (3) t (cid:17)i h π ⊺ t Σ t π t + 2 ǫ ω ⊺ t Σ t π t i(cid:19) dt + h F π , (1) t + ǫI ω t (cid:16) F π , (1) t + F π , (2) t (cid:17)i ( π t + ǫ ω t ) ⊺ d M t + o ( ǫ )= dZ π t + ǫ (cid:26) F π , (1) t ω ⊺ t θ t + I ω t (cid:16) F π , (1) t + F π , (2) t (cid:17) ( r t + π ⊺ t θ t )+ F π , (2) t ω ⊺ t Σ t π t + I ω t (cid:16) F π , (2) t + F π , (3) t (cid:17) π ⊺ t Σ t π t (cid:17)(cid:27) dt + ǫ (cid:26) F π , (1) t ω ⊺ t + I ω t (cid:16) F π , (1) t + F π , (2) t (cid:17) π ⊺ t (cid:27) d M t + o ( ǫ ) . Since Z π = Z π + ǫ ω = U ( x ), integrating both sides of the equation above from 0 to T ∧ τ π + ǫ ω andtaking expectations yields the desired result provided that the stochastic integral on the RHS ofthe equation has zero mean, which we prove in the following lemma:9 emma 4. For any constrained admissible controls π , ω ∈ A K we have F π , (1) , F π , (2) ∈ L ∞ ,MT ( R ) and E "Z T ∧ τ π + ǫ ω (cid:26) F π , (1) t ω ⊺ t + (cid:16) F π , (1) t + F π , (2) t (cid:17) I ω t π ⊺ t (cid:27) d M t = 0 . Proof.
The first statement follows from the definition of the constrained admissible set. Namely,the fact that portfolios are stopped at a certain wealth threshold and that the risk-free rate isbounded implies that wealth is bounded on the interval [0 , T ]. Furthermore, since F π , ( k ) are con-tinuous functions of wealth they must also be bounded on this interval.Next, we rewrite the stochastic integral as Z T π + ǫ ω t (cid:26) F π , (1) t ω ⊺ t + (cid:16) F π , (1) t + F π , (2) t (cid:17) I ω t π ⊺ t (cid:27) d M t . The goal is to show that the stochastic integral under the expectation, which we denote V t , is alocal martingale. If this is the case then there exists a sequence of stopping times T n ↑ ∞ a.s. suchthat V T n ∧ t is a martingale for each n . By choosing n ∗ = inf { n : T n > T } so that T n ∗ ∧ T = T and V T n ∗ ∧ t is a martingale which would give0 = V = V T n ∗ ∧ = E [ V T n ∗ ∧ T ] = E [ V T ]as required.To show that V is a local martingale we begin with the following observation: any integral withrespect to M where the integrand is predictable and in L T ( R n ) is a continuous local martingaleby Theorem 30, Ch. IV of Protter (2005). Applying this to (2.8) it follows that the wealth process X π has continuous paths. Furthermore, since U ( k ) is continuous, F π , ( k ) has continuous paths for all k ∈ N as well. Since F π , ( k ) is also F -adapted we can conclude that it is F -predictable. The indicatoris also F -predictable by the continuity of the wealth paths. Additionally, since F π , (1) ∈ L ∞ ,MT and ω ∈ L T ( R n ) is predictable R T π + ǫ ω t F π , (1) t ω ⊺ t d M t is a (continuous) local martingale.Next, we show that W t := R T π + ǫ ω t (cid:16) F π , (1) t + F π , (2) t (cid:17) I ω t π ⊺ t d M t is also a (continuous) localmartingale. By similar reasoning as above I ω is continuous as it is the sum of an ordinary integraland an integral with respect to a continuous martingale, M , with a predictable integrand, ω , thatis in L T ( R n ). Since I ω is continuous and adapted it is also predictable and hence the integrand in W is predictable. Furthermore, the quadratic variation of W satisfies Z t π + ǫ ω t ( I ω t ) π ⊺ t Σ t π t dt ≤ C (cid:18) sup s ∈ [0 ,t ] ( I ω s ) (cid:19) (cid:18)Z t k π t k dt (cid:19) < ∞ a.s. for all t ≥ . The RHS of the inequality follows because I ω is continuous and hence bounded on compact setsand because E hR t k π t k dt i < ∞ implies that R t k π t k dt < ∞ a.s. It follows by Theorem 30,Ch. IV of Protter (2005) that W t (and hence V t ) is a continuous local martingale and the proof iscomplete. 10his completes the proof of the proposition.The following result is used to simplify the expression for the Gˆateaux derivative, particularly,to handle the term I ω t h π t . Lemma 5.
Let a = ( a t ) t ≥ , b = ( b t ) t ≥ , ℓ = ( ℓ t ) t ≥ be processes with a, ℓ ∈ L T ( R ) and b ∈ L T ( R n ) and F -predictable and let τ be an F -stopping time with τ ≤ T . Then, E (cid:20)Z τ ℓ t (cid:18)Z t a u du + Z t b ⊺ u d M u (cid:19) dt (cid:21) = E (cid:20)Z τ a t (cid:18) M t − Z t ℓ u du (cid:19) dt (cid:21) + E (cid:20)Z τ b ⊺ t d hM , M i t (cid:21) , (3.14) where M t = E t (cid:20)Z τ ℓ u du (cid:21) := E (cid:20)Z τ ℓ u du (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) and d hM , M i is a vectorized version of d hM , M i i . Proof.
We treat the two integrals on the LHS of the equation above separately. For the firstintegral we begin by demonstrating that the integral is finite. Let λ denote the Lebesgue measureon ( R , B ( R )). Then we have E (cid:20)Z τ ℓ t (cid:18)Z t a u du (cid:19) dt (cid:21) = Z Ω × [0 ,τ ] (cid:26) ℓ t (cid:18)Z t a u du (cid:19)(cid:27) ( P × λ )( dω, dt ) ≤ "Z Ω × [0 ,T ] ( ℓ t ) ( P × λ )( dω, dt ) Ω × [0 ,T ] (cid:18)Z t a u du (cid:19) ( P × λ )( dω, dt ) Cauchy-Schwarz inequality= E "Z T ( ℓ t ) dt E "Z T (cid:18)Z t a u du (cid:19) dt = E "Z T ( ℓ t ) dt T E "(cid:18)Z t a u du (cid:19) dt ! Tonelli’s theorem ≤ E "Z T ( ℓ t ) dt T E (cid:20)Z t ( a u ) du (cid:21) dt ! Jensen’s inequality < ∞ since a, ℓ ∈ L T ( R ). This allows us to change the order of integration by applying Fubini’s to write E "Z τ Z t ℓ t a u du dt = E "Z τ Z τu ℓ t a u dt du change order of integration= E "Z τ (cid:18)Z τu ℓ t dt (cid:19) a u du = E "Z τ E u (cid:20)Z τu ℓ t dt (cid:21) a u du tower property and Fubini’s theorem= E "Z τ (cid:18) M u − Z u ℓ t dt (cid:19) a u du . E (cid:20)Z τ ℓ t (cid:18)Z t b ⊺ u d M u (cid:19) dt (cid:21) = Z Ω × [0 ,τ ] (cid:26) ℓ t (cid:18)Z t b ⊺ u d M u (cid:19)(cid:27) ( P × λ )( dω, dt ) ≤ "Z Ω × [0 ,T ] ( ℓ t ) ( P × λ )( dω, dt ) Ω × [0 ,T ] (cid:18)Z t b ⊺ u d M u (cid:19) ( P × λ )( dω, dt ) Cauchy-Schwarz inequality= E "Z T ( ℓ t ) dt E "Z T (cid:18)Z t b ⊺ u d M u (cid:19) dt = E "Z T ( ℓ t ) dt T E "(cid:18)Z t b ⊺ u d M u (cid:19) dt Tonelli’s theorem= E "Z T ( ℓ t ) dt T E (cid:20)Z t b ⊺ u Σ u b u du (cid:21) dt Itˆo’s isometry ≤ E "Z T ( ℓ t ) dt T E (cid:20)Z t C k b u k du (cid:21) dt< ∞ since ℓ ∈ L T ( R ), b ∈ L T ( R n ). This allows us to once again change the order of integration to write E " Z τ ℓ t (cid:18)Z t b ⊺ u d M u (cid:19) dt = E " Z τ (cid:18)Z τt ℓ u du (cid:19) b ⊺ t d M t = E " Z τ (cid:18)Z τ ℓ u du − Z t ℓ u du (cid:19) b ⊺ t d M t = E " Z τ (cid:18) M τ − Z t ℓ u du (cid:19) b ⊺ t d M t . Now we have two terms to consider: Z = E " Z τ M τ b ⊺ t d M t and Z = E " Z τ (cid:18)Z t ℓ u du (cid:19) b ⊺ t d M t . Denote L t := R t ℓ u du and note that it is continuous in t . The integrand of the stochastic integralappearing in Z is predictable since it is the product of a predictable process and a continuousadapted process. The quadratic variation of the stochastic integral is Z T { τ ≤ T } L t b ⊺ t Σ t b t dt ≤ C Z T L t k b t k dt ≤ C sup t ∈ [0 ,T ] L t ! (cid:18)Z T k b t k dt (cid:19) < ∞ a.s. since b ∈ L T ( R n ) and L is continuous. Z = 0. For Z we have E " Z τ M τ b ⊺ t d M t = E " M τ Z τ b ⊺ t d M t = E " (cid:18)Z τ d M t (cid:19) (cid:18)Z τ b ⊺ t d M t (cid:19) = E " Z τ b ⊺ t d hM , M i t The last step follows by Itˆo’s isometry since both M and M are square integrable martingales and b ∈ L T ( R n ).We are now ready to compute the Gˆateaux derivative for our performance criteria. Proposition 2.
The functional H K : A K → R is Gˆateaux differentiable for all π , ω ∈ A K withGˆateaux derivative H ′ K ( π ) given by (cid:10) ω , H ′ K ( π ) (cid:11) = E " Z T ∧ τ π ω ⊺ t (cid:26)(cid:18) g π t + (cid:18) M π t − Z t h π u du (cid:19) ( θ t − Σ t π t ) (cid:19) dt + d hM π , M i t (cid:27) , (3.15) where M π t := E t (cid:20)Z T ∧ τ π h π t dt (cid:21) is an F -martingale with E [( M π t ) ] < ∞ for all t. Proof.
First, notice that lim ǫ → E [ Z π T ∧ τ π + ǫ ω ] = E [ Z π T ∧ τ π ] = E [ Z π T ] , with the last equality following from the fact that portfolios are stopped at τ π . This allows us towrite lim ǫ → H K ( π + ǫ ω ) − E [ Z T ∧ τ π + ǫ ω ] ǫ = lim ǫ → H K ( π + ǫ ω ) − H K ( π ) ǫ = h ω , H ′ K ( π ) i . So by rearranging (3.9) and taking the limit as ǫ tends to 0 we find that h ω , H ′ K ( π ) i = E (cid:20)Z T ∧ τ π (cid:16) ω ⊺ t g π t + I ω t h π t (cid:17) dt (cid:21) . Next, we use the fact that h as well as the integrands in I ω t of (3.8b) are in L T ( R ) to apply Lemma 5and simplify to obtain the expression in (3.15). Finally, notice that since M π is a Doob martingaleand h ∈ L T ( R ) it is in fact a true martingale with finite second moment.The next lemma gives an explicit representation of M π t − R t h π u du and d hM π , M i t which willallow us to simplify the Gˆateaux derivative and eventually solve the optimal control problem.13 emma 6. Define the processes q π = ( q π t ) t ≥ and Y π = ( Y π t ) t ≥ as q π t := F π , (1) t + F π , (2) t , and (3.16) Y π t := E t (cid:20) exp (cid:18)Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u (cid:19)(cid:21) > . (3.17) Further, write Y π as the solution to the SDE dY π t = Y π t µ π t dt + Y π t ( σ π t ) ⊺ d M t . (3.18) Then we have the following: ( i ) M π t − Z t h π u du = F π , (1) t ( Y π t −
1) (3.19)( ii ) d hM π , M i t = Σ t h ( Y π t − q π t π t + F π , (1) t Y π t σ π t i dt (3.20) Proof.
To demonstrate the first statement, we apply Itˆo’s lemma and product rule to obtain dF π , (1) t = (cid:20) (cid:16) F π , (1) t + F π , (2) t (cid:17) ( r t + π ⊺ t θ t ) + (cid:16) F π , (2) t + F π , (3) t (cid:17) π ⊺ t Σ t π t (cid:21) dt + (cid:16) F π , (1) t + F π , (2) t (cid:17) π ⊺ t d M t = h π t dt + q π t π ⊺ t d M t . Next, we write dF π , (1) t F π , (1) t = 1 F π , (1) t h h π t dt + q π t π ⊺ t d M t i , and therefore F π , (1) T ∧ τ π = F π , (1) t exp (cid:18)Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u (cid:19) . Then, noting that E t hR T ∧ τ π t q π u π ⊺ u d M u i = 0 since q π is bounded and π ∈ L T ( R n ) and F -predictable, we have M π t − Z t h π u du = E t (cid:20)Z T ∧ τ π t h π u du (cid:21) = E t (cid:20)Z T ∧ τ π t h π u du + Z T ∧ τ π t q π u π ⊺ u d M u (cid:21) = E t (cid:20)Z T ∧ τ π t dF π , (1) u (cid:21) = E t h F π , (1) T ∧ τ π − F π , (1) t i = F π , (1) t ( Y π t − , which completes the proof of the first statement.14ext, we are interested in the quadratic covariation process hM π , M i t . For this we first write M π t = Z t h π u du + F π , (1) t ( Y π t − ⇒ d M π t = h π t dt + h d (cid:16) F π , (1) t ( Y π t − (cid:17)i = h π t dt + h ( Y π t − dF π , (1) t + F π , (1) t dY π t + d h Y π , F π , (1) i t i = (cid:16) ( Y π t − q π t π ⊺ t + F π , (1) t Y π t ( σ π t ) ⊺ (cid:17) d M t , (3.21)where the drift term is zero since M π is a martingale. This allows us to identify hM π , M i t as theexpression given in (3.20).Next, we provide a simplified expression for the Gˆateaux derivative using the last two results. Corollary 1.
The Gˆateaux derivative given in Proposition 2 can be written as h ω , H ′ K ( π ) i = E " Z T ∧ τ π ω ⊺ t Y π t (cid:16) g π t + F π , (1) t Σ t σ π t (cid:17) dt . (3.22) Proof.
Using Lemma 6 and Proposition 2 we can write the Gˆateaux derivative as h ω , H ′ K ( π ) i = E " Z T ∧ τ π ω ⊺ t n g π t + F π , (1) t ( Y π t −
1) ( θ t − Σ t π t ) + Σ t h ( Y π t − q π t π t + F π , (1) t Y π t σ π t io dt . Recalling that g π t = F π , (1) t θ t + F π , (2) t Σ t π t and q π t = F π , (1) t + F π , (2) t , substituting these terms intothe expression above and simplifying yields the result.The final result we require before deriving the optimal control is a statement concerning theprocess pair ( Y π , σ π ) that appear in Lemma 6. Lemma 7.
The pair ( Y π , σ π ) defined in the Lemma 6 satisfy the backward stochastic differentialequation (BSDE) d log Y π t = − π t (cid:16) h π t F π , (1) t + q π t F π , (1) t ( σ π t ) ⊺ Σ t π t + ( σ π t ) ⊺ Σ t σ π t (cid:17) dt + π t ( σ π t ) ⊺ d M t log Y π T = 0 (3.23) Furthermore, σ π ∈ L T ( R n ) and is F -predictable. roof. We begin by writing Y π t = E t (cid:20) exp (cid:18)Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u (cid:19)(cid:21) = E t (cid:20) exp (cid:18)Z Tt π u (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + Z Tt π u q π u F π , (1) u π ⊺ u d M u (cid:19)(cid:21) = E t (cid:20) Γ T Γ t (cid:21) = ⇒ Γ t Y π t = E t [Γ T ] , where Γ satisfies the SDE d Γ t = Γ t π t (cid:16) h π u F π , (1) u dt + q π t F π , (1) t π ⊺ t d M t (cid:17) , Γ = 1 . Also, since Y π stops once τ π is reached we may write dY π t = π t Y π t µ π t dt + π t Y π t ( σ π t ) ⊺ d M t , Y π T = 1 . To find µ π we apply Itˆo’s product rule to obtain d ( Y π t Γ t ) = Y π t d Γ t + Γ t dY π t + d [Γ , Y π ] t = Y π t Γ t π t (cid:16) h π t F π , (1) t dt + q π t F π , (1) t π ⊺ t d M t (cid:17) + Γ t π t (cid:16) Y π t µ π t dt + Y π t ( σ π t ) ⊺ d M t (cid:17) + Y π t Γ t π t q π t F π , (1) t ( σ π t ) ⊺ Σ t π t dt = Y π t Γ t (cid:16) π t µ π t + π t h π t F π , (1) t + π t q π t F π , (1) t ( σ π t ) ⊺ Σ t π t (cid:17) dt + π t Y π t Γ t (cid:16) σ π t + q π t F π , (1) t π t (cid:17) ⊺ d M t . Since Y π Γ is a martingale, the drift term in the SDE above must be equal to zero. Therefore, µ π is given by µ π t = − (cid:16) h π t F π , (1) t + q π t F π , (1) t ( σ π t ) ⊺ Σ t π t (cid:17) . Substituting back into the SDE satisfied by Y π and applying Itˆo’s lemma yields the result.Next, we have that E "Z T ( Y π t ) dt = E Z T ( E t " exp Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt ≤ E "Z T E t " exp Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + 2 Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt = Z T E " E t " exp Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + 2 Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt = Z T E " exp Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + 2 Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt = Z T E " exp Z T ∧ τ π t (cid:20) h π u F π , (1) u − (cid:16) q π u F π , (1) u (cid:17) π ⊺ u Σ u π u (cid:21) du + 2 Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt = Z T E " exp Z T ∧ τ π t h π u F π , (1) u ! E Z T ∧ τ π t q π u F π , (1) u π ⊺ u d M u ! dt< ∞ L T ( R n ), respectively.This implies that log Y π T ∈ L T ( R ) by Jensen’s inequality. However, we also have thatlog Y π t = Z t (cid:0) µ π u − h Y π , Y π i u (cid:1) du + Z t ( σ π u ) ⊺ d M u , which is in L T ( R ) and F -adapted only if σ π ∈ L T ( R n ) and F -predictable. We proceed to finding the optimal control for the stochastic control problem (3.1). To thisend, we use the results of the previous section to find the unique control that causes the Gˆateauxderivative to vanish and relate it to the solution of a FBSDE. We begin by providing a necessaryand sufficient condition for the Gˆateaux derivative to vanish in the constrained problem.
Proposition 3.
The Gˆateaux derivative (3.22) vanishes in all directions, i.e. h ω , H ′ K ( π ) i = 0 forall ω ∈ A K , if and only if g π t + F π , (1) t Σ t σ π t = 0 t -a.e. in the interval [0 , T ∧ τ π ] , P -a.s.. Proof.
The Gˆateaux derivative is: h ω , H ′ K ( π ) i = E " Z T ∧ τ π ω ⊺ t Y π t (cid:16) g π t + F π , (1) t Σ t σ π t (cid:17) dt . Clearly, if g π t + F π , (1) t Σ t σ π t = 0 then h ω , H ′ K ( π ) i = 0 for all ω ∈ A K .We prove necessity by contradiction. Assume that h ω , H ′ K ( π ) i = 0 for all ω ∈ A K . Assumefurther that B = n ( ω, t ) ∈ Ω × [0 , T ∧ τ π ] : (cid:16) g π t + F π , (1) t Σ t σ π t (cid:17) ( ω ) = 0 o has positive measure. Now define the process ω t = h(cid:16) g π t + F π , (1) t Σ t σ π t (cid:17) B i ω t This is an admissible portfolio since it is F -predictable, F π , (1) is bounded, g π , σ π ∈ L T ( R n ) andthe process is stopped once the wealth threshold is reached. It follows that h ω , H ′ K ( π ) i = E " Z T ∧ τ π ∧ τ ω Y π t ω ⊺ t ω t B dt > Y π t > ω ⊺ t ω t > B . This gives our contradiction and hence B must havezero measure, which completes the proof.We now present our main theorem which characterizes the optimal portfolio for our stochasticcontrol problem. 17 heorem 1. Define the processes ζ = ( ζ t ) t ≥ and φ = ( φ t ) t ≥ by ζ t = − F π , (1) t F π , (2) t , φ t = F π , (3) t F π , (2) t , (3.24) and the portfolio process π ∗ = ( π ∗ t ) t ≥ by π ∗ t = ζ t (cid:0) Σ − t θ t + σ π ∗ t (cid:1) , (3.25) along with the FBSDE dX π ∗ t = X π ∗ t (cid:0) r t + ζ t θ ⊺ t Σ − t θ t + ζ t θ ⊺ t σ π ∗ t (cid:1) dt + X π ∗ t ζ t (cid:0) Σ − t θ t + σ π ∗ t (cid:1) ⊺ d M t X π ∗ = xd log Y π ∗ t = (cid:16) A t r t + B t θ ⊺ Σ − t θ t + B t θ ⊺ t σ π ∗ t + (cid:0) B t + ζ t φ t (cid:1) ( σ π ∗ t ) ⊺ Σ t σ π ∗ t (cid:17) dt + ( σ π ∗ t ) ⊺ d M t log Y π ∗ T = 0 (3.26) where A t = ζ t − and B t = 1 + ζ t φ t .If one of the two following conditions holds:(a) ζ ∈ L ∞ ,MT ( R ) , or(b) ζ t X π ∗ t is P -a.s. continuous with E " sup t ∈ [0 ,T ] ( ζ t X π ∗ t ) < ∞ and the wealth equation (3.7) corresponding to e π ∗ t = π ∗ t X π ∗ t has a unique square-integrable solution with E [( X π ∗ T ) ] < ∞ ,then π ∗ is an admissible portfolio and is the unique solution to the stochastic control problem (3.1) and, furthermore, the FBSDE (3.26) has a unique solution. Remark 1.
The process ζ t = − F π , (2) t F π , (1) t = − U ′′ ( X π t )( X π t ) U ′ ( X π t ) is the ArrowPratt measure of relative riskaversion (or coefficient of relative risk aversion). Remark 2.
The BSDE component of (3.26) is a quadratic BSDE. The existence and uniquenessof solutions to BSDEs of this type are discussed in Kobylanski et al. (2000) when the noise processis a Brownian motion and in Morlais (2009) for the case of more general martingale noise pro-cesses. When the equations are fully coupled the reader is referred to Luo and Tangpi (2015) andLuo and Tangpi (2017) for existence and uniqueness results. roof. First, we show that π ∗ is an admissible portfolio for the unconstrained problem (3.1) andhence its stopped counterpart π ∗ K is an admissible portfolio for the constrained problem (3.3).Since all the processes appearing in (3.25) are either continuous and adapted or predictable, wehave that π ∗ is predictable under either condition (a) or (b). Next, recall that θ is bounded and,from Appendix A of Al-Aradi and Jaimungal (2018), the elements of Σ − are also bounded andwe have that σ π ∈ L T ( R n ). Thus, when ζ is bounded π ∗ ∈ L T ( R n ) and is clearly an admissibleportfolio. To show admissibility under the alternative condition on ζ we work with the auxiliarycontrol problem where the control process is given in terms of dollar amounts as in the proof ofProposition 1. The dollar amount control process corresponding to the weight process (3.25) is e π ∗ t = ζ t X π ∗ t (cid:0) Σ − t θ t + σ π ∗ t (cid:1) This process is locally square-integrable, i.e. R T k e π ∗ t k < ∞ P -a.s., due to the assumptions made incondition (b). Therefore, by Lemma 3.1 of Lim (2004), e π ∗ is an admissible dollar amount processand hence π ∗ must be an admissible weight process.Next, it is easy to verify that π ∗ K is the unique portfolio that satisfies g π ∗ K t + F π ∗ K , (1) t Σ t σ π ∗ K t = 0 on [0 , T ∧ τ π ] . This in turn implies by Proposition 3 that π ∗ K is the unique portfolio at which the Gˆateaux derivative(3.22) vanishes in all directions, i.e. h ω K , H ′ K ( π ∗ ) i = 0 for all ω K ∈ A K . Now, we take the limitas K tends to infinity to conclude that π ∗ is the unique portfolio such that h ω , H ′ ( π ∗ ) i = 0 for all ω ∈ A . To do this we consider the limitlim K →∞ lim n →∞ H K ( π ∗ K + ǫ n ω K ) − H K ( π ∗ K ) ǫ n where { ǫ n } n ∈ N is a sequence of real numbers tending to zero. Denoting the ratio in the expressionabove by a nk , we note that we can interchange the order of the limits if lim K →∞ a nk exists for all fixed n and lim n →∞ a nk exists for all fixed K . The latter is true since lim n →∞ a nk = h ω K , H ′ K ( π ∗ ) i = 0. For theother limit notice that π ∗ K → π ∗ in the L norm as K → ∞ and so H K ( π ∗ K ) → H ( π ∗ ) as K → ∞ by the fact that H K ( π ∗ K ) = H ( π ∗ K ) and the continuity of H . Therefore,lim K →∞ a nk = lim K →∞ H K ( π ∗ K + ǫ n ω K ) − H K ( π ∗ K ) ǫ n = lim K →∞ H ( π ∗ K + ǫ n ω K ) − H ( π ∗ K ) ǫ n = H ( π ∗ + ǫ n ω ) − H ( π ∗ ) ǫ n is a well-defined limit and thus we can interchange the order of the limits to obtainlim K →∞ lim n →∞ H K ( π ∗ K + ǫ n ω K ) − H K ( π ∗ K ) ǫ n = lim n →∞ H ( π ∗ + ǫ n ω ) − H ( π ∗ ) ǫ n = h ω , H ′ ( π ∗ ) i . K ∈ N . Since π ∗ is the only stationary point for the functional H and since we haveshown that there exists a unique global maximizer, which must also be a stationary point for H , π ∗ must be the unique global maximizer.Finally, the FBSDE (3.26) is obtained by substituting the optimal control process (3.25) in theforward SDE (2.7) and the limiting version of the BSDE (3.23) when K → ∞ and simplifying. Theexistence and uniqueness of the optimal control and its dependence on σ π ∗ in turn implies that theFBSDE (3.26) must have a unique solution, which completes the proof.
4. Specific Utility Functions U ( x ) = log x It is easy to check that F π , ( k ) t = ( − k +1 ( k − ζ t = 1. Sincethis process is bounded, we may apply Theorem 1 to derive the optimal control. Furthermore, theprocesses A t and B t given in Theorem 1 vanish and hence the pair Y π ∗ t = 1 and σ π t = 0 solve theBSDE in (3.26), which is now decoupled from the forward SDE. The optimal portfolio is thereforegiven by π ∗ t = Σ − t θ t . Note that this portfolio is the well-known growth optimal portfolio . U ( x ) = x η η , η < F π , ( k ) t = ( X π t ) η Q k − i =1 ( η − i ) and therefore ζ t = − η is once againbounded. Substituting the relevant expressions in the FBSDE (3.26) gives the decoupled system dX π ∗ t = X π ∗ t (cid:0) r t + − η θ ⊺ t Σ − t θ t + − η θ ⊺ t σ π ∗ t (cid:1) dt + − η X π ∗ t (cid:0) Σ − t θ t + σ π ∗ t (cid:1) ⊺ d M t X π ∗ = xd log Y π ∗ t = − (cid:16) ηr t + η − η θ ⊺ t Σ − t θ t + η − η θ ⊺ t σ π ∗ t +
12 11 − η ( σ π ∗ t ) ⊺ Σ t σ π ∗ t (cid:17) dt + ( σ π ∗ t ) ⊺ d M t log Y π ∗ T = 0 (4.1) Next, notice that if we apply Itˆo’s lemma to rewrite the BSDE in geometric form this gives dY π ∗ t Y π ∗ t = − (cid:16) ηr t + η − η θ ⊺ t Σ − t θ t + η − η θ ⊺ t σ π ∗ t + η − η ( σ π ∗ t ) ⊺ Σ t σ π ∗ t (cid:17) dt + ( σ π ∗ t ) ⊺ d M t Y π ∗ T = 1which is identical to the BSDE derived for the power utility case in Ferland and Watier (2008)when the noise process is a Brownian motion; with Y π ∗ t playing the role of p t and σ π ∗ t replacing Σ − / t Λ t p t . Therefore, we can lean on the results obtained in that paper, in particular, the form ofthe optimal solution when the model parameters are deterministic.20 .3. Exponential Utility - U ( x ) = − e − γx In this case, we have F π , ( k ) t = ( − k +1 γ k ( X π t ) k e − γX π t and therefore ζ t = γX π t and ζ t φ t = 1. Inthis case, we find that ζ t X π t = γ is constant. Moreover, the FBSDE in (3.26) reduces to dX π ∗ t = (cid:0) r t X π ∗ t + γ θ ⊺ t Σ − t θ t + γ θ ⊺ t σ π ∗ t (cid:1) dt + γ (cid:0) Σ − t θ t + σ π ∗ t (cid:1) ⊺ d M t X π ∗ = xd log Y π ∗ t = − (cid:16) (1 − γX π ∗ t ) r t − θ ⊺ t Σ − t θ t − θ ⊺ t σ π ∗ t (cid:17) dt + ( σ π ∗ t ) ⊺ d M t log Y π ∗ T = 0 (4.2) and it is not difficult to verify that the forward SDE has a unique square-integrable solutionso that condition (b) of Theorem 1 is satisfied. Now, if we define a new process by log e Y π ∗ t =log Y π ∗ t + R Tt r s ds , we find that FBSDE system for X π ∗ and e Y π ∗ is dX π ∗ t = (cid:0) r t X π ∗ t + γ θ ⊺ t Σ − t θ t + γ θ ⊺ t σ π ∗ t (cid:1) dt + γ (cid:0) Σ − t θ t + σ π ∗ t (cid:1) ⊺ d M t X π ∗ = xd log e Y π ∗ t = (cid:16) γX π ∗ t r t + θ ⊺ t Σ − t θ t + θ ⊺ t σ π ∗ t (cid:17) dt + ( σ π ∗ t ) ⊺ d M t log e Y π ∗ T = 0 (4.3)which can be mapped to Equation (16) of Ferland and Watier (2008) when the noise process is aBrownian motion. Once again we refer the reader to that paper for detailed results.
5. Conclusions
In this paper we have solved the Merton problem of maximizing the expected utility of terminalwealth using variational analysis techniques, providing sufficient and necessary conditions for theoptimal solution in terms of the solution to a quadratic FBSDE. One extension in which thisapproach would be useful is in the setting of partial information where the noise drivers of assetprices are general martingales. In this case, the usual approach using the stochastic maximumprinciple would not lead to a solution. Some of the present results would also be useful in extendingthe results of the outperformance and tracking problem discussed in Al-Aradi and Jaimungal (2018)and Al-Aradi and Jaimungal (2019) to the partial information setting where more general utilityfunctions are considered. It should also be possible to extend the results of the current paper tothe case where the noise processes may jump.
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