Backward SDEs for Control with Partial Information
aa r X i v : . [ q -f i n . M F ] J u l Backward SDEs for Control with Partial Information
A. Papanicolaou ∗ July 24, 2018
Abstract
This paper considers a non-Markov control problem arising in a financial market where assetreturns depend on hidden factors. The problem is non-Markov because nonlinear filtering isrequired to make inference on these factors, and hence the associated dynamic program effec-tively takes the filtering distribution as one of its state variables. This is of significant difficultybecause the filtering distribution is a stochastic probability measure of infinite dimension, andtherefore the dynamic program has a state that cannot be differentiated in the traditional sense.This lack of differentiability means that the problem cannot be solved using a Hamilton-Jacobi-Bellman (HJB) equation. This paper will show how the problem can be analyzed and solvedusing backward stochastic differential equations (BSDEs), with a key tool being the problem’sdual formulation.
Keywords:
Non-Markov Control, Backward Stochastic Differential Equations, Portfolio Opti-mization, Partial information.
Subject classifications:
Consider an investor who seeks to optimally allocate among ( d + 1) -many assets: a risk-free in-strument (e.g., a money-market or bank account) that pays interest rate r ≥ , and d-many riskyexchange-traded funds (ETFs) denoted S = ( S , S , . . . , S d ) ⊤ where S i ( t ) = time- t price of the i th ETF . These prices are continuous processes on a filtered probability space (Ω , ( F t ) t ≤ T , P ) . Let W and B denote a pair of F t Brownian motions where W ∈ C ([0 , T ]; R d ) and B ∈ C ([0 , T ]; R q ) for a positiveinteger q < ∞ , and with dW ( t ) dW ( t ) ⊤ = I d × d dt, dB ( t ) dB ( t ) ⊤ = I q × q dt dW ( t ) dB ( t ) ⊤ = 0 , where I ( · ) denotes an identity matrix, and ( · ) ⊤ denotes matrix/vector transpose. The ETFs’ priceprocess S ∈ C ([0 , T ]; R d ) has returns that depend on a stochastic factor Y ∈ C ([0 , T ]; R q ) , as givenby the following hidden Markov model, dS i ( t ) S i ( t ) = h i ( Y ( t )) dt + d X j =1 σ ij w dW j ( t ) + q X j =1 σ ij y dB j ( t ) (observed) , (1) dY ( t ) = b ( Y ( t )) dt + a ( Y ( t )) dB ( t ) (hidden) , (2) ∗ Department of Finance and Risk Engineering, NYU Tandon School of Engineering, 6 MetroTech Center, BrooklynNY 11201 [email protected] . Part of this research was performed while the author was visiting the Institute for Pureand Applied Mathematics (IPAM), which is supported by the National Science Foundation. Y (0) is unobserved and independent of W and B . In order to ensureexistence and uniqueness of strong solutions to the SDEs, the coefficients a, b , and h are assumed tobe C and Lipschitz continuous, with matrix a ∈ R q × q satisfying the condition inf y ∈ R q aa ⊤ ( y ) > (i.e., positive definiteness). The matrices σ w and σ y combine for the total covariance σ = (cid:16) σ w σ ⊤ w + σ y σ ⊤ y (cid:17) / ∈ R d × d , where it is assumed there is a constant ǫ such that < ǫ ≤ σσ ⊤ ≤ ǫ < ∞ , (3)i.e., σσ ⊤ is positive definite and bounded.Let F St denote the σ -algebra generated by { S ( u ) : u ≤ t } for any time t ∈ [0 , T ] . The investormust decide upon an F St -adapted allocation vector π ( t ) ∈ R d where for each iπ i ( t ) = time- t proportion of wealth in i th ETF . Clearly F St ⊂ F t , and in particular Y ( t ) is not observable given F St . Hence the investor will needto filter Y ( t ) given F St , and then use this filter to make an optimal investment decision. For a givenstrategy π the investor’s wealth is the process X π ∈ C ([0 , T ]; R ) that is a semi-martingale with dX π ( t ) X π ( t ) = rdt + d X i =1 π i ( t ) (cid:18) dS i ( t ) S i ( t ) − rdt (cid:19) , (4)where π is considered admissible if it is S-integrable, i.e., d X i =1 Z T (cid:12)(cid:12) π i ( t ) X π ( t ) (cid:12)(cid:12) dt < ∞ almost surely,(see [KK07, KS99]). The investor has a concave utility function U ( x ) and finds an optimal π bysolving for her optimal value function, V ( t, x ) = sup π E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } i , where the supremum is taken over all F St -adapted π ’s. This is a non-Markov control problembecause the optimal π ( t ) will depend on the entire history F St . In particular, the filter for Y ( t ) is a non-Markov process, and as the optimal control will depend in this filter it causes the entireproblem to be non-Markov.This paper analyzes this non-Markov problem with a specialized focus on the effects of partialinformation. As F St ⊂ F t , the investor with only F St is said to be partially informed, and naturallythere is a disadvantage by not having the full information of F t . In particular, all processes in (1)and (2) would be observed if the investor had the information contained in F t , in which case itstands to reason that there would be an improvement from her optimal F St -adapted value function.An investor who observes the information in F t is said to be fully informed. The partially-informed investor will compute the posterior distribution of Y ( t ) given F St , whichshe could use to write her optimal strategy in feedback form , but such a characterization is a functionof a probability measure, which means it is a function of an infinite-dimensional input. Functions The definition of ‘feedback form’ is given in [Car15][Chapter 2] and in [Bjö09][Chapter 19]. dS i ( t ) S i ( t ) = ˆ h i ( t ) dt + dν i ( t ) , where ˆ h i ( t ) = E [ h i ( Y ( t )) |F St ] , and ν i ( t ) is the innovation given by ν i ( t ) = Z t (cid:18) dS i ( u ) S i ( u ) − ˆ h i ( u ) du (cid:19) , such that σ − ν ( t ) is a d-dimensional F St -adapted Brownian motion. Completeness of the marketleads to considerable simplification, as there is a unique equivalent martingale measure (EMM) (i.e.,an equivalent measure where e − rt S ( t ) is a local martingale), making the dual function a straight-forward conditional expectation (i.e., the dual problem’s infimum over the set of EMM is trivialbecause the set is a singleton containing the unique EMM). As conditional expectations can berepresented as solutions to BSDEs, it follows that the dual value function is the solution to aBSDE, from which the primal value function and optimal strategy can be computed as well.In contrast to the partially-informed investor, the fully-informed does not need to filter becauseshe observes the full F t , and therefore chooses an optimal F t -adapted π that is obtained froma finite-state HJB equation and is written as a function of X π ( t ) and Y ( t ) . However, the fullinformation model remains an incomplete market model because the Y process cannot be bought orsold, thereby making it somewhat technical to solve the full-information HJB equation. Existenceand regularity of solutions to this HJB equation can be shown when the SDE coefficients meetspecific assumptions (see [Pha02]). If they exist then HJB-based solutions are convenient, but itis still useful to solve the full-information problem using BSDEs because it allows for comparisonswith the BSDEs from partial information.The investor’s quantification of factor-latency is the so-called information premium , or the ex-pected loss in utility due to partial information. From the perspective of partial information, fullinformation is an improvement in the sense that E h V full ( t, x, Y ( t )) (cid:12)(cid:12)(cid:12) F St i ≥ V ( t, x ) , ∀ x ≥ and ∀ t ∈ [0 , T ] , where V full ( t, x, y ) is the fully-informed investor’s value function. This inequality is consistent withcommon-sense intuition that it is better to know the exact values of the Y factors, but it is interestingto point out that this inequality shows how a complete-market investor can expect an improvementif she were allowed to switch to an incomplete-market. It should also be pointed out that this is anexpectation, and it may be possible for V full ( t, x, Y ( t )) < V ( t, x ) (see Example 4.3). Quantificationof the information premium is an important question that is addressed in this paper using BSDEs. Portfolio optimization builds on control theory and relies on concepts such as duality and concav-ity, which are presented in many books and papers including [KS99] and [Rog02]. Initial works3n consumption-portfolio choice and asset pricing under partial information include [Gen86] whopresents a separation theorem: agents first filter then optimize; [Det86] with results on an economywith Gaussian information structures under partial information wherein a Kalman filter applies;[Bas00, Bas05, DM94] with results on markets with multiple heterogeneous agents who updatetheir beliefs with the arrival of financial innovations; [DF86] shows how equilibrium interest rateswith partial information have a trade-off between latent-variable persistence and the parameterscontrolling inference; and also [Fel89] which shows that the expectations hypothesis holds only ifrates are non-stochastic.In [KX91] the partial information portfolio optimization problem is shown to reduce to a com-plete market problem, a result which is also shown in [BDL10]. Portfolio optimization with partialinformation and filtering is done in [Bre06, Car09, LP16, WW08], but only for linear Gaussian cases.Greater generality and the role of martingales and duality theory are considered in [Lak98, Pha01].There is also substantial literature dealing with partial information and (unobserved) regimes fol-lowing finite-state Markov chains, such as [BR05, SH04]. The linear version of the full-informationproblem is addressed in [KO96], with attention given to the so-called nirvana cases where investors’expected utility is infinite. The role of forward-backward dynamics in portfolio optimization isshown in [DZ91] with a novel use of Malliavan calculus. Partial information with nonlinear filtering,BSDEs, and indifference pricing are considered in [MS10], but under the assumption of a bound on σ − h , an assumption which is not made in this paper.Backward SDEs are covered in [CSTV07, CDET13, Car15, EKPQ97, Kob00, PR14, Pha09],including important results for existence and uniqueness of solutions, and in [EKR00] and [HIM05]BSDEs are applied to problems of stochastic control for utility maximization. There is also anapplication of BSDEs in [MPZ15] to robust utility maximization under volatility uncertainty. Pathdependence and HJB equations with stochastic coefficients are considered in [Pen92], which canbe compared to the BSDEs in non-Markov control problems. Another possibility is to write par-tial information’s infinite-dimensional program using the master equation, similar to [BFY15]; themaster equation uses Gâteaux derivatives in an HJB-type equation with differentiation done overmeasure-valued inputs. Two important resources for control theory are [FS05], and [Ben92] forcontrol problems with partial information. There is also [BKS09] where the application of partial-information control methods are used to optimize in a market where the price on a basket of goodsis noisy, and a modified Mutual Fund Theorem is obtained. Finally, a review of nonlinear filter-ing is found in [Ben92, FL91] for the Zakai and Kushner-Stratonovich equations, and Monte-Carlomethods for approximation (i.e., the particle filter) are presented in [CMR05]. This paper brings together results from filtering, duality, and BSDE theory, and uses them to solvethe nonlinear partial-information optimal portfolio problem. This paper’s application of BSDEsis significant because consideration is given to the case of unboundedness of the function h ( y ) in(1). If h were bounded then the results from [HIM05, MS10] would apply. Unboundedness of h is of considerable interest because it allows for extreme behavior among investors with low riskaversion, but it introduces some technical difficulty in proving existence and uniqueness of BSDEsolutions; the proofs are provided in this paper and rely on some of the specific features of thepartial-information finance problem.The BSDE approach is also used when comparing with full information and quantifying theinformation premium. The full-information problem is solved using BSDEs and the informationpremium is represented dynamically using the BSDE coefficients. The information premium is im-portant because it gives quantitative evidence that information matters; partially-informed investors4re at a disadvantage to the fully informed.The rest of the paper is organized as follows: Section 2 formalizes the filtering and controlproblem, and introduces the dual formulation; Section 3 shows how the problem can be solved usingBSDEs when U ( x ) is a power utility, with verification that the solution π obtained from the BSDEsis in fact optimal –both for partial and full information; Section 4 provides insight by consideringthe example of the Gaussian linear case; Section 5 gives a nonlinear example with a simulation ofthe BSDEs. Appendices A, B, and C contain technical proofs for the propositions and theorems ofSection 3. It will be assumed throughout that h satisfies the Novikov condition Condition 2.1 (Novikov) . The function h is such that E exp (cid:18) ǫ Z T k h ( Y ( t )) − r k dt (cid:19) < ∞ , (5) where r = ( r, r, . . . , r ) ⊤ ∈ R d , k · k denotes the Euclidean norm, and ǫ > is the bounding constantin (3) . Clearly (5) holds for h bounded, but it will be interesting to consider h unbounded along withlow risk aversion (these ideas will become clearer in later sections). In matrix/vector form, the observations are given by dS ( t ) S ( t ) = h ( Y ( t )) dt + σ w dW ( t ) + σ y dB ( t ) . The filter is defined for an appropriate test function g as ˆ g ( t ) = E h g ( Y ( t )) (cid:12)(cid:12)(cid:12) F St i , for any g such that sup t ∈ [0 ,T ] E k g ( Y ( t )) k < ∞ . Using ˆ h ( t ) = E [ h ( Y ( t )) |F St ] , an important featurefrom filtering theory is the innovations process ν ( t ) = Z t (cid:18) dS ( u ) S ( u ) − ˆ h ( u ) du (cid:19) , (6)which is a Gaussian process, namely ζ ( t ) = σ − ν ( t ) is F St -adapted d-dimensional Brownian motion. The innovations process is used to re-write equation(1) in a complete-market form, dS ( t ) S ( t ) = ˆ h ( t ) dt + σdζ ( t ) . (7)5his is a complete market because there is a unique equivalent martingale measure (EMM), namely d Q d P = Z ( t ) that is given by the Dolean-Dade exponent (due to Condition 2.1), d Q d P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F St = Z ( t )= exp (cid:18) − Z t (cid:13)(cid:13)(cid:13) σ − (ˆ h ( u ) − r ) (cid:13)(cid:13)(cid:13) du − Z t ( σ − (ˆ h ( u ) − r )) ⊤ dζ ( u ) (cid:19) . (8) The investor chooses an F St -adapted strategy ( π ( t )) t ≤ T and has a self-financing wealth process, asgiven in equation (4), that can be written using the innovations process, dX π ( t ) X π ( t ) = rdt + d X i =1 π i ( t )(ˆ h i ( t ) − r ) dt + d X i =1 π i ( t ) dν i ( t ) . The investor’s strategy is selected from an admissible set A given by A = (cid:26) F St -adapted π : [0 , T ] × Ω → R d , s.t. Z T (cid:12)(cid:12)(cid:12) X π ( t ) k π ( t ) k (cid:12)(cid:12)(cid:12) dt < ∞ a.s. (cid:27) , (9)(see [KK07, KS99]). For any π ∈ A the wealth process is almost surely non-negative, which rulesout arbitrage from doubling strategies.The investor has a utility function U : R + → R + that is concave and satisfies the Inadaconditions: Condition 2.2.
The utility function U ( x ) is continuously differentiable with U ′ ( x ) > and U ′′ ( x ) < for all x ≥ , and satisfies the Inada conditions, lim x ր∞ U ′ ( x ) = 0 and lim x ց U ′ ( x ) = ∞ . The utility function used throughout this paper is of constant relative risk aversion (CRRA), orsimply the power utility, U ( x ) = 11 − γ x − γ , for γ > and γ = 1 . The investor seeks to maximize expected terminal utility of discounted wealth,with her control being selected from the class of admissible strategies given in (9). This leads theinvestor to find her optimal value function V ( t, x ) , which is formally written as a supremum overstrategies in A , V ( t, x ) = sup π ∈A E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } i for all x > .The nonlinearity introduced by the supremum can be avoided by considering the dual formu-lation of this problem. Let V ⋆ denote the solution to the dual value function (see [Lak98, Rog02,KS99]), V ⋆ ( t, p ) = inf Q ≪ P E " U ⋆ (cid:18) pe − r ( T − t ) d Q d P (cid:12)(cid:12)(cid:12) F ST (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F St = E " U ⋆ (cid:18) pe − r ( T − t ) Z ( T ) Z ( t ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F St , (10)for all p > where Q ≪ P denotes the family of equivalent probability measures under which e − rt S ( t ) is an F St (local) martingale. Clearly, completeness of the market and the unique EMM6iven by equation (8) are the reason why the infimum is dropped in (10). The dual value function V ⋆ is also a non-Markov process, yet it will be seen in Section 3 that it can be expressed usingBSDEs, and hence it will be possible to obtain tractable representations of the solution to thepartial-information control problem.For continuous processes driven by Brownian motions, the general relationship between V and V ⋆ is discussed in [Rog02], namely that V ⋆ ( t, p ) = sup x> ( V ( t, x ) − xp ) for all p > . In general V ( t, x ) ≤ inf p ( V ⋆ ( t, p ) + xp ) for all x > , but if Condition 2.2 holds and if V ( t, x ) < ∞ for some x > , then V and V ⋆ are conjugates (i.e., they are Fenchel-Legendre transforms of oneanother), V ( t, x ) = inf p> ( V ⋆ ( t, p ) + xp ) for all x > . For the power utility, first-order conditions yield the transform U ⋆ ( p ) = γ − γ p − − γγ , and the expression in (10) can be rewritten as V ⋆ ( t, p ) = U ⋆ (cid:16) pe − r ( T − t ) (cid:17) ξ ( t ) = γ − γ (cid:16) pe − r ( T − t ) (cid:17) − − γγ ξ ( t ) , (11)where ξ ( t ) = Z ( t ) − γγ E h Z ( T ) − − γγ (cid:12)(cid:12)(cid:12) F St i . For p > it is clear that V ⋆ is a finite and strictly convex function if | ξ ( t ) | < ∞ almost surely.To ensure finiteness of ξ ( t ) , the model parameters and the risk aversion must permit the followingcondition: Condition 2.3.
The model parameters in (1) , (2) , and the power utility’s risk aversion γ , are suchthat E exp (cid:18) | γ − || γ − | ǫγ Z T k ˆ h ( t ) k dt (cid:19) < ∞ , where ǫ > is the bounding constant given in (3) , with the derivation of this bound following fromProposition A.3. This bound ensures | V ⋆ ( t, p ) | < ∞ for all p ∈ (0 , ∞ ) . The set of h functions for which Condition 2.3 holds is not empty, as shown in the followingremark. Remark 1 (Nonlinear h Satisfying Condition 2.3) . Partial information can be reduced to a condi-tion on full information through multiple applications of Jensen’s inequality, E exp (cid:18) | γ − || γ − | ǫγ Z T k ˆ h ( t ) k dt (cid:19) ≤ T Z T E exp (cid:18) T | γ − || γ − | ǫγ k h ( Y ( t )) k (cid:19) dt . Inada conditions and concavity are the main requirements for conjugacy in a complete market. In comparison,conjugacy in an incomplete market requires the additional condition of asymptotic elasticity, lim x →∞ xU ′ ( x ) /U ( x ) < as shown in [KS99]. ence, Condition 2.3 is satisfied for any h such that sup t ∈ [0 ,T ] E exp (cid:18) T | γ − || γ − | ǫγ k h ( Y ( t )) k (cid:19) dt < ∞ . Certainly this includes bounded nonlinear functions. An explicit example in one dimension involves Y being a Cox-Ingersoll-Ross (CIR) process, dY ( t ) = κ ( ¯ Y − Y ( t )) dt + a p Y ( t ) dB ( t ) where κ > , ¯ Y > , < a ≤ Y κ , and h ( y ) = √ y , with a sufficient condition for Condition 2.3being T | γ − || γ − | ǫγ < κa . Note that this example does not have the condition of inf y ∈ R q aa ⊤ ( y ) > ,but this does not pose an issue because the SDE for Y ( t ) is well defined for a ≤ Y κ . Section 5will explore this example further.
The need for Condition 2.3 is seen in the proof of Proposition A.3 in Appendix A, from whichit is seen that E sup t ∈ [0 ,T ] | ξ ( t ) | ≤ E sup t ∈ [0 ,T ] (cid:18) Z ( T ) Z ( t ) (cid:19) − − γγ ≤ E exp (cid:18) | γ − || γ − | ǫγ Z T ( k ˆ h ( t ) k + k r k ) dt (cid:19) < ∞ , (12)and it will be important in Section 3 to have E sup t ∈ [0 ,T ] | ξ ( t ) | < ∞ as part of the existence anduniqueness theory for ξ to be a solution to a BSDE. Furthermore, defining G ( t ) to be G ( t ) = ξ ( t ) γ , due to Condition 2.3, the finiteness in equation (12) implies conjugacy of the Legendre transforms, V ( t, x ) = inf p> ( V ⋆ ( t, p ) + xp )= inf p> (cid:18) γ − γ (cid:16) pe − r ( T − t ) (cid:17) − − γγ ξ ( t ) + xp (cid:19) = U (cid:16) xe r ( T − t ) (cid:17) G ( t ) . (13)Equation (13) allows for optimal solutions to be obtained by solving the dual problem, with the op-timal V being obtained via straightforward (numerical) calculation of a Fenchel-Legendre transformon V ⋆ . Regardless of the chosen function to be computed, nonlinear filtering causes V and V ⋆ torequire specially-designed backward recursive algorithms because of infinite dimensionality in theconditioning. To be more precise, the conditioning on F St is an infinite-dimensional object and fornumerical methods will need to be replaced with a finite-dimensional approximation. Sometimesthere are ways to write the filtering distribution in a finite-dimensional form (e.g., using a Kalmanfilter [Bre06], or finite-dimensional Markov chains [BR05]), but general nonlinear filtering doesn’thave such forms.Before starting the next section it is important to define the concept of investor nirvana. Nirvanais defined in [KO96] as follows:
Definition 2.1 (Investor Nirvana) . For unbounded U , an investor achieves nirvana at ( t, x ) if V ( t, x ) = ∞ . For bounded U , nirvana is achieved when V ( t, x ) = max x U ( x ) . V ⋆ ( t, p ) = ∞ because it may be unclear whether there is nirvana or a duality gap (i.e., strict inequality such that V ( t, x ) < inf p> ( V ⋆ ( t, p ) + xp ) = ∞ for some x > ). Proposition 2.1.
In the partial-information case, investor nirvana cannot occur for γ ∈ (0 , ifCondition 2.3 holds, and cannot occur for γ > given (5) .Proof. For γ > it follows from equations (11) and (13) that U (cid:16) xe r ( T − t ) (cid:17) ≤ V ( t, x ) ≤ inf p ( V ⋆ ( t, p ) + xp ) = U (cid:16) xe r ( T − t ) (cid:17) ξ ( t ) γ ≤ , for all x ∈ (0 , ∞ ) , implying that ≤ ξ ( t ) ≤ . From equation (5) it follows that P (cid:16) inf t ∈ [0 ,T ] log( Z ( T ) /Z ( t )) = −∞ (cid:17) = 0 , so that P (cid:16) ( Z ( T ) /Z ( t )) − − γγ > |F St (cid:17) > almost surely. This implies ξ ( t ) = E "(cid:18) Z ( T ) Z ( t ) (cid:19) − − γγ (cid:12)(cid:12)(cid:12) F St > almost surely.For γ ∈ (0 , with Condition 2.3 not being violated, it follows that equation (12) holds, andso ξ ( t ) < ∞ almost surely for all t ∈ [0 , T ] . Hence, V ( t, x ) ≤ inf p ( V ⋆ ( t, p ) + xp ) < ∞ for all x ∈ (0 , ∞ ) . Remark 2 (Other Utility Functions) . This paper considers the problem only for power utilityfunction. However, for exponential utility there should be results similar to power utility with γ > ,although there may be some technical difficulties in adapting the Inada conditions and wealth processto the entire real line. Log utility is a simple case that does not require BSDEs, as the optimalsolution is simply the myopic strategy (see [GKSW14, Lak98]). The partial-information dual function V ⋆ ( t, p ) can be obtained by solving a BSDE. Solutions toBSDEs are constructed in the following function spaces, P d = n the set of d-dimensional F St -adapted measurable processes on Ω × [0 , T ] o H T ( P d ) = (cid:26) y ∈ P d s.t. E Z T k y ( t ) k dt < ∞ (cid:27) S T ( P d ) = ( y ∈ P d ∩ C ([0 , T ]; R d ) s.t. E sup t ∈ [0 ,T ] k y ( t ) k < ∞ ) S ∞ T ( P d ) = ( y ∈ P d ∩ C ([0 , T ]; R d ) s.t. sup t ∈ [0 ,T ] k y ( t ) k < ∞ a.s. ) . (14)9his section has the derivation of the BSDE for V ⋆ ( t, p ) given by (11), and will give the conditionsfor existence and uniqueness. Define the martingale M ( t ) = E h Z ( T ) − − γγ (cid:12)(cid:12)(cid:12) F St i for ≤ t ≤ T , so that ξ ( t ) = Z ( t ) − γγ M ( t ) . Condition 2.3 ensures M ( t ) is square integrable, E M ( t ) < E M ( T ) = E Z ( T ) − − γγ < ∞ , andallows for a unique representation of M ( t ) as M ( t ) = E h Z ( T ) − − γγ i + d X i =1 Z t M ( u ) θ i ( u ) dζ i ( u ) for ≤ t ≤ T , (15)where θ ( t ) is the unique F St -adapted process with E R T M ( u ) k θ ( u ) k du < ∞ (see [BDL10]). In fact, θ is square-integrable by itself, θ ∈ H T ( P d ) (see Proposition A.1 in Appendix A). The representationin (15) should not be confused with the standard martingale representation theorem because thefiltration generated by ζ may be smaller than F St .Using the representation of (15), the dual value function V ⋆ ( t, p ) is given by the ansatz (11) andthe pair ( ξ, α ) ∈ S T ( P ) × H T ( P d ) that solves a BSDE. Theorem 3.1.
Assume Condition 2.3. The process ξ ( t ) in the representation of V ⋆ in (11) isgiven by the unique pair ( ξ, α ) ∈ S T ( P ) × H T ( P d ) that solves the BSDE, − dξ ( t ) = β ( t, α ( t ) , ξ ( t )) dt − d X i =1 α i ( t ) dζ i ( t ) , (16) ξ ( T ) = 1 , where β ( t, α ( t ) , ξ ( t )) = 1 − γγ d X i =1 (cid:16) σ − (ˆ h ( t ) − r ) (cid:17) i α i ( t ) + 12 1 − γγ (cid:13)(cid:13)(cid:13) σ − (ˆ h ( t ) − r ) (cid:13)(cid:13)(cid:13) ξ ( t )= 1 − γ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) σ − ˆ h ( t ) − r γ + α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ξ ( t ) − − γ | ξ ( t ) | k α ( t ) k . Some remarks are in order before starting the proof of Theorem 3.1.
Remark 3 (Existence of Solutions to (16)) . It should be noted that existence of a solution to (16) is due to Condition 2.3, as it allows for the martingale representation in (15) , from which a solutionis constructed in terms of θ and ˆ h , ξ ( t ) = M (0) exp − Z t β ( u, α ( u ) , ξ ( u )) ξ ( u ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( u ) ξ ( u ) (cid:13)(cid:13)(cid:13)(cid:13) ! du + Z t α ( u ) ⊤ ξ ( u ) dζ ( u ) ! (17) α ( t ) ξ ( t ) = θ ( t ) − − γγ σ − (cid:16) ˆ h ( t ) − r (cid:17) , (18)10 here it can be checked that ξ ( t ) = Z ( t ) − γγ M ( t ) and α ( t ) is the diffusion term from the Itô differ-ential of d (cid:16) Z ( t ) − γγ M ( t ) (cid:17) , and hence it follows from Condition 2.3 that ( ξ, α ) ∈ S T ( P ) × H T ( P d ) .However, it should also be pointed out that θ is not easily obtained from the martingale represen-tation theorem, but rather is found by solving the BSDE. On the other hand, BSDEs have explicitsolution in very few cases, and so numerical methods should be used to find ( ξ, α ) and θ . Remark 4 (Uniqueness of Solutions to (16)) . Formulas (17) and (18) show the existence of asolution to equation (16) when Condition 2.3 holds. If the function h is bounded, then the coefficient β is uniformly Lipschitz and uniqueness follows from an application of the existing theory (see[Car15, EKPQ97, Pha09]). The proof for h unbounded uses a truncation argument to show thatsolutions are a unique limit from a sequence of bounded problems (see Propositions A.2 and A.3). Remark 5.
Condition 2.3 may be violated for γ near zero, in which case formulas (17) and (18) do not provide a solution and there could be investor nirvana. It follows from (17) that in terms of α ( t ) , investor nirvana means P (log ξ ( t ) = ∞ )= P − Z t β ( u, α ( u ) , ξ ( u )) ξ ( u ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( u ) ξ ( u ) (cid:13)(cid:13)(cid:13)(cid:13) ! du + Z t α ( u ) ⊤ ξ ( u ) dζ ( u ) = ∞ ! > , for some t ∈ [0 , T ) , which is certainly not the case for any θ ∈ H T ( P d ) .Proof of Theorem 3.1. The martingale representation in (15) is used to write a forward SDE dξ ( t )= M ( t ) d (cid:16) Z ( t ) − γγ (cid:17) + Z ( t ) − γγ dM ( t ) + dM ( t ) · d (cid:16) Z ( t ) − γγ (cid:17) = Z ( t ) − γγ M ( t ) d X i =1 (cid:18) − − γγ (cid:16) σ − (ˆ h ( t ) − r ) (cid:17) i + θ i ( t ) (cid:19) dζ i ( t ) − − γγ Z ( t ) − γγ M ( t ) d X i =1 (cid:18)(cid:16) σ − (ˆ h ( t ) − r ) (cid:17) i θ i ( t ) (cid:19) dt + 12 (1 − γ )(1 − γ ) γ Z ( t ) − γγ M ( t ) (cid:13)(cid:13)(cid:13) σ − (ˆ h ( t ) − r ) (cid:13)(cid:13)(cid:13) dt = − ξ ( t ) d X i =1 (cid:18) − γγ (cid:16) σ − (ˆ h ( t ) − r ) (cid:17) i − θ i ( t ) (cid:19)| {z } = − α i ( t ) dζ i ( t ) − − γγ ξ ( t ) d X i =1 (cid:16) σ − (ˆ h ( t ) − r ) (cid:17) i θ i ( t ) − (cid:18) − γγ (cid:19) (cid:13)(cid:13)(cid:13) σ − (ˆ h ( t ) − r ) (cid:13)(cid:13)(cid:13) !| {z } = β ( t,α ( t ) ,ξ ( t )) dt , which is (16) with α ( t ) and β ( t, α, ξ ) given accordingly. Equation (16) has non-Lipschitz coefficientsif h is not bounded, and therefore uniqueness of solutions is not covered by the general theoryfor solutions to BSDEs given in [Car15, EKPQ97, Pha09]. Instead, uniqueness is shown using atruncation argument and the probabilistic representation of ξ given in (11).11or some positive K < ∞ , define the truncated filter, ˆ h K ( t ) = K ˆ h ( t ) k ˆ h ( t ) k , if k ˆ h ( t ) k ≥ K ˆ h ( t ) , otherwise,and consider the bounded BSDE − dξ K ( t ) = β K ( t, α K ( t ) , ξ K ( t )) dt − d X i =1 α iK ( t ) dζ i ( t ) , (19) ξ K ( T ) = 1 , where β K is the same drift function from (16) but with ˆ h K ( t ) replacing the unbounded ˆ h ( t ) . Thisdrift parameter is linear with uniform linear growth bounds, | β K ( t, α K ( t ) , ξ K ( t )) |≤ | − γ | γ (cid:13)(cid:13)(cid:13) σ − (ˆ h K ( t ) − r ) (cid:13)(cid:13)(cid:13) k α K ( t ) k + | − γ | γ (cid:13)(cid:13)(cid:13) σ − (ˆ h K ( t ) − r ) (cid:13)(cid:13)(cid:13) | ξ K ( t ) |≤ C K ( k α K ( t ) k + | ξ K ( t ) | ) , which also serves as a uniform Lipschitz constant. Therefore, equation (19) fits into the frameworkof [Car15, EKPQ97, Pha09] and has solution ( ξ K , α K ) that is unique in the space S T ( P ) × H T ( P d ) .Now define the stopping time τ K = inf n t ≥ k ˆ h ( t ) k ≥ K o , and notice that τ K ր ∞ almost-surely as K ր ∞ because ˆ h ( t ) is integrable (due to the Novikov Condition in (5)). Then us-ing the fact that ( | ξ ( t ) − ξ K ( t ) | ) [ τ K ≥ T ] = 0 from Proposition A.2, and also using the bound sup K> E sup t ∈ [0 ,T ] | ξ K ( t ) | < ∞ from Proposition A.3, it is shown that ξ K converges in mean, E sup t ∈ [0 ,T ] | ξ ( t ) − ξ K ( t ) | = E sup t ∈ [0 ,T ] | ξ ( t ) − ξ K ( t ) | [ τ K Let Σ = σσ ⊤ . The optimal strategy is π ∗ ( t ) = Σ − ˆ h ( t ) − r γ + ( σ − ) ⊤ α ( t ) ξ ( t ) , (20) where Σ − h ( t ) − r γ is the so-called myopic strategy and ( σ − ) ⊤ α ( t ) /ξ ( t ) is a dynamic hedging com-ponent due to stochasticity in the drift (see [DRM03, Mer71]).Proof. Due to the properties of power utility, notice that (1 − γ ) V ( t, x ) ≥ for all x ≥ and all γ > , γ = 1 . 13or any π ∈ A the SDE for V ( t, X π ( t )) is dV ( t, X π ( t ))= d (cid:16) U ( X π ( t )) e r (1 − γ )( T − t ) ξ ( t ) γ (cid:17) = V ( t, X π ( t )) (1 − γ ) π ( t ) ⊤ (ˆ h ( t ) − r ) − γ (1 − γ ) k σ ⊤ π ( t ) k γ (1 − γ ) π ( t ) ⊤ σ α ( t ) ξ ( t ) − γ β ( t, α ( t ) , ξ ( t )) ξ ( t ) − γ ( γ − (cid:13)(cid:13)(cid:13)(cid:13) α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ! ! dt + V ( t, X π ( t )) (cid:18) (1 − γ ) π ( t ) ⊤ σ + γ α ( t ) ⊤ ξ ( t ) (cid:19) dζ ( t ) ≤ (1 − γ ) γV ( t, X π ( t )) sup π ( t ) π ( t ) ⊤ ˆ h ( t ) − r γ − k σ ⊤ π ( t ) k π ( t ) ⊤ σ α ( t ) ξ ( t ) − − γ β ( t, α ( t ) , ξ ( t )) ξ ( t ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ! ! dt + V ( t, X π ( t )) (cid:18) (1 − γ ) π ( t ) ⊤ σ + γ α ( t ) ⊤ ξ ( t ) (cid:19) dζ ( t )= V ( t, X π ( t )) (cid:18) (1 − γ ) π ( t ) ⊤ σ + γ α ( t ) ⊤ ξ ( t ) (cid:19) dζ ( t ) . (21)The maximized dt term is obtained by maximizing the quadratic form, π ∗ ( t ) = arg max π ( t ) − k σ ⊤ π ( t ) k + 2 σ − ˆ h ( t ) − r γ + α ( t ) ξ ( t ) ! ⊤ σ ⊤ π ( t ) − β ( t, α ( t ) , ξ ( t ))(1 − γ ) ξ ( t ) − (cid:13)(cid:13)(cid:13)(cid:13) α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ! , (22)from which first-order conditions yield π ∗ ( t ) shown in (20). This maximizer is written in terms of α ( t ) , the filter ˆ h ( t ) , and the model parameters, and it is straightforward to check that the right-handside of (22) is equal to zero when evaluated at π ( t ) = π ∗ ( t ) with β ( t, α ( t ) , ξ ( t )) given by Theorem3.1. Hence, V ( t, X π ∗ ( t )) is a supermartingale, and if it can be shown to be a true martingale thenit is verified that π ∗ is an optimal strategy (see [BMZ11]).Inserting the expression (20) for π ∗ ( t ) into (21), and then using expression (18) for α ( t ) in termsof θ ( t ) , there is the SDE dV ( t, X π ∗ ( t )) = V ( t, X π ∗ ( t )) (cid:18) (1 − γ ) π ∗ ( t ) ⊤ σ + γ α ( t ) ⊤ ξ ( t ) (cid:19) dζ ( t )= V ( t, X π ∗ ( t )) θ ( t ) ⊤ dζ ( t ) , where θ ( t ) is the martingale representation from (15). Solving this SDE yields V ( t, X π ∗ ( t )) = V (0 , X π ∗ (0)) exp (cid:18) − Z t k θ ( u ) k du + Z t θ ( u ) ⊤ dζ ( u ) (cid:19) = V (0 , X π ∗ (0)) M ( t ) M (0) , M ( t ) is a true martingale. Hence, E h V ( T, X π ∗ ( T )) (cid:12)(cid:12)(cid:12) F St ∨ { X π ∗ ( t ) = x } i = V ( t, X π ∗ ( t )) + E (cid:20)Z Tt V ( u, X π ∗ ( u )) θ ( u ) ⊤ dζ ( u ) (cid:12)(cid:12)(cid:12) F St ∨ { X π ∗ ( t ) = x } (cid:21)| {z } =0 = V ( t, X π ∗ ( t )) . This verifies that π ∗ is an optimal strategy. Investment under ‘full information’ means that the information in F t is available to market partic-ipants; there are no hidden states because ( W ( u ) , B ( u )) u ≤ t ∈ F t . With full information the wealthprocess is dX π ( t ) X π ( t ) = rdt + d X i =1 π i ( t )( h i ( Y ( t )) − r ) dt + d X i =1 d X j =1 π i ( t ) σ ij w dW j ( t ) + d X i =1 q X j =1 π i ( t ) σ ij y dB j ( t ) , where π is selected from among the set of full-information strategies A full = (cid:26) F t -adapted π : [0 , T ] × Ω → R d s.t. Z T (cid:12)(cid:12)(cid:12) X π ( t ) k π ( t ) k (cid:12)(cid:12)(cid:12) dt < ∞ a.s. (cid:27) . (23)Then the optimal investment is a Markov control problem, V full ( t, x, y ) = sup π ∈A full E h U ( X ( T )) (cid:12)(cid:12)(cid:12) X ( t ) = x, Y ( t ) = y i . (24) Proposition 3.1. Given (5) , investor nirvana cannot occur in the full-information case for γ > .Proof. The market is incomplete but the Novikov condition in (5) means that a possible equivalentmartingale measure is the one having Radon-Nikodym derivative E ( t ) = exp (cid:18) − Z t (cid:13)(cid:13) σ − ( h ( Y ( u )) − r ) (cid:13)(cid:13) du − Z t ( h ( Y ( u )) − r ) ⊤ (cid:16) ( σ − w ) ⊤ dW ( u ) + ( σ − y ) ⊤ dB ( u ) (cid:17)(cid:19) , i.e., the minimal-entropy martingale measure. Now, it should be clear that E ( t ) can be non-zero,namely P (cid:16) E ( T ) / E ( t ) > (cid:12)(cid:12)(cid:12) Y ( t ) = y (cid:17) > , and so E h E ( T ) γ − γ (cid:12)(cid:12)(cid:12) Y ( t ) = y i > , V full ( t, x, y ) ≤ inf p (cid:18) E (cid:20) U ⋆ (cid:18) pe − r ( T − t ) E ( T ) E ( t ) (cid:19) (cid:12)(cid:12)(cid:12) Y ( t ) = y (cid:21) + xp (cid:19) = inf p U ⋆ ( pe − r ( T − t ) ) E "(cid:18) E ( T ) E ( t ) (cid:19) γ − γ (cid:12)(cid:12)(cid:12) Y ( t ) = y + xp ! < . Hence, nirvana in the sense of Definition 2.1 does not occur.The full-information value function satisfies a Hamilton-Jacobi-Bellman (HJB) equation, (cid:18) ∂∂t + rx ∂∂x + L (cid:19) V full + sup π (cid:18) x π ⊤ Σ π ∂ ∂x V full + xπ ⊤ ( h ( y ) − r ) ∂∂x V full + xπ ⊤ σ y a ( y ) ⊤ ∂∂x ∇ V full (cid:19) = 0 (25) V full (cid:12)(cid:12)(cid:12) t = T = U , where Σ = σσ ⊤ , ∇ denotes the gradient in y , and L = 12 q X i,j =1 (cid:16) aa ⊤ ( y ) (cid:17) ij ∂ ∂y i ∂y j + q X i =1 b i ( y ) ∂∂y i . If (25) has a classical solution then the optimal strategy is written in feedback form, π ∗ ( t, x, y ) = − Σ − ( h ( y ) − r ) ∂∂x V full ( t, x, y ) x ∂ ∂x V full ( t, x, y ) − σ y a ( y ) ⊤ ∂∂x ∇ V full ( t, x, y ) x ∂ ∂x V full ( t, x, y ) ! . (26)By Theorem 8.1 in Chapter III.8 of [FS05], if π ∗ given by (26) is an admissible strategy in A full ,then strict concavity of the objective inside the supremum implies that a classical solution to (25)will satisfy a verification lemma.For the case of power utility there is a simplifying ansatz for the solution to equation (25), V full ( t, x, y ) = U (cid:16) xe r ( T − t ) (cid:17) G full ( t, y ) , (27)which means G full satisfies the equation (cid:18) ∂∂t + L (cid:19) G full + (1 − γ ) max π ∈ R d f (cid:16) y, π, G full , a ⊤ ∇ G full (cid:17) = 0 (28) G full (cid:12)(cid:12)(cid:12) t = T = 1 , where the objective function f is strictly concave in π for any ( y, π, g, η ) ∈ R q × R d × R + × R q , as f ( y, π, g, η ) = (cid:16) − γ π ⊤ Σ π + π ⊤ ( h ( y ) − r ) (cid:17) g + π ⊤ σ y η . (29)16he objective in (29) can be maximized with first-order conditions, where the maximizer is π ∗ ( t, y ) = Σ − (cid:18) h ( y ) − r γ + σ y ηγg (cid:19) , (30)from which it is seen that the maximized objective is F ( y, g, η ) = max π ∈ R d f ( y, π, g, η )= g γ (cid:18) ( h ( y ) − r ) + σ y ηg (cid:19) ⊤ Σ − (cid:18) ( h ( y ) − r ) + σ y ηg (cid:19) ≥ . (31)If equation (28) has a classical solution, then an optimal strategy is found by evaluating (30) at ( g, η ) = ( G full , a ⊤ ∇ G full ) , π ∗ ( t, y ) = Σ − h ( y ) − r γ + 1 γG full ( t, y ) σ y a ( y ) ⊤ ∇ G full ( t, y ) ! , which can be seen as being comprised of two components: a myopic component given by theoptimal from the standard Merton problem, plus a dynamic hedging term motivated by stochasticfluctuations in Y ( t ) . Remark 6 (Examples of Other Nonlinear HJB Equations) . Some examples in the finance literaturewhere there occurs a nonlinear HJB equation like (28) include: optimal portfolio allocation withconsumption and an unhedgeable income stream [DFSZ97]; a generalization of problem (24) butwith scalar Y ( t ) in [SZ05]. Other examples include the linear case (i.e., h ( y ) and b ( y ) linear, a ( y ) constant in y ) where the solution to (28) can be found with an affine ansatz (see [Ben92, Bre06]);these linear models can have investor nirvana if there is low risk aversion (see [KO96] or Section 4of this paper). Equation (28) is a semi-linear PDE with uniformly elliptic operator, for which classical solutionshave been shown to exist under relatively general circumstances. Existence of smooth solutions areshown [Pha02], and for scalar cases it is shown in [Zar01] that the PDE for G full reduces to a powertransform of a solution to a linear PDE. Specifically, for the case of a ( y ) constant in y , [Pha02] givesa sufficient condition for smooth solutions to the HJB, Condition 3.1. If the diffusion matrix a in equation (2) is constant in y , with b ( y ) and h ( y ) being C and Lipschitz in y , and k σ − h ( y ) k being C and Lipschitz in y ,then there exists a function ϕ ( t, y ) differentiable in t and twice differentiable in y such that G full ( t, y ) = exp( − ϕ ( t, y )) , i.e., there is a classical solution to equation (28) , and with |∇ ϕ ( t, y ) | ≤ C (1 + | y | ) for all t ∈ [0 , T ] and for all y ∈ R q . emark 7. For non-constant a ( y ) , [Pha02] explains how to reparameterize the SDE for Y ( t ) sothat the Condition 3.1 applies, namely by looking for a function φ ( y ) with ∇ φ ( y ) = a ( y ) − i.e., the inverse of matrix a ( y ) , so that Y ( t ) = φ − ( ˜ Y ( t )) with d ˜ Y ( t ) = ˜ b ( ˜ Y ( t )) dt + dB ( t ) , where ˜ b (˜ y ) = (cid:16) ∇ φ ( y ) ⊤ b ( y ) + trace (cid:2) a ( y ) ⊤ (cid:0) ∇∇ ⊤ φ ( y ) (cid:1) a ( y ) (cid:3) (cid:17)(cid:12)(cid:12)(cid:12) y = φ − (˜ y ) . From here it must bechecked that ˜ b is C and Lipschitz. The solution G full is the value function G full ( t, y ) = 1 + (1 − γ ) × sup π ∈A full E "Z Tt f (cid:16) Y ( u ) , π ( u ) , G full ( Y ( u )) , a ( Y ( u )) ⊤ ∇ G full ( u, Y ( u )) (cid:17) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y ( t ) = y . As explained on page 143 in Chapter 6 of [Pha09], there is a nonlinear Feynman-Kac representationfor G full , with G full ( t, Y ( t )) = χ ( t ) where χ ( t ) solves the following BSDE, − d χ ( t ) = (1 − γ ) F ( Y ( t ) , χ ( t ) , ψ ( t )) dt − ψ ( t ) ⊤ dB ( t ) , for t ≤ T χ ( T ) = 1 . (32)A solution to (32) is a pair ( χ , ψ ) ∈ S T ( P full ) × H T ( P fullq ) with P fullq = n the set of q-dimensional F Bt -adapted measurable processes on Ω × [0 , T ] o , where S T and H T are the same as those defined in (14) except with P fullq . Given the solution to(32), the optimal strategy is π ∗ ( t ) = π ∗ ( t, Y ( t ) , χ ( t ) , ψ ( t )) = Σ − (cid:18) h ( Y ( t )) − r γ + σ y ψ ( t ) γ χ ( t ) (cid:19) , (33)which is similar to the formula in (30), and is an admissible strategy (i.e., is S-integrable) because χ ( t ) > a.s. by a comparison principle as explained in Theorem 6.2.2 on page 142 in Chapter 6 of[Pha09].Existence of solutions to (32) are not covered by the general theory in [Car15, EKPQ97, Pha09]because F ( t, y, g, p ) does not have a uniform Lipschitz constant, and is not covered by [Kob00]because F has a g term. However, a classical solution G full to (28) can be evaluated at Y ( t ) toobtain the solution to the BSDE, χ ( t ) = G full ( t, Y ( t )) (34) ψ ( t ) = a ( Y ( t )) ⊤ ∇ G full ( t, Y ( t )) , provided that this solution is in S T ( P full ) × H T ( P fullq ) . Proposition 3.2. Suppose Condition 3.1. If E exp (cid:18) δ | γ − || γ − | ǫγ Z T k h ( Y ( t )) k dt (cid:19) < ∞ , and E Z T k Y ( t ) k δ dt < ∞ , T (35) for some δ , δ > with δ + δ = 1 , then the pair given by equation (34) is in S T ( P full ) × H T ( P full q ) ,and hence a solution to BSDE (32) . roof. If Condition 3.1 holds then the gradient of log G full has a linear growth bound, and hencethe integrability condition E Z T k a ( Y ( t )) ⊤ ∇ G full ( t, Y ( t )) k dt ≤ C E Z T | G full ( t, Y ( t )) | (1 + k Y ( t ) k ) dt ≤ C (cid:18) E Z T | G full ( t, Y ( t ))) | δ dt (cid:19) /δ (cid:18) E Z T k Y ( t ) k δ dt (cid:19) /δ , (36)where δ , δ ≥ with δ + δ = 1 . From the duality bound V full ( t, x, y ) ≤ U ⋆ ( pe − r ( T − t ) ) E "(cid:18) E ( T ) E ( t ) (cid:19) γ − γ (cid:12)(cid:12)(cid:12) Y ( t ) = y + xp , we have E sup t ∈ [0 ,T ] | G full ( t, Y ( t ))) | δ < ∞ if E sup t ∈ [0 ,T ] (cid:18) E (cid:20)(cid:16) E ( T ) E ( t ) (cid:17) γ − γ (cid:12)(cid:12)(cid:12) Y ( t ) (cid:21)(cid:19) δ < ∞ , and sotaking steps similar to those in the proof of Proposition A.3 it follows that a sufficient conditionfor finiteness of inequality (36) are the inequalities of (35); because δ ≥ it follows from the firstinequality of (35) that G full ( t, Y ( t ))) ∈ S T ( P full ) .In the literature, Proposition 6.3.2 in Chapter 6.3 of [Pha09] shows equation (34) to be in S T ( P full ) × H T ( P fullq ) if G full ( t, y ) has at most linear growth in y and if the gradient has a bound ofpolynomial growth k a ( y ) ⊤ ∇ G full ( t, y ) k ≤ C (1 + k y k n ) for some C ≥ and n ≥ . Proposition 3.3. If a unique solution to (32) exists, then π ∗ ( t ) = π ∗ ( t, Y ( t ) , χ ( t ) , ψ ( t )) given by (33) is such that U ( X π ∗ ( t )) χ ( t ) satisfies a verification lemma, and hence π ∗ is the optimal strategy.Proof. (See Appendix B).If there exists a solution to BSDE (32) then it is unique: Theorem 3.3. If there exists ( χ , ψ ) ∈ S T ( P full ) × H T ( P full q ) that is a solution to BSDE (32) , thenit is also the unique solution.Proof. (See Appendix C). Remark 8. Condition 3.1, Proposition 3.2, Proposition 3.3, and Theorem 3.3 contributed towardexistence of BSDE solutions to solve the full-information control problems. Sections 4 and 5 providefinancial examples with explicit formulae for classical solutions. Remark 9 (Existence in the Absence of Classical Solutions) . The solution to (32) can existwithout the existence of a classical solution to (28) . A solution ( χ , ψ ) has associated with it aviscosity solution to (28) , i.e., there is a deterministic function G full such that χ ( t ) = G full ( t, Y ( t )) almost surely, where G full is a viscosity solution of (28) ,(see Proposition 6.3.3 in Chapter 6.3 of [Pha09]). However, existence of a viscosity solution maynot be sufficient for existence of a solution to (32) , as (i) G full must be square integrable and (ii)it is not clear how to construct ψ from the viscosity solution. Moreover, a solution to (32) might e identified if the viscosity solution is unique, but the current theory for uniqueness of viscositysolutions requires the PDE to satisfy a strong comparison principle and also some growth conditions[Kob00, Pha09] that are not satisfied by the nonlinear term F ( t, y, g, η ) . Lastly, it should be pointedout that the operator L is what is called degenerate elliptic, and so a classical solution to (28) isalso a viscosity solution (see [CIL92]), reaffirming that (34) is the appropriate formula if there isregularity. Intuitively it would seem that full information is better than partial –or at least that it cannothurt investment. This is correct, but the full-information market is incomplete because Y ( t ) isnot tradeable, and cannot be reduced to a complete market like that given in (7). Generallyspeaking, there is added premium and lowered utility when a model is incomplete. However, partialinformation is an exception, as it turns out that the partially-informed investor expects the fullyinformed to have an advantage.From the perspective of the partially-informed investor, the information premium (i.e., the lossin utility due to partial information) is, Π( t, x ) , E h V full ( t, x, Y ( t )) − V ( t, x ) (cid:12)(cid:12)(cid:12) F St i = U ( xe − r ( T − t ) ) E h G full ( t, Y ( t )) − G ( t ) (cid:12)(cid:12)(cid:12) F St i . This is similar to the loss of information quantified in [Bre06, Car09] for the linear Gaussian problem,but is quantified with BSDEs for the general nonlinear case. Proposition 3.4. The information premium is equal to Π( t, x )= (1 − γ ) U ( xe r ( T − t ) ) × E " Z Tt F ( Y ( u ) , χ ( u ) , ψ ( u )) − γ β ( u, α ( u ) , ξ ( u ))(1 − γ ) ξ ( t ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( u ) ξ ( u ) (cid:13)(cid:13)(cid:13)(cid:13) ! G ( u ) | {z } ≥ ! du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F St ≥ , (37) where (1 − γ ) U ( x ) ≥ by definition for all x ≥ .Proof. The fully-informed investor has the option to follow the partially-informed optimal strategy,hence, E h V full ( t, x, Y ( t )) (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } i = E " sup π ∈A full E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) F t ∨ { X π ( t ) = x } i (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } ≥ E (cid:20) sup π ∈A E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) F t ∨ { X π ( t ) = x } i (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } (cid:21) ≥ sup π ∈A E h E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) F t ∨ { X π ( t ) = x } i (cid:12)(cid:12)(cid:12) F St ∨ { X π ( t ) = x } i = V ( t, x ) , (38)20or all t ∈ [0 , T ] and all x ≥ , and Π( t, x ) ≥ for all x > and t ∈ [0 , T ) .Using the BSDEs of (32) and the inequality shown in (38), the information premium is written as Π( t, x )= U ( xe r ( T − t ) ) E (cid:2) G full ( t, Y ( t )) − G ( t ) |F St (cid:3) = (1 − γ ) U ( xe r ( T − t ) ) × E " Z Tt F ( Y ( u ) , χ ( u ) , ψ ( u )) − γ β ( u, α ( u ) , ξ ( u ))(1 − γ ) ξ ( t ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( u ) ξ ( u ) (cid:13)(cid:13)(cid:13)(cid:13) !| {z } ≥ G ( u ) ! du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F St ≥ , where (1 − γ ) U ( x ) ≥ by definition for all x ≥ , F ( Y ( t ) , χ ( t ) , ψ ( t )) ≥ for all t ∈ [0 , T ] , and β ( t, α ( t ) , ξ ( t ))(1 − γ ) ξ ( t ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13) = 12 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) σ − ˆ h ( t ) − r γ + α ( t ) ξ ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ by the formula for β ( t, α ( t ) , ξ ( t )) given in Theorem 3.1.The importance of (37) is that it shows how the information premium incrementally grows withtime. Alternatively, one could look at d (cid:0) E (cid:2) G full ( t, Y ( t )) − G ( t ) |F St (cid:3)(cid:1) , but the BSDEs provide adifferent perspective because the coefficients provide a breakdown of the premium’s growth.Before moving to the next section, it should be pointed out how the information premium canbe either infinite or undefined. The obvious lower bound V ( t, x ) ≥ U ( xe r ( T − t ) ) is obtained with π ≡ , and leads to the upper bound Π( t, x ) ≤ U (cid:16) xe r ( T − t ) (cid:17) (cid:16) E h G full ( t, Y ( t )) (cid:12)(cid:12)(cid:12) F St i − (cid:17) . These bounds depend on finiteness of the full-information value function, and so investor nirvanafor full information occurring with non-zero probability results in either • Π( t, x ) = ∞ because V ( t, x ) < ∞ and E [ V full ( t, x, Y ( t )) |F St ] = ∞ , • Π( t, x ) = ∞ − ∞ (undefined) because V ( t, x ) = ∞ and E [ V full ( t, x, Y ( t )) |F St ] = ∞ .These two cases are considered at the end of Section 4. It should also be pointed out that theinformation premium is usually positive, as shown numerically in [FPS15, FPS17, Pap13]. Consider the linear case with h ( y ) = µ + y . Suppose that Y ( t ) ∈ R is an Ornstein-Uhlenbeckprocess, and there is only one risky asset so that S ( t ) ∈ R . The SDEs are dS ( t ) S ( t ) = ( µ + Y ( t )) dt + σ (cid:16)p − ρ dW ( t ) + ρdB ( t ) (cid:17) (39) dY ( t ) = − κY ( t ) dt + adB ( t ) , (40)21ith κ, a, σ > , ρ ∈ ( − , , and µ ∈ R being the long-term mean rate of return. The wealthprocess is dX π ( t ) X π ( t ) = rdt + π ( t ) (cid:18) dS ( t ) S ( t ) − rdt (cid:19) = (cid:16) π ( t )( µ + Y ( t )) + r (1 − π ( t )) (cid:17) dt + π ( t ) σ (cid:16)p − ρ dW ( t ) + ρdB ( t ) (cid:17) . For simplicity take r = µ = 0 . This model is the scalar version of the model considered in [Bre06,Car09, WW08], except that they avoided nirvana situations by considering the case of γ > .Indeed, this section considers γ < and examines the stability of a scalar Riccati equation, whereasstability of the matrix Riccati equation in [Bre06, Car09] would require a significantly more difficultanalysis. The optimal investment problem for full information is V ( t, x, y ) = sup π E h U ( X ( T )) (cid:12)(cid:12)(cid:12) X ( t ) = x, Y ( t ) = y i , which is the solution V ( t, x, y ) to the HJB equation V t + a V yy − κyV y − ( yV x + ρσaV xy ) σ V xx = 0 V (cid:12)(cid:12)(cid:12) t = T = U , where the optimal portfolio is π ∗ = − x yV x + ρσaV xy σ V xx . For power utility U ( x ) = − γ x − γ the solution of the HJB equation is given by the ansatz V ( t, x, y ) = U ( x ) G ( t, y ) , which yields the following equation for G : G t + a G yy − κyG y + 1 − γγ ( yG + ρσaG y ) σ G = 0 G (cid:12)(cid:12)(cid:12) t = T = 1 , where π ∗ = yγσ + ρaG y γσG . We now apply another ansatz, G ( t, y ) = exp (cid:16) A ( t ) y + H ( t ) (cid:17) , for which there are the ordinary differential equations A ′ ( t ) + 2 a (cid:18) − γ ) ρ γ (cid:19) A ( t ) − (cid:18) κ − (1 − γ ) ρaγσ (cid:19) A ( t ) + 1 − γ γσ = 0 (41) H ′ ( t ) + a A ( t ) = 0 , (42)22ith terminal conditions A ( T ) = 0 = H ( T ) apply. Then the optimal control is π ∗ ( t ) = yγσ + 2 ρayA ( t ) γσ . Let A ± be the roots of the polynomial a (cid:16) (1 − γ ) ρ γ (cid:17) A ( t ) − (cid:16) κ − (1 − γ ) ρaγσ (cid:17) A ( t ) + − γ γσ . Fromthe quadratic equation, these roots are found to be A ± = 2 (cid:16) κ − (1 − γ ) ρaγσ (cid:17) ± r (cid:16) κ − (1 − γ ) ρaγσ (cid:17) − (1 − γ ) a γσ (cid:16) (1 − γ ) ρ γ (cid:17) a (cid:16) (1 − γ ) ρ γ (cid:17) , (43)and the Riccati equation (41) is written as A ′ ( t ) = − c A ( t ) − A + )( A ( t ) − A − ) , (44)where c = 4 a (cid:16) (1 − γ ) ρ γ (cid:17) . The roots A ± given by equation (43) are real iff ≤ (cid:18) κ − (1 − γ ) ρaγσ (cid:19) − (1 − γ ) a γσ (cid:18) − γ ) ρ γ (cid:19) = κ − (1 − γ ) aγσ (cid:16) κρ + aσ (cid:17) . (45)Instabilities can arise if the roots are complex. The best way to understand why is to look atthe linearization of the Riccati equation (41). Letting v ( t ) be the solution to the following linearequation, v ′′ − (cid:18) κ − (1 − γ ) ρaγσ (cid:19) v ′ + (1 − γ ) a γσ (cid:18) − γ ) ρ γ (cid:19) v = 0 , with the appropriate terminal conditions v ′ ( T ) = 0 and v ( T ) = 0 , the solution to Riccati equation(41) is A ( t ) = v ′ ( t ) / (2 a v ( t )) . For a characteristic equation with complex roots, the solution is v ( t ) = e (cid:16) κ − (1 − γ ) ρaγσ (cid:17) ( T − t ) (cid:16) C cos(Ξ( T − t )) + C sin(Ξ( T − t )) (cid:17) , where Ξ is the absolute value of the imaginary component Ξ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s κ − (1 − γ ) aγσ (cid:16) κρ + aσ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and where the constants are chosen to match the terminal conditions, so that C (cid:16) κ − (1 − γ ) ρaγσ (cid:17) = C .Investor nirvana occurs because it may be that v ( t ) = 0 for some t ∈ [0 , T ] . If this is the case, then A ( t ) will blow up at some finite time ≤ t < T . An example of such an instability is κρ + aσ > and γ tending toward zero. Another instability occurs for κ tending toward zero with constant γ ∈ (0 , . 23 .2 The Partially-Informed Investor Letting b Y ( t ) = E [ Y ( t ) |F St ] , the innovations process is ν ( t ) = Z t (cid:18) dS ( u ) S ( u ) − b Y ( u ) du (cid:19) , for which σ ν ( t ) is a Brownian motion. Furthermore, letting Σ( t ) = E ( Y ( t ) − b Y ( t )) , the investortracks the hidden process Y ( t ) using the Kalman filter d b Y ( t ) = − κ b Y ( t ) dt + 1 σ (Σ( t ) + σaρ ) dν ( t ) ddt Σ( t ) = − κ (cid:18) Σ( t ) − a (1 − ρ )2 κ (cid:19) − aρσ Σ( t ) − (cid:18) σ Σ( t ) (cid:19) , where for t large there is the asymptotic Σ( t ) → Σ as t ր ∞ with Σ = − ( κσ + aρσ ) + r ( κσ + aρσ ) + (cid:16) aσ p − ρ (cid:17) , (46)Assuming Σ(0) = Σ , then ddt Σ( t ) = 0 for all t > and the partial-information model is written withconstant coefficients and the innovations, dS ( t ) S ( t ) = b Y ( t ) dt + σdζ ( t ) d b Y ( t ) = − κ b Y ( t ) dt + ¯ adζ ( t ) , where ζ ( t ) = σ ν ( t ) is a Brownian motion and ¯ a = σ (cid:0) Σ + σaρ (cid:1) . Hence, the partial-informationmodel is equivalent to the full-information model in equations (39) and (40) having ρ = 1 anddiffusion coefficient ¯ a .An investor achieves nirvana when V ( t, x, y ) = ∞ for γ ∈ (0 , , and when V ( t, x, y ) = 0 for γ > (see Definition 2.1 or [KO96]). Propositions 2.1 and 3.1 showed nirvana cannot occur for γ > for both partial and full information, respectively. For partial-information this can be verifiedfor the linear model by investigating the parameters. Similar to the condition set forth in (45), thepartial-information ansatz involves a real root iff κ − (1 − γ )¯ aγσ (cid:16) κ + ¯ aσ (cid:17) ≥ . (47)For γ > a minimum of zero is achieved in (47) when ¯ a = − κσ . Indeed, from (46) it is seen that ¯ a = σ (cid:0) Σ + σaρ (cid:1) ≥ − kσ , so Proposition 2.1 is verified for γ > because there cannot be a complexroot.For γ ∈ (0 , there are some interesting cases of investor-nirvana occurrence: Example 4.1 (Infinite Information Premium) . Suppose − < ρ < , − ρ < κ < , σ ≥ , and a such that − ρκσ < a < √− κρσ . Then from (47) it is seen that the partially-informed investorwill never achieve nirvana for γ ∈ (0 , because κ + Σ + σaρσ < , ut from (45) it is seen that the fully-informed investor will achieve nirvana as γ tends toward zerobecause κρ + aσ > . Hence, the information premium is infinite if the investors are given enough time. Example 4.2 (Undefined Information Premium) . The parameters can be selected so that the in-formation premium from Section 3.4 is undefined (i.e., equal to the difference ∞ − ∞ ). Suppose − √ < ρ < and a = − κσρ . Then Σ = aσ p − ρ , and the partially-informed investor will achievenirvana as γ tends toward zero because (47) is violated, κ + Σ + σaρσ = 2 κ + a ( p − ρ + ρ ) σ > . The fully-informed investor will also achieve nirvana because (45) is violated κρ + aσ > . Hence, the information premium is undefined if both investors have a long enough investment period. Example 4.3 (Simulation of Paths) . For γ > the Riccati equation for A ( t ) can be solved explicitly,which allows for easy simulation of the BSDE solutions and the G functions under both partial andfull information. For γ > it follows that A + > > A − , so A − is the long-term equilibrium of A ( t ) , and equations (41) and (42) have explicit solutions, A ( t ) = A − − e − D ( T − t ) − A − A + e − D ( T − t ) H ( t ) = a A − ( T − t ) − cA − log A + − A − e − D ( T − t ) A + − A − !! , where A ± is given by (43) and D = 2 r(cid:16) κ − (1 − γ ) ρaγσ (cid:17) − (1 − γ ) a γσ (cid:16) (1 − γ ) ρ γ (cid:17) , and where c is thesame as that used in (44) . As γ > it follows that D > , and so the solution is stable for large T .Figure 1 shows a simulation of the linear model with the parameters given in Table 1. Thesimulation is informative because it shows how paths of G ( t ) and G full compare; in particular itshows how it is possible for G ( t ) < G full ( t, Y ( t )) even though Proposition 3.4 has shown G ( t ) ≥ E [ G full ( t, Y ( t )) |F St ] for γ > . Parameter Values κ a ρ σ T γ The parameters for the simulation shown in Figure 1. Asset Price t Drift Process E[Y|F]Y t G Process G partialG full t × -3 -202468101214 Info Premium Figure 1: Simulation of the linear model using the parameters of Table 1, with µ = r = 0 . Top Left: The simulated asset price S ( t ) . Top Right: The simulated Y ( t ) and the filter. Bottom Left: TheBSDE solutions G ( t ) and G full ( t, Y ( t )) . Bottom Right: The difference G ( t ) − G full ( t, Y ( t )) , for whichthere are a few times t ∈ [0 , T ] when G ( t ) < G full ( t, Y ( t )) even though Proposition 3.4 has shown G ( t ) ≥ E [ G full ( t, Y ( t )) |F St ] for γ > . A Nonlinear Example Recall the example from Remark 1. Suppose that Y ( t ) ∈ R is a CIR process, and there is only onerisky asset so that S ( t ) ∈ R . The SDEs are dS ( t ) S ( t ) = c p Y ( t ) dt + σ (cid:16)p − ρ dW ( t ) + ρdB ( t ) (cid:17) (48) dY ( t ) = κ ( ¯ Y − Y ( t )) dt + a p Y ( t ) dB ( t ) , (49)with < a ≤ κ ¯ Y , ρ ∈ ( − , , c ∈ R , and ¯ Y > being the long-term level of Y ( t ) . The wealthprocess is dX π ( t ) X π ( t ) = rdt + π ( t ) (cid:18) dS ( t ) S ( t ) − rdt (cid:19) = (cid:16) cπ ( t ) p Y ( t ) + r (1 − π ( t )) (cid:17) dt + π ( t ) σ (cid:16)p − ρ dW ( t ) + ρdB ( t ) (cid:17) . In this example take γ > to avoid nirvana situations. For simplicity take r = 0 and ρ = 0 so thatthe model is affine. The value function for power utility has an explicit solution. Similar to the fully-informed investorin the linear example of Section 4, it is shown in [Zar01] for ansatz V full ( t, x, y ) = U ( x ) G full ( t, y ) , that G solves the PDE (in this case for ρ = 0 ) G full t + a y G full yy + κ ( ¯ Y − y ) G full y + c (1 − γ )2 γσ yG full = 0 G full (cid:12)(cid:12)(cid:12) t = T = 1 . Using the ansatz, G full ( t, y ) = exp (cid:16) A ( t ) y + H ( t ) (cid:1) , the solution uses functions A and H satisfying the equations A ′ ( t ) + a A ( t ) − κA ( t ) + c (1 − γ )2 γσ = 0 (50) H ′ ( t ) + κ ¯ Y A ( t ) = 0 . (51)Similar to Example 4.3, equations (50) and (51) have explicit solutions, A ( t ) = A − − e − D ( T − t ) − A − A + e − D ( T − t ) H ( t ) = κ ¯ Y A − ( T − t ) − a A − log A + − A − e − D ( T − t ) A + − A − !! , A ± = κ ± q κ − c (1 − γ ) γσ a a D = s κ − c (1 − γ ) γσ a . Direct simulation of Z ( t ) from equation (8) allows for a numerical approximation of the first com-ponent of the solution to BSDE (16). Namely, an approximation of ξ from the dual value functionin (11) with a Monte Carlo expectation, where the expectation to be approximated is simplifiedusing Itô’s lemma as done in the proof of Proposition A.3, ξ ( t ) = E "(cid:18) Z ( T ) Z ( t ) (cid:19) − − γγ (cid:12)(cid:12)(cid:12) F St = E (cid:20) exp (cid:18) (1 − γ ) c γ σ Z Tt b Y ( u ) du (cid:19) (cid:12)(cid:12)(cid:12) F St (cid:21) ≈ N N X ℓ =1 exp (cid:18) (1 − γ ) c γ σ Z Tt b Y ( ℓ,t ) ( u ) du (cid:19) , for sample size N , where for each ℓ there is an independent sample ( b Y ( ℓ,t ) ( u )) u ∈ [ t,T ] conditional on F St . Samples of b Y ( ℓ,t ) ( t ) are obtained from forward sequential Monte Carlo (SMC) and computationof the filter. To compute the filter, one can either compute a particle filter for each trajectory of S , or one can approximate Y with a finite-state Markov chain and then use a repeated applicationof Bayes rule over a small time step. The latter approach has been taken here because it is bothfast and accurate (i.e., because Y does not have a heavy tail) for this model. Note that simulationof ξ ( t ) is like a branching process: for two times t, t + ∆ t ∈ [0 , T ] the particles initialized at time t cannot be reused for the simulation of particles to be initialized at t + ∆ t (see [HLTT14] for moreon branching processes’ relation to BSDEs).The optimal value function is V ( t, x ) = U ( x ) ξ ( t ) γ , and so the information premium is seen by comparing G ( t ) = ξ ( t ) γ to G full ( t, Y ( t )) . Using Jensen’sinequality in the same manner as in Remark 1, Condition 2.1 (Novikov) is satisfied if c T σ < κa , in which case Z ( t ) is a true F St martingale.Figure 2 shows a comparison of full and partial information for realizations obtained with pa-rameters from Table 2. Noteworthy aspects in this example are: • Compared to the filters in Figure 1, the filters in Figure 2 do not do as good of a job trackingthe hidden drift p Y ( t ) . The reason is because the linear example has a strong correlation of − . , which increases the signal-to-noise ratio (SNR) . In contrast, this nonlinear example haszero correlation and hence much lower SNR.28 Compared to the coefficients G shown in Figure 1, the partial-information G in Figure 2 issmoother. This is due to the lack of tracking in the filer (see previous bullet point). • The coefficients G in Figure 2 have steeper slopes than those in Figure 1. This is because thefilter b Y ( t ) is almost constant in time, b Y ( t ) ≈ ¯ Y = . , which means positive average portfolioreturn, and G ( t ) ≈ exp (cid:16) (1 − γ ) ¯ Y t γσ (cid:17) . Comparatively, the linear example has parameters chosenso that b Y ( t ) ≈ for a net-zero average return. In other words, the parameters are such thatthe Sharpe ratios are higher in this nonlinear example. • The optimal π for partial information has not been computed because no numerical methodwas proposed.This fourth point is a reiteration of a comment in Remark 3, where it was pointed out that θ from the martingale representation is difficult to compute and requires a special numerical method;a numerical method for α would accomplish as much. In general, these bullets points highlightpossible topics for future exploration in the area of numerical BSDE.Finally, it should be pointed out that the information premium is low in this nonlinear example,which is seen by observing that G and G full are close together in Figure 2. The reason for this isbecause ¯ Y > r = 0 with Sharpe ratios ˆ Y t /σ ≈ ¯ Y /σ = . and equal to . for σ = . and σ = . ,respectively, and so both the partially and fully-informed investors are placing a significant portionof their wealth into the risky asset. Comparatively, the linear case of Example 4.3 would have amore pronounced premium if ρ = 0 ; this would be the case because of low SNR, in which case thefilter remains close to zero (i.e., ˆ Y t ≈ for all t ) causing the Sharpe ratio to be very close zero, andtherefore the partially-informed investor would invest very little in the risky asset and experiencenone of the improved portfolio returns. Parameter Values c κ ¯ Y a T γ .25 8 .05 .4 1 1.2Table 2: Parameter values for the nonlinear example in equations (48) and (49). Different values of σ aretested, namely a low value of . and a high value of . . Note the if the value of σ is too low then Condition5 will fail and it is possible for Z ( t ) to have E [ Z ( T ) /Z ( t ) |F St ] < . Investment with filtering under partial information is a non-Markov control problem, but also hassome simplicity because the model can be reduced to a complete market. For the case of investorswith a power utility function, the dual value function is shown to be the solution to a BSDE. Theoptimal strategy is also shown to be expressed in terms of the solution to the BSDE, and canbe broken into two components: a myopic component where point estimate of Y t is inserted intothe standard Merton problem, and a hedging term due to stochastic drift. In comparison withfull information, the information premium is defined to be the expected loss in utility (from theperspective of the partially informed investor), and quantified in terms of the coefficients of theBSDEs.A possible direction for future work on this problem is on the development of numerical methodsfor solving the partial-information BSDE; the proposed Monte Carlo approximation of Section 5 is29 Drift Process for σ = 0.026 cE[Y |F]cY Drift Process for σ = 0.15 cE[Y |F]cY G(t) Function for σ = 0.026 G partialG full t G(t) Function for σ = 0.15 G partialG full Figure 2: A low-noise simulation with σ = . and a high-noise example with σ = . . Top Left: Thesimulated low-noise Y ( t ) and its filter. Top Right: The simulated high-noise Y ( t ) and its filter. BottomLeft: The low-noise G ( t ) ’s with sample size N = 10 . Bottom Right: A high-noise G ( t ) ’s with sample size N = 10 . 30 small step towards this goal. Monte Carlo and particle filtering will be useful, but there is likelyto be an exponentially growing number of states taken by the filter, and so further innovation isneeded. A Proofs for Section 3.1 Proposition A.1. If Condition 2.3 holds, then θ ∈ H T ( P d ) where θ is the martingale representationin (15) .Proof. A stochastic integral is a local martingale, so there is an increasing family of stopping times ( τ j ) j =1 , , ,... such that τ j ր ∞ almost surely and R t ∧ τ j θ ( u ) ⊤ dζ ( u ) is a true martingale. Then E Z T ∧ τ j k θ ( t ) k dt = − E (cid:20) − Z T ∧ τ j k θ ( t ) k dt + Z T ∧ τ j θ ( t ) ⊤ dζ ( t ) (cid:21) = − E log (cid:16) M ( T ∧ τ j ) . M (0) (cid:17) (because dM ( t ) = M ( t ) θ ( t ) ⊤ dζ ( t ) in (15)) , = − E log (cid:16) E h Z ( T ) − − γγ (cid:12)(cid:12)(cid:12) F ST ∧ τ j i . M (0) (cid:17) ≤ − E log (cid:16) Z ( T ) − − γγ (cid:17) + 2 log M (0) (Jensen’s inequality) = 2(1 − γ ) γ E log Z ( T ) + 2 log M (0)= − − γγ E Z T (cid:13)(cid:13)(cid:13) σ − (ˆ h ( t ) − r ) (cid:13)(cid:13)(cid:13) dt + 2 log M (0) < ∞ . This implies E R T k θ ( u ) k du ≤ lim inf j E R T ∧ τ j k θ ( u ) k du < ∞ . Proposition A.2. Let ( ξ, α ) ∈ S T ( P ) × H T ( P d ) be a solution to (16) , and let ( ξ K , α K ) ∈ S T ( P ) × H T ( P d ) be the unique solution in S T ( P ) × H T ( P d ) for the bounded BSDE in (19) (in fact ξ K ∈ S ∞ T ( P ) as shown in [Kob00]). For the stopping time τ K = inf n t > s.t k ˆ h ( t ) k ≥ K o , the solutions are equal for all ω ∈ Ω such that τ K ≥ T . That is, ( ξ ( t ) − ξ K ( t )) [ τ K >T ] = 0 for all t ∈ [0 , T ] , and ( α ( t ) − α K ( t )) [ τ K >T ] = 0 for all t ∈ [0 , T ] .Proof. The proof is by contradiction. Letting O = { ω ∈ Ω s.t τ K ≥ T } . Suppose ( ξ, α ) = ( ξ K , α K ) for some ω ∈ O . Then there is another solution to (19), ( ˜ ξ K , ˜ α K ) = (cid:26) ( ξ, α ) for ω ∈ O ( ξ K , α K ) for ω / ∈ O , with ( ˜ ξ K , ˜ α K ) = ( ξ K , α K ) , but the solution to (19) is unique. Hence there is a contradiction. Proposition A.3. Let ( ξ K , α K ) ∈ S ∞ T ( P ) × H T ( P d ) be the unique solution to the BSDE in (19) .If Condition 2.3 holds, then sup K> E sup t ∈ [0 ,T ] | ξ K ( t ) | < ∞ . roof. Recall the notation ˆ h K ( t ) and Z K ( t ) from the proof of Theorem 3.1. Applying Itô’s lemmato Z K ( t ) − − γγ yields a forward SDE, d (cid:16) Z K ( t ) − − γγ (cid:17) = ( γ − γ − γ Z K ( t ) − − γγ k σ − (ˆ h K ( t ) − r ) k dt − γ − γ Z K ( t ) − − γγ (ˆ h K ( t ) − r )( σ − ) ⊤ dζ ( t ) . This SDE is a true semi-martingale because ˆ h K is bounded, and so using variation of constants(i.e., integrating factor) and taking expectations yields an upper bound, E Z K ( T ) − − γγ = E exp (cid:18) ( γ − γ − γ Z T k σ − (ˆ h K ( t ) − r ) k dt (cid:19) ≤ E exp (cid:18) | γ − || γ − | ǫγ Z T (cid:16) k ˆ h ( t ) k + k r k (cid:17) dt (cid:19) , where ǫ > is the constant from (3) that bounds σ . Now notice the solution to BSDE (19) has thefollowing martingale bound, ξ K ( t ) = E "(cid:18) Z K ( T ) Z K ( t ) (cid:19) − − γγ (cid:12)(cid:12)(cid:12) F St = E (cid:20) exp (cid:18) − γ − γ Z Tt k σ − (ˆ h K ( u ) − r ) k du (cid:19) (cid:12)(cid:12)(cid:12) F St (cid:21) ≤ E (cid:20) exp (cid:18) − γ − γ Z T k σ − (ˆ h K ( u ) − r ) k du (cid:19) (cid:12)(cid:12)(cid:12) F St (cid:21) = E h Z K ( T ) − − γγ (cid:12)(cid:12)(cid:12) F St i , for which the last quantity is a continuous martingale, with continuity because it has a martingale-type-representation like that in equation (15). Hence, from the Doob maximal inequality it is seenthat E sup t ∈ [0 ,T ] | ξ K ( t ) | ≤ E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) E h Z K ( T ) − − γγ (cid:12)(cid:12)(cid:12) F St i (cid:12)(cid:12)(cid:12) ≤ E Z K ( T ) − − γγ ≤ E exp (cid:18) | γ − || γ − | ǫγ Z T (cid:16) k ˆ h ( u ) k + k r k (cid:17) du (cid:19) < ∞ , where the second inequality is from Doob and where finiteness is given by Condition 2.3, and hencethe supremum over K is finite. B Verification Lemma for Full Information This Appendix contains the verification proof for Proposition 3.3 from Section 3.3. For any ad-missible π ∈ A full consider the stopped SDE for U ( X π ( t ) e r ( T − t ) ) χ ( t ) , and let τ k be an increasing32equence of stopping times with τ k ∧ T → T a.s. and for which the stochastic integral is a truemartingale. The expectation satisfies E h U (cid:16) X π ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i = U (cid:0) xe rT (cid:1) χ (0)+ (1 − γ ) E "Z T ∧ τ k U (cid:16) X π ( u ) e r ( T − u ) (cid:17) χ ( u ) π ( u ) ⊤ (cid:16) h ( Y ( u )) − r − γ π ( u ) (cid:17) + π ( u ) ⊤ σ y ψ ( u ) − F ( Y ( u ) , χ ( u ) , ψ ( u )) ! du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y + E "Z T ∧ τ k U (cid:16) X π ( u ) e r ( T − u ) (cid:17) χ ( u ) (1 − γ ) π ( u ) ⊤ ( σ w dW ( u ) + σ y dB ( t ))+ ψ ( u ) χ ( u ) dB ( u ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y = U (cid:0) xe rT (cid:1) χ (0)+ (1 − γ ) E "Z T ∧ τ k U (cid:16) X π ( u ) e r ( T − u ) (cid:17) χ ( u ) π ( u ) ⊤ (cid:16) h ( Y ( u )) − r − γ π ( u ) (cid:17) + π ( u ) ⊤ σ y ψ ( u ) − F ( Y ( u ) , χ ( u ) , ψ ( u )) ! du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y ≤ U (cid:0) xe rT (cid:1) χ (0) , where the inequality becomes an equality by inserting F from (31) and π ( u ) = π ∗ ( u, Y ( u ) , χ ( u ) , ψ ( u )) given by equation (33). Hence, U (cid:0) xe rT (cid:1) χ (0)= E h U (cid:16) X π ∗ ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i ≤ sup π ∈A full E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i = V full (0 , x, y ) . Verification is to show inequality in the other direction for the limit. B.1 Case < γ < For < γ < , using Fatou’s lemma in the limit as k → ∞ yields E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i = E (cid:20) lim inf k U (cid:16) X π ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y (cid:21) ≤ lim inf k E h U (cid:16) X π ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i ≤ U (cid:0) xe rT (cid:1) χ (0) . t ∈ [0 , T ] , and hence V full ( t, x, y ) = sup π ∈A full E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) X ( t ) = x, Y ( t ) = y i ≤ U (cid:16) xe r ( T − t ) (cid:17) χ ( t ) , which completes the verification for γ ∈ (0 , . B.2 Case γ > In this case U ( x ) < so Fatou lemma does not apply directly. Let X π ∗ ( T ) = inf ≤ t ≤ T X π ( t ) andassume E U ( X π ∗ ( T )) > −∞ . Then ≤ E h U ( X π ( T )) − U ( X π ∗ ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i = E (cid:20) lim inf k (cid:16) U (cid:16) X π ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) − U ( X π ∗ ( T )) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y (cid:21) ≤ lim inf k E h(cid:16) U (cid:16) X π ( T ∧ τ k ) e r ( T − T ∧ τ k ) (cid:17) − U ( X π ∗ ( T )) (cid:17) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i = U (cid:0) xe rT (cid:1) χ (0) + lim inf k E h − U ( X π ∗ ( T )) χ ( T ∧ τ k ) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i ≤ U (cid:0) xe rT (cid:1) χ (0) − E h U ( X π ∗ ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i . Now E h U ( X π ∗ ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i cancels from both sides and there is the bound E h U ( X π ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i < U (cid:0) xe rT (cid:1) χ (0) . If it cannot be shown that E h U ( X π ∗ ( T )) (cid:12)(cid:12)(cid:12) X (0) = x, Y (0) = y i < ∞ , then a truncation argumentcan be used to show the bound up to an arbitrarily small constant. C Proof of Theorem 3.3 General existence and uniqueness theory for BSDEs can be applied if the problem is truncated tohave Y ( t ) and π ( t ) confined to compact sets. For some positive K < ∞ define the truncated set ofadmissible strategies A full K = A full ∩ ( π : [0 , T ] × Ω → R d s.t. sup t ∈ [0 ,T ] k π ( t ) k < K a.s. ) . Also define the stopping time τ K = inf { t > s.t k Y ( t ) k ≥ K } , and consider the truncated BSDE: − d χ K ( t ) = (1 − γ ) F K ( Y ( t ) , χ K ( t ) , ψ K ( t )) dt − ψ K ( t ) ⊤ dB ( t ) , for t ≤ τ k χ K ( T ∧ τ K ) = 1 , (52)where F K ( y, g, η ) = max k π k≤ K f ( y, π, g, η ) and is well defined because f given by (29) is a concavefunction of π . There is a uniform Lipschitz constant for F K for all t ≤ τ K , and so (52) has a34nique solution ( χ K , ψ K ) ∈ S T ( P full ) × H T ( P fullq ) . The solution to the BSDE is associated witha viscosity solution, χ K ( t ) = G full K ( t, Y ( t )) and ψ K ( t ) = a ( Y ( t )) ⊤ ∇ G full K ( t, Y ( t )) , where G full K is aviscosity solution of the boundary value problem, (cid:18) ∂∂t + L (cid:19) G full K + (1 − γ ) F K (cid:0) y, G full K , σ y ∇ G full K (cid:1) = 0 (53) G full K (cid:12)(cid:12)(cid:12) t = T = 1 G full K (cid:12)(cid:12)(cid:12) k y k = K = 1 . Equation (53) has a unique classical solution, as it meets the criterions for application of Theorem4.1 from Chapter IV.4 of [FS05]. Moreover, as L is degenerate elliptic and the Hessian ∇∇ ⊤ G full K isnot present in the nonlinearity of (53), the unique solution to (53) is also a viscosity solution (see[CIL92]). Hence χ K ( t ) = G full K ( t, Y ( t )) is a viscosity solution, and is the value function χ full K ( t ) = 1 + (1 − γ ) × sup π ∈A full K E "Z T ∧ τ K t ∧ τ K f (cid:16) Y ( u ) , π ( u ) , G full K ( u, Y ( u )) , σ y ∇ G full K ( u, Y ( u )) (cid:17) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . This truncated value function can be used to show uniqueness of solutions to (32). The proof isbased on the following two propositions, Proposition C.1. Suppose there exists ( χ , ψ ) ∈ S T ( P full ) × H T ( P full q ) a solution to (32) , inparticular that E sup t ∈ [0 ,T ] | χ ( t ) | < ∞ . Then sup K> E sup t ∈ [0 ,T ] | χ K ( t ) | ≤ E sup t ∈ [0 ,T ] | χ ( t ) | < ∞ , where ( χ K , ψ K ) ∈ S T ( P full ) × H T ( P full q ) is a solution to (52) .Proof. Start with the case γ ∈ (0 , . For any ( t, y, g, p ) ∈ [0 , T ] × R q × R + × R q , F K ( y, g, p ) ≤ F ( y, g, p ) . Hence, ≤ χ K ( t ) = χ K ( t ∧ τ K ) ≤ χ ( t ∧ τ K ) by a comparison principle (see Proposition2.9 in [Kob00]), and sup K> E sup t ∈ [0 ,T ] | χ K ( t ) | ≤ sup K> E sup t ∈ [0 ,T ] | χ ( t ∧ τ K ) | ≤ E sup t ∈ [0 ,T ] | χ ( t ) | < ∞ , because sup t ∈ [0 ,T ] | χ ( t ∧ τ K ) | ≤ sup t ∈ [0 ,T ] | χ ( t ) | .For γ > the comparison is made by looking at ≥ (1 − γ ) F K ( y, g, p ) ≥ (1 − γ ) F ( y, g, p ) ,which implies ≥ χ K ( t ) = χ K ( t ∧ τ K ) ≥ χ ( t ∧ τ K ) . Taking expectations of squares yields sup K> E sup t ∈ [0 ,T ] | χ K ( t ) | ≤ E sup t ∈ [0 ,T ] | χ ( t ∧ τ K ) | ∨ ≤ E sup t ∈ [0 ,T ] | χ ( t ) | ∨ < ∞ . Proposition C.2. Suppose there exists ( χ , ψ ) ∈ S T ( P full ) × H T ( P full q ) a solution to (32) . Then ( χ ( t ) − χ K ( t )) [ τ K ≥ T ] = 0 almost surely for all t ∈ [0 , T ] ,where ( χ K , ψ K ) ∈ S T ( P full ) × H T ( P full q ) is a solution to (52) . roof. The proof is by contradiction and (similar to that of Proposition A.2). 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