Forward BSDEs and backward SPDEs for utility maximization under endogenous pricing
aa r X i v : . [ q -f i n . M F ] M a y Forward BSDEs and backward SPDEs for utility maximizationunder endogenous pricing ∗ Thai Nguyen † and Mitja Stadje ‡ May 12, 2020
Abstract
We study the expected utility maximization problem of a large investor who is allowedto make transactions on a tradable asset in a financial market with endogenous perma-nent market impacts as suggested in [24] building on [6, 7]. The asset price is assumedto follow a nonlinear price curve quoted in the market as the utility indifference curveof a representative liquidity supplier. We show that optimality can be fully character-ized via a system of coupled forward-backward stochastic differential equations (FBSDEs)which is equivalent to a highly non-linear backward stochastic partial differential equation(BSPDE). Existence results can be achieved in the case where the driver function of therepresentative market maker is quadratic or the utility function of the large investor isexponential. Explicit examples are provided when the market is complete or the driverfunction is positively homogeneous.
This paper studies a stochastic optimal control problem with feedback effect using forwardbackward stochastic differential equations (FBSDEs) and backward stochastic partial differ-ential equations (BSPDEs). Consider a large trader or a financial investor who trades riskyassets like a financial derivative in a financial market. Classical financial mathematics sup-poses that the trader is a price taker in the sense that the prices of the risky assets (and thestochastic process representing these) at every time instance are exogenously given and arein particular not affected by the trades of the investor. However, it is well known that thebuying and selling of larger positions actually influences prices by affecting the supply or thevolatility of the underlying asset, see for example [12, 57, 29]. We consider a model with per-manent market impacts and analyze a utility maximization problem for a large investor whois trading risky assets in a financial markets where the trading influences the future prices,and price curves are non-linear in volume. Our model captures endogenously such phenomenaas nonlinearity in liquidation and market contractions due to illiquidity. ∗ The first author would like to express his gratitude to the Institute of Insurance Science, University of Ulmfor the support while he was a postdoctoral research associate at the University of Ulm. † Universit´e Laval, ´Ecole d’Actuariat, Qu´ebec city, Canada and School of Economic Mathematics-Statistics,University of Economics Hochiminh city, Vietnam. Email: [email protected], [email protected] ‡ University of Ulm, Institute of Insurance Science and Institute of Financial Mathematics, Ulm, Germany.Email: [email protected] g -expectation and the pricing and hedging can be done by solving a semi-linearPDE in the special case of a Markovian setting. The latter paper also provides a completenesscondition under which any derivative can be perfectly replicated by a dynamic trading strategy.Contrary to exogenously based liquidity models, an endogenously based model may give abetter economic understanding of the liquidity risk. The closest price impact model to oursis the one introduced by Bank and Kramkov [6, 7] who allow the Market Makers to haveutility functions of von Neumann-Morgenstern type (see also the earlier works [57, 29]). Theyshow the existence of the representative liquidity supplier and construct a nonlinear stochasticintegral to describe the Large Trader’s P&L. Our utility function stems from time-consistentconvex risk measures represented as a g -expectation, and in terms of decision theory representsan ambiguity averse preference with the exponential utility being the only intersection to vonNeumann-Morgenstern utility functions. The focus of this paper is on optimal portfolio choicewhich is not addressed in [6, 7, 24]. The special case of an exponential utility function forthe investor and for the representative liqudidity supplier is analyzed in [4] without coupledFBSDEs or BSPDEs.We remark that contrary to the works on utility maximization above, the terminal wealthin our case is a space of feasible terminal conditions of g -expectations and depends on thestrategy in a complex and non-convex way. We give solutions of the optimal solutions interms of solutions of FBSDEs and of BSPDEs and point out the connection between theseapproaches. Due to the non-linearity in the target function, the control variable and thefeedback effects, the mathematical proofs are delicate. In the case of no market impacts and2inear pricing, our results reduce to the classical utility maximization problem.The remainder of the paper is organized as follows. Section 2 presents a model setupunder permanent market impact where hedging is represented in terms of g -expectations.Solutions of the expected utility maximization are characterized by FBSDEs in Section 3.Existence results are discussed in Section 4. Some explicit examples are given in Section 5.The connection to BSPDEs is studied in Section 6. Extra technical results are reported in theappendix. In a limit order book the roles of suppliers and demanders of liquidity are not symetric.Every liquidity supplier submits a price quote for a specific asset for a specific volume andtrades with the other liquidity suppliers until an equilibrium is achieved. The remaining limitorder form a price curve which is a nonlinear function in volume. Taking a Bertrand-typecompetition among liquidity suppliers into account, it would then be reasonable to beginwith modelling the price curve as the utility indifference curve of a representative liquiditysupplier. While Bank and Kramkov [6, 7] used Neumann-Morgenstern utility functions for therepresentative agent we will as in [24] use time-consistent convex risk measures instead. Theexponential utility function assumed by Anthropelos et al. [4] is in the intersection of these twoframeworks. Modulo a compactness assumption using a time-consistent convex risk measureis equivalent to using a g -expectation providing a powerful stochastic calculus tool. A furtheradvantage of our approach from an economic point of view is that ambiguity aversion is takeninto account. In the present paper, we therefore simply assume that there is a representativeliquidity supplier, called the Market, who quotes a price for each volume based on the utilityindifference principle and her utility is a g -expectation with a cash-invariance property. Theexistence of the representative agent under such utility functions follows from Horst et al. [31].While a cash invariance axiom might not be realistic for an individual investor it is much morereasonable for financial institutions which might be reluctant to specify a utility function, butare exactly the entities which typically act as market makers. If the driver of the g -expectationis a linear function, then the price curve becomes linear in volume and we recover the standardframework of financial engineering. For the individual investor we assume a strictly concaveNeumann-Morgenstern utility function U : R → R satisfying standard conditions.We assume zero risk-free rates meaning in particular that cash is risk free and does nothave any bid-ask spread. Let T > T . Consider a security whose value at T is exogenouslydetermined. We denote the value by S and regard it as an F T -measurable random variabledefined on a filtered probability space (Ω , F , P , ( F t ) t ∈ [0 ,T ] ) with ( F t ) being the completion ofthe filtration generated by a d -dimensional Browninan motion W = ( W , . . . , W d ) satisfyingthe usual conditions. The security S can for instance be a zero-coupon bond, an asset backedsecurity, or a derivative with an underlyer which is not traded. The price of this security at T is trivially S , but the price at t < T should be F t -measurable and will be endogenouslydetermined by a utility-based mechanism. There are two agents in our model: A Large Traderand a Market Maker. The Market Maker quotes a price for each volume of the security. Shecan be risk-averse and so her quotes can be nonlinear in volume and depend on her inventoryof this security. The Large Trader refers to the quotes and makes a decision. She cannot avoid3ffecting the quotes by her trading due to the inventory consideration of the Market Maker,and seeks an optimal strategy under this endogenous market impact.As the pricing rule of the Market Maker, our model adopts the utility indifference principle,using an evaluation Π given by the solution of a g -expectation defined as follows. Let g = { g ( t, ω, z ) } : [0 , T ] × Ω × R d → R be a P ⊗ B ( R d ) measurable function, where P is theprogressively measurable σ field, such that z g ( t, ω, z ) is a convex function with g ( t, ω,
0) = 0for each ( t, ω ) ∈ [0 , T ] × Ω. As usual the ω will typically be suppressed. For a stopping time τ we denote by D τ a linear space of F τ -measurable random variables to be specified below. Wedenote by L ( F τ ) the space of F τ -measurable random variables, by L ( F τ ) = L (Ω , F τ , P )the space of F τ -measurable random variables which are square integrable with respect to P ,and by L (d P × d t ) the space of progressively measurable proceses which are square integrablewith respect to d P × d t . Products of vectors will be understood as vectorproducts. Equalitiesand inequalities are understood a.s. In the sequel we always assume either one of the followingconditions: (Hg) g ( t, z ) = α | z | + l ( t, z ) is convex in z , l is Lipschitz in z (also uniformly in t and ω ) andcontinuously differentiable in z with l ( t,
0) = 0 . Furthermore, D τ = (cid:8) X ∈ L ( F τ ) | E [exp( a | X | )] < ∞ for all a > (cid:9) . (HL) g ( t, z ) is convex, Lipschitz in z (also uniformly in t and ω ) with g ( t,
0) = 0 and D τ = L (Ω , F τ , P ) . The subgradient of g is defined as ∇ g ( t, z ) := { y ∈ R d | g (˜ z ) − g ( z ) ≥ y (˜ z − z ) for all ˜ z ∈ R d } . If g is differentiable in z then the subgradient has only one element henceforth, denoted by g z . Sometimes we will additionally discuss the case that g is positively homogeneous (in z )meaning that g ( t, λz ) = λg ( t, z ) for λ >
0. In this case it follows from (Hg) or (HL) , that g must be Lipschitz continuous.It is well-known, see e.g. [37, 42, 16] that under (HL) or (Hg) for each X ∈ D T , there existsprogressively measurable processes (Π( X ) , Z ( X )) = (Π( X ) , Z ( X ) , . . . , Z d ( X )) such that E (cid:20)Z T | Z t ( X ) | d t (cid:21) < ∞ , sup ≤ t ≤ T | Π t ( X ) | ∈ D T , and for all t ∈ [0 , T ], X = Π t ( X ) + Z Tt g (cid:0) s, Z s ( X ) (cid:1) d s − Z Tt Z s ( X )d W s , (2.1)where Z Tt Z s ( X )d W s = d X j =1 Z Tt Z js ( X )d W js . Equation (2.1) is also called a backward stochastic differential equation (BSDE) with driverfunction g , terminal condition X and solution (Π t ( X ) , Z t ( X )). Sometimes we will also writeΠ t ( X ) = E gt ( X ) and call Π( X ) a g -expectation of X .4 xample ( g -expectation) .
1. The simplest example of a g -expectation is if g = 0 and thus Π t ( X ) = E [ X |F t ] (2.2) with D t = L (Ω , F t , P ) . This evaluation can be interpreted as the orthogonal projectionof future cash-flows.2. Another example is if g ( s, z ) = ν s z for a bounded progressively measurable process ν ,and Π t ( X ) = E Q [ X |F t ] (2.3) with Q having the Radon-Nikodym derivative with stochastic logarithm ν , i.e., d Q d P :=exp { R T ν s dW s − R T | ν s | ds } . In this case D t = L (Ω , F t , P ) .3. An third example of a g -expectation is an exponential utility Π t ( X ) = − γ log E [exp ( − γX ) |F t ] , (2.4) with D t = { X ∈ L ( F t ) | E [exp( a | X | )] < ∞ for all a > } , where γ > is a parameterof risk-aversion. In this case Π is a g -expectation with driver function g ( t, z ) = γ | z | ,see Barrieu and El Karoui [9]. By letting γ → , we recover the previous example. Byletting γ → ∞ , we have Π t ( X ) = inf Q (cid:8) E Q [ X |F t ] : Q ∼ P , Q = P on F t (cid:9) , which is the essential infimum value of X under the conditional probability given F t .4. We remark that there is a close connection between g -expectations and convex risk mea-sures, see e.g., Barrieu and El Karoui [9], or Delbaen et al. [22]. A convex risk measuresatisfies the following axioms. For any X, Y ∈ D T ,(a) Normalization: ρ τ (0) = 0 ,(b) Cash-Invariance: ρ τ ( X + Y ) = ρ τ ( X ) + Y if Y ∈ D τ ,(c) Convexity: ρ τ ( λX + (1 − λ ) Y ) ≤ for all λ ∈ [0 , if ρ τ ( X ) ≤ and ρ τ ( Y ) ≤ ,(d) Time-Consistency: ρ τ ( X ) ≥ ρ τ ( Y ) if there exists σ ≥ τ such that ρ σ ( X ) ≥ ρ σ ( Y ) .A g -expectation is a convex risk measure and satisfies Bellman’s principle if and only if − g is convex and g ( t,
0) = 0 , see Jiang [34]. In particular, − Π( − X ) satisfies axioms (a)-(d). It is worth noting that under additional compactness or domination assumptions,every evaluation satisfying axioms (a)-(d) corresponds to a g -expectation. For these andother related results, see Coquet et al. [18] and Hu et al. [33].5. In the theory of no-arbitrage pricing, attempts have been made to narrow the no-arbitragebounds by restricting the set of pricing kernels considered. One of these approaches isthe good-deal bounds ansatz introduced in Cochrane and Sa´a-Requejo [17] which corre-sponds to excluding in the no-arbitrage bounds pricing kernels which induce a too highSharpe ratio, see also Bj¨ork and Slinko [11]. Upper good-deal bounds can be modelled via g -expectations with positively homogeneous drivers. We refer to [24] for further discus-sions. T , represented by H M . Assume S and H M are either square integrable if ( HL ) holdsor bounded if ( Hg ) holds. If the Market Maker at time t ∈ [0 , T ] is holding z units of thesecurity in question besides H M , then her utility is measured as Π t ( H M + zS ). According tothe utility indifference principle, the Market Maker quotes a selling price for y units of thesecurity by P t ( z, y ) := inf { p ∈ R : Π t ( H M + zS − yS + p ) ≥ Π t ( H M + zS ) } =Π t ( H M + zS ) − Π t ( H M + ( z − y ) S ) . (2.5)For the equality we have used cash invariance. Note that in the risk-neutral case (2.3), P t ( z, y ) = y E Q [ S |F t ].Let Θ be the set of the simple progressively measurable processes θ with θ = 0. TheLarge Trader is allowed to take any element θ ∈ Θ as her trading strategy. The price for the y units of the security at time t is P t ( − θ t , y ). This is because the Market Maker holds − θ t unitsof the security due to the preceding trades with the Large Trader. Then the profit and loss attime T associated with θ ∈ Θ (i.e., the terminal wealth corresponding to the self-financingstrategy θ ) is given by I ( θ ) := θ T S − X ≤ t By standard results on BSDEs, see for instance Barrieu and El-Karoui [9] and Briandand Hu [15, 16], the mapping y → Z y is continuous in the L (d P × d t ) norm. Set Z y,nt = Z k/nt for kn ≤ y < k +1 n . Then Z y,nt −−−→ n →∞ Z yt in L (d P × d t ). Theorem 5.1 in [3] entails that thereexists a constant K > y , y ∈ R , E (cid:2) Z T |Z y t − Z y t | d t (cid:3) ≤ K | y − y | . (2.6)It follows by (2.6) that E (cid:2) R T |Z y, k t − Z yt | d t (cid:3) ≤ K k . Hence, by the Borel-Cantelli lemma, Z y, k t → Z yt in d P × d t a.s. as k → ∞ . Note that as Z y, k t for fixed ( t, ω ) is piecewise constant as a function of y , it clearly is P ⊗ B ( R )-measurable. Thus, its a.s. limit, Z yt , is P ⊗ B ( R )-measurable as well.6 roposition 2.1. Suppose that y → Z ( t, ω, y ) is continuous d P × d t a.s. Then if θ n ∈ Θ , θ n → θ in L (d P × d t ) as n → ∞ and |Z θ n | is uniformly integrable, we have L - lim n →∞ I ( θ n ) = I ( θ ) := H M − Π ( H M ) − Z T g ( t, Z θt )d t + Z T Z θt d W t where Z θt ( ω ) := Z ( ω, t, θ t ( ω )) . Proof. By the continuity of Z in y we have that Z θ n → Z θ d P × d t a.s. Since by assumption |Z θ n | is uniformly integrable, convergence holds actually in L (d P × d t ). Now by Lemma 1in [24] we have for θ n ∈ Θ , I ( θ n ) = H M − Π ( H M ) − Z T g ( t, Z θ n t )d t + Z T Z θ n t d W t . Passing to the limit yields the proposition.Motivated by the above proposition we define the set of admissible strategies asΘ := (cid:26) θ : Ω × [0 , T ] → R predictable with E (cid:20) Z T |Z θt | d t (cid:21) < ∞ (cid:27) . As can be seen from Proposition 2.1 and the definition of Θ, Z ( t, ω, y ) plays a crucial rolefor our analysis. Below we will thus state some further useful properties of Z and give threetractable examples. Definition 2.1 (Derivative of Z ) . For any y ∈ R , we define ∂ Z y ( t, y ) as the progressivelymeasurable L (d P × d t ) limit of Z ( t,y + ǫ ) −Z ( t,y ) ǫ as ǫ tends to zero, i.e., lim ε → E (cid:20) Z T (cid:12)(cid:12)(cid:12)(cid:12) Z ( t, y + ǫ ) − Z ( t, y ) ǫ − ∂ Z y ( t, y ) (cid:12)(cid:12)(cid:12)(cid:12) d t (cid:21) = 0 . (2.7)By Theorem 2.1 in [3] the limit exists. Note that for these results differentiability as-sumptions for the driver g are needed which in case of positive homogeneity are not satisfied(except if g is linear). However in this case the existence of the limit may be verified directlyif H M = 0; see Example 1.In what follows, we will provide three examples where the continuity assumption of Propo-sition 2.1 holds. Example 1 (Positive Homogenity) . If g is positively homogeneous and H M = 0 then Z yt = yZ t ( − S ). Example 2 (Smoothness) . Suppose that g does not depend on ω , and define the diffusionprocess R t,r to be the solution to the following SDE:d R t,ru = b ( u, R t,ru ) d s + σ d W u , t ≤ u ≤ T, (2.8) R t,ru = r, ≤ u ≤ t, (2.9)where b : [0 , T ] × R n → R n is continuously differentiable with respect to r with boundedderivative b r , and σ ∈ R n × d is a constant (matrix).Let us consider S = s ( R t,rT ) and H M = h M ( R t,rT ) with s, h M : R n → R in C with thederivatives s r and h M r growing at most polynomially.7 roposition 2.2. Suppose that the first and second order derivatives of g with respect to z exist and are bounded and for the first and second order derivatives of h M and s either a) h M rr and s rr or b) h M r and s r exist and are bounded. Then y → Z ( t, ω, y ) is continuous d P × d t a.s. Furthermore, Z in case a) grows at most linearly in y (uniformly in t, ω ), and in case b)is uniformly bounded in t, ω and y , and the differential quotient from Definiton 2.1 convergesnot only in L (d P × d t ) but also d P × d t a.s. Proof. The proof is reported in Appendix A. Definition 2.2. A market is complete if for any − H L ∈ D T at T , a perfect replication ispossible, in the sense that there exists ( a, θ ) ∈ R × Θ such that − H L = a + I ( θ ) . Example 3 (Market Completeness) . Suppose we are in the Markov-setting above with d = n = 1, and σ in (2.8) may depend on ( u, R t,ru ), and satisfy standard Lipschitz and lineargrowth conditions. Assume that :1. s is Lipschitz continuous,2. g is differentiable with bounded derivative and D t = L ( F t ) for t ∈ [0 , T ], or g ( t, z ) = α | z | + η t z for a progressively measurable process ( η t ) and D t = (cid:8) X ∈ L ( F t ) | E [exp( a | X | )] < ∞ for all a > (cid:9) , for t ∈ [0 , T ] , h ′ M is of exponential growth and σ and µ are bounded, or h ′ M is of polynomial growth,4. inf x ∈ R s ′ ( x ) > ǫ . Proposition 2.3. Under the above assumptions Z · t ( ω ) = Z ( t, ω, · ) is continuous, strictlyincreasing and one-to-one. Furthermore, the market is complete. Proof. The proof is reported in Appendix B.In order to complete Example 3, we remark that since Z yt ( ω ) is monotone, for θ n ↑ θ ∈ Θwe clearly have that |Z θ n | ≤ |Z θ | ∈ L (d P × d t ) is uniformly integrable. Furthermore, y → Z ( ω, t, y ) is continuous in y a.s. The reason is that Z y is continuous in y in the L (d P × d t )norm which implies for every sequence, a.s. convergence along a subsequence. But thenmonotonicity entails that the whole sequence must converge a.s. Hence, the assumptions ofProposition 2.1 hold. For simplicity we assume througout the rest of this paper that the Market Maker has zeroinventory, i.e., H M = 0. Let us define Z θt ( ω ) = Z ( ω, t, θ t ( ω )). The portfolio value at time t isthen given by X θt = x + I θt , where x = − Π ( − X θT ) and the random part I θt := − Z t g ( s, Z θs )d s + Z t Z θs d W s , , t ].We study the following utility maximization problem for the Large Tradersup θ ∈ Θ E [ U ( X θT )] , (3.1)where θ t is the number of risky asset held at time t and X θ = x is the initial endowment.Here we assume that the utility function U is a strictly increasing, concave and three timesdifferentiable function. Problem (3.1) can be restated assup θ ∈ Θ E [ U ( x + I θT )] . (3.2) Theorem 3.1. Assume that θ ∗ is an optimal strategy of Problem (3.2) and E [ | U ( X θ ∗ T ) | ] < ∞ and E [ | U ′ ( X θ ∗ T ) | ] < ∞ . We assume furthermore that1. θ ε := θ ∗ + εh ∈ Θ for sufficiently small ε > if h is progressively measurable and takesvalues in { , ± } . 2. (i) lim ε → Z ( t,y + ε ) −Z ( t,y ) ε = ∂ Z y ( t, y ) d P × d t a.s.,(ii) sup ≤ δ ≤ ε | ∂ Z y ( t, θ ∗ t + δh ) | ∈ L (d P × d t ) for ε sufficiently small if h is progressivelymeasurable and takes values in { , ± } . 3. Either g is continuously differentiable or g is positively homogeneous in the second ar-gument and Z t ( − S ) = 0 , d P × d t a.s. respectively.4. For any progressively measurable h taking values in { , ± } , the family (cid:26) ε − ( X θ ε T − X θ ∗ T ) Z U ′ ( X θ ∗ T + l ( X θ ε T − X θ ∗ T ))d l, ε ∈ (0 , (cid:27) , (3.3) with X θ ε T being the wealth process induced by θ ε , is uniformly integrable.Then there exists a continuous adapted process ζ with ζ T = 0 such that U ′ ( X θ ∗ + ζ ) is asquare integrable martingale process and the optimality is characterized by the existence of aprogressively measurable Y t ∈ ∇ g ( t, Z θ ∗ t ) such that ∂ Z y ( t, θ ∗ t )( − U ′ ( X θ ∗ t + ζ t ) Y t + U ′′ ( X θ ∗ t + ζ t )( Z θ ∗ t + M ζt )) = 0 , (3.4) where M ζt := d h ζ, W i t / d t . Proof. Let θ ∗ be an optimal strategy. Then the wealth process is given by X θ ∗ t = x − Z t g ( s, Z θ ∗ s )d s + Z t Z θ ∗ s d W s . Define R t := E [ U ′ ( X θ ∗ T ) |F t ] and ζ t := I ( R t ) − X θ ∗ t . Then ζ is progressively measurable and R is a square integrable martingale. By the martingale representation theorem there exists asquare integrable progressively measurable process β such that R t = U ′ ( X θ ∗ T ) − Z Tt β s d W s , 9r d R t = β t d W t with terminal condition R T = U ′ ( X θ ∗ T ). Note that I ( R t ) = X θ ∗ t + ζ t and ζ T = 0. Applying Itˆo’s formula we have the following backward representation X θ ∗ t + ζ t = X θ ∗ T − Z Tt d R s U ′′ ( X θ ∗ s + ζ s ) + 12 Z Tt U (3) ( U ′′ ) ( X θ ∗ s + ζ s )d h R, R i s = X θ ∗ T − Z Tt β s d W s U ′′ ( X θ ∗ s + ζ s ) + 12 Z Tt U (3) ( U ′′ ) ( X θ ∗ s + ζ s ) | β s | d s, where U ′′ and U (3) are the second and the third order derivatives of U , respectively. Hence, ζ t is a solution of the following BSDE with zero terminal condition (i.e., ζ T = 0) ζ t = − Z Tt (cid:18) β s U ′′ ( X θ ∗ s + ζ s ) − Z θ ∗ s (cid:19) d W s + 12 Z Tt (cid:18) | β s | U (3) ( U ′′ ) ( X θ ∗ s + ζ s ) − g ( s, Z θ ∗ s ) (cid:19) d s. (3.5)By construction the marginal utility process U ′ ( X θ ∗ t + ζ t ) = R t is a martingale and β t = U ′′ ( X θ ∗ t + ζ t )( Z θ ∗ t + M ζt ) . (3.6)Plugging the last equation into the dynamics of ζ we get ζ t = − Z Tt M ζs d W s + 12 Z Tt (cid:18) |Z θ ∗ s + M ζs | U (3) U ′′ ( X θ ∗ s + ζ s ) − g ( t, Z θ ∗ t ) (cid:19) d s. (3.7)Let us now characterize the optimality. Let h be any bounded progressively measurable processtaking values in { , ± } and let ε > θ ε := θ ∗ + εh is still admissible by our assumptions. Due to the optimality of θ ∗ we have E [ U ( X θ ε T )] − E [ U ( X θ ∗ T )] ≤ . Now, using U ( b ) − U ( a ) = ( b − a ) Z U ′ ( a + l ( b − a ))d l we can represent ε − ( U ( X θ ε T ) − U ( X θ ∗ T )) = ε − ( X θ ε T − X θ ∗ T ) Z U ′ ( X θ ∗ T + l ( X θ ε T − X θ ∗ T ))d l. We introduce the process χ ǫt ( h ) := − Z t ∂ Z y ( s, θ ∗ s ) g ′ ( s, Z θ ∗ s , h s )d s + Z t h s ∂ Z y ( s, θ ∗ s )d W s , where the directional derivative of g is given by g ′ ( t, Z θ ∗ t , h t ) := h t g z (cid:0) t, Z θ ∗ t (cid:1) if g is continuously differentiable in z,h t sgn( θ ∗ t ) g (cid:0) t, sgn( θ ∗ t ) Z t ( − S ) (cid:1) if g is positively homogeneous and θ ∗ t = 0 , | h t | g (cid:0) t, Z t ( − S )sgn( h t ) (cid:1) if g is positively homogeneous and θ ∗ t = 0 , x ) = x > , − x < ,a with a ∈ {− , , } if x = 0 . In the rest of the proof we use the convention sgn(0)=0. We have used above that if g ispositively homogeneous Z θs = θZ s ( − S ) (see Example 1) and for θ ∗ s = 0 ∂g ( s, Z θ ∗ + εh s s ) ∂ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = ∂∂ε | θ ∗ s + εh s | g ( s, Z ( − S )sgn( θ ∗ s + εh s )) (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = h s sgn( θ ∗ s ) g ( s, sgn( θ ∗ s ) Z s ( − S )) . By Assumption 2), we now have ǫ − ( Z θ ε t − Z θ ∗ t ) → h t ∂ Z y ( t, θ ∗ t ) d P × d t a.s. and (by themean value theorem) also in L (d P × d t ) . Hence, we can conclude that ε − ( U ( X θ ε T ) − U ( X θ ∗ T ))converges to U ′ ( X θ ∗ T ) χ ǫT ( h ) a.s. d P × d t . As by assumption the family (cid:26) ε − ( X θ ε T − X θ ∗ T ) Z U ′ ( X θ ∗ T + l ( X θ ε T − X θ ∗ T ))d l, ε ∈ (0 , (cid:27) is uniformly integrable, we can infer that E (cid:2) U ′ ( X θ ∗ T ) χ ǫT ( h ) (cid:3) = lim ε → E [ U ( X θ ε T )] − E [ U ( X θ ∗ T )] ε ≤ . (3.8)Applying Itˆo’s lemma to the product U ′ ( X θ ∗ t + ζ t ) χ ǫt ( h ) and noting that ζ T = 0 we have U ′ ( X θ ∗ T ) χ ǫT ( h ) = Z T ∂ Z y ( t, θ ∗ t ) (cid:18) β t h t − U ′ ( X θ ∗ t + ζ t ) g ′ ( t, Z θ ∗ t , h t ) (cid:19) d t + Z T (cid:18) h t U ′ ( X θ ∗ t + ζ t ) ∂ Z y ( t, θ ∗ t ) + β t χ ǫt ( h ) (cid:19) d W t . (3.9)Using the BDG inequality and the square integrability of U ′ ( X t + ζ t ) , χ ǫ , ∂ Z y , and R we canconclude that E (cid:20) sup u ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z u (cid:18) h t U ′ ( X θ ∗ t + ζ t ) ∂ Z y ( t, θ ∗ t ) + β t χ ǫt ( h ) (cid:19) d W t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) < ∞ . (3.10)Hence, the stochastic integral in (3.9) is a square integrable martingale. Taking expectationson both sides leads to E (cid:20) Z T ∂ Z y ( t, θ ∗ t ) (cid:0) β t h t − U ′ ( X θ ∗ t + ζ t ) g ′ ( t, Z θ ∗ t , h t ) (cid:1) d t (cid:21) ≤ , (3.11)for any bounded progressively measurable h taking values in { , ± } .Assume now that g is continuously differentiable in z . Using the definition of g ′ andreplacing h by − h we obtain − E (cid:20) Z T ∂ Z y ( t, θ ∗ t ) (cid:18) β t h t − U ′ ( X θ ∗ t + ζ t ) h t g z ( t, Z θ ∗ t ) (cid:19) d t (cid:21) ≤ . 11t follows from the last two inequalities that0 = E (cid:20) Z T h t ∂ Z y ( t, θ ∗ t )( β t − U ′ ( X θ ∗ t + ζ t ) g z ( t, Z θ ∗ t ))d t (cid:21) . (3.12)Now let A t := ∂ Z y ( t, θ ∗ t )( β t − U ′ ( X θ ∗ t + ζ t ) g z ( t, Z θ ∗ t )) and choose h t = A t > . From (3.12)we get A t ≤ 0, d P × d t almost everywhere. Similarly choosing h t = A t < we get the reverseinequality and can conclude that ∂ Z y ( t, θ ∗ t )( β t − U ′ ( X θ ∗ t + ζ t ) g z ( t, Z θ ∗ t )) = 0 , d P × d t a.s. Plugging (3.6) into the last equation leads to the desired conclusion.Finally let us consider the case where g is positively homogeneous. In this case let usfirst remark that θ g ( t, Z θt ) = | θ | g ( t, sgn( θ ) Z t ( − S )) is continuously differentiable except at θ = 0. Hence, on the set { ( t, ω ) | θ ∗ t ( ω ) = 0 } the argument for (3.12) to hold is analogous asabove. On the other hand, on θ ∗ t = 0 (3.11) entails for h t = ˜ h t I θ ∗ t =0 that we have E (cid:20) Z T I θ ∗ t =0 (cid:26) ˜ h t Z t ( − S ) β t − U ′ ( X t + ζ t ) | ˜ h t | g ( t, Z t ( − S )sgn(˜ h t )) (cid:27) d t (cid:21) ≤ . (3.13)Set A t = Z t ( − S ) β t − U ′ ( X t + ζ t ) g ( t, Z t ( − S )) and ˜ h t = I A t > . Then it follows from (3.13) thatwe actually must have I θ ∗ t =0 (cid:8) Z t ( − S ) β t − U ′ ( X t + ζ t ) g ( t, Z t ( − S )) (cid:9) ≤ P × d t a.s.Next, set ˜ A t = {− Z t ( − S ) β t − U ′ ( X t + ζ t ) g ( t, − Z t ( − S )) } and ˜ h t = − I ˜ A t > . Then E (cid:20) Z T I θ ∗ t =0 , ˜ A t > (cid:26) − Z t ( − S ) β t − U ′ ( X t + ζ t ) g ( t, − Z t ( − S )) (cid:27) d t (cid:21) ≤ . Thus, I θ ∗ t =0 {− Z t ( − S ) β t − U ′ ( X t + ζ t ) g ( t, − Z t ( − S )) } ≤ P × d t a.s.Overall, on θ ∗ t = 0 we have B t := U ′ ( X t + ζ t ) g ( t, Z t ( − S )) ≥ β t Z t ( − S ) ≥ − U ′ ( X t + ζ t ) g ( t, − Z t ( − S )) =: C t . Define λ t := β t Z t ( − S ) − C t B t − C t (with the convention 0 / ≤ λ t ≤ 1. Recallingthat for g positively homogeneous, ∂ y Z ( t, θ ) = Z t ( − S ), our proof is completed by noting thaton θ ∗ t = 0, β t Z t ( − S ) = λ t U ′ ( X t + ζ t ) g ( t, Z t ( − S )) + (1 − λ t ) U ′ ( X t + ζ t )( − g ( t, − Z t ( − S ))) ∈ U ′ ( X t + ζ t ) Z t ( − S ) ∇ g ( t, 0) = U ′ ( X t + ζ t ) ∂ Z y ( t, θ ∗ t ) ∇ g ( t, Z θ ∗ t ) , where we used that for appropriate ¯ Y , ˜ Yg ( t, Z t ( − S )) = max Y ∈∇ g ( t, Z t ( − S ) Y = Z t ( − S ) ¯ Y ∈ Z t ( − S ) ∇ g ( t, − g ( t, − Z t ( − S )) = − max Y ∈∇ g ( t, − Z t ( − S ) Y = − ( − Z t ( − S )) ˜ Y = Z t ( − S ) ˜ Y ∈ Z t ( − S ) ∇ g ( t, , and the convexity of the subgradient. Noting that by a measurable selection theorem (e.g.[5]), we can choose the element of the subgradient progressively measurable completes theproof. Example 4 (Continuation of Example 1) . As already mentioned in the proof, in Example 1, ∂ Z y = yZ ( − S ) and therefore clearly Assumptions 1) to 2) of Theorem 3.1 are satisfied. Example 5 (Continuation of Example 2) . The following proposition together with the a.s.existence of the differential quotient ensured by Proposition 2.2, provides sufficient conditionsunder which Assumptions 1) and 2) of Theorem 3.1 above are satisfied in the Markovian-setting. Proposition 3.1. Assume in the Markovian-setting of Example 2 that g zz is bounded, andadditionally that either s p or g z is bounded. Then ∂ Z y is uniformly bounded and continuousin y . Proof. The proof is reported in Appendix C. Example 6 (Continuation of Example 3) . As in the case without price impact (see e.g. [55]),it can be observed that Assumptions 1), 2) and 4) can be dropped for a complete market as inExample 3. Since the market is complete there then exists θ ε ∈ Θ such that Z θ ε t = Z θ ∗ t + εh t . Ifthis were not the case, the terminal payoff X θ ε T := x − R T g ( s, Z θ ∗ s + εh s )d s + R T ( Z θ ∗ s + εh s )d W s would not be replicable which would be a contradiction to market completeness. Next, define φ ( ε ) := U ( X θ ε T ). In the sequel let us assume that g is continuously differentiable in the secondargument. We can conclude that φ ( ε ) is concave since the wealth process is a.s. concave in Z , and U is increasing. Moreover φ ′ ( ε ) = U ′ ( X θ ε T ) (cid:18) − Z T g z ( t, Z θ ε t ) h t d t + Z T h t d W t (cid:19) . This implies that the function Φ( ε ) defined byΦ( ε ) := φ ( ε ) − φ (0) ε = U ( X θ ε T ) − U ( X θ ∗ T ) ε , is decreasing with respect to ε ∈ (0 , 1] andlim ε → Φ( ε ) = U ′ ( X θ ∗ T ) b χ θ ∗ T ( h ) a.s. , where b χ θ ∗ s ( h ) := − Z s g z ( t, Z θ ε t ) h t d t + Z s h t d W t . By the Cauchy-Schwarz inequality we obtain E [ | U ′ ( X θ ∗ T ) χ θ ∗ T ( h ) | ] ≤ ( E [ | U ′ ( X θ ∗ T ) | ]) / ( E [ | b χ θ ∗ T ( h ) | ]) / < ∞ , ε ) increasingly tends to the limit U ′ ( X θ ∗ T ) b χ θ ∗ T ( h ) as ε ց 0. Note that E [Φ( ε )] is well-defined as E [Φ( ε ) + ] ≤ E [( U ′ ( X θ ∗ T ) b χ θ ∗ T ( h )) + ] < ∞ for all ε > 0. Hence, by themonotone convergence theorem we conclude thatlim ε → E [Φ( ε )] = lim ε → E (cid:20) U ( X θ ε T ) − U ( X θ ∗ T ) ε (cid:21) = E [ U ′ ( X θ ∗ T ) b χ θ ∗ T ( h )] ≤ , which is an equivalent form of (3.8) obtained in the proof of Theorem 3.1. Repeating argumentssimilar as in (3.9)-(3.12) we get β t − U ′ ( X θ ∗ t + ζ t ) g z ( t, Z θ ∗ t ) = 0 , d P × d t a.s. , which is in fact an even stronger condition than (3.4).We are now able to characterize the optimal strategy in terms of a fully-coupled forward-backward system. For that we need the following assumption. Assumption 3.1. There exist a unique P ⊗ B ( R d +2 ) -measurable function H : [0 , T ] × Ω × R d +2 → R d taking values in Image ( Z ( t, ω, · )) such that ∈ ∂ Z y ( t, H ( t, X, ζ, M )) (cid:0) − U ′ ( X + ζ ) ∇ g ( t, H ( t, X, ζ, M )) + U ′′ ( X + ζ )( H ( t, X, ζ, M ) + M ) (cid:1) , (3.14) for each ( ω, t, X, ζ, M ) ∈ Ω × [0 , T ] × R d +2 . Furthermore, H grows at most linearly in M . Example 7. Assumption 3.1 is satisfied under the assumptions of Theorem 3.1 if d = 1, ∂ Z y = 0, ψ := U ′ /U ′′ is bounded and g is continuously differentiable in z since in this casewe can divide both sides in (3.14) by ∂ Z y , and note that the functionˆ U ( H ) := − U ′ ( X + ζ ) g z ( t, H ) + U ′′ ( X + ζ )( H + M )is bijective. Furthermore, H has at most linear growth by Lemma 3.2 below. Finally byTheorem 3.1 and the bijectivity of ˆ U we actually must have Z θ ∗ t = H ( t, X, ζ, M ).Another example where Assumption 3.1 holds is given by the case that g is positively homo-geneous and Z ( − S ) = 0. In fact (3.14) becomes then0 ∈ − U ′ ( X + ζ )sgn( θ t ) g ( t, sgn( θ t ) Z t ( − S )) + Z t ( − S ) U ′′ ( X + ζ )( θ t Z t ( − S ) + M ) , as the image space of Z ( t, ω, · ) is given by { θZ t ( − S ) | θ ∈ R } . We get then H ( t, X, ζ, M ) = θ ( t, Xζ, M ) Z t ( − S ) with θ ( t, X, ζ, M ) given by − Z t ( − S ) M + U ′ U ′′ ( X + ζ ) g ( t, Z t ( − S )) | Z t ( − S ) | =: θ ( t, X, ζ, M ) (3.15)if the left-hand side of (3.15) is positive, and − Z t ( − S ) M − U ′ U ′′ ( X + ζ ) g ( t, − Z t ( − S )) | Z t ( − S ) | =: θ ( t, X, ζ, M ) (3.16)if the left-hand side of (3.16) is negative, and else a convex combination of both left-hand sidesin (3.15)-(3.16) adding up to zero. 14 emma 3.1. If g is positively homogeneous, or if d = 1 , ∂ Z y = 0 and ψ := U ′ /U ′′ isbounded, then any function H satisfying (3.14) grows at most linearly in M . In other words,there exists C > : |H ( t, X, ζ, M ) | ≤ C (1 + | M | ) . Proof. If g is positively homogeneous, the claim follows immediately from (3.15)-(3.16). Letus consider the other case. Omitting the argument ( t, X, ζ, M ) and dividing both sides in(3.14) by U ′′ ( X + ζ ) ∂ Z y ( t, H ), we get H = − M + ψ Y , for Y ∈ ∇ g ( t, H ) . Using that ψ ≤ ∇ g ( z ) − ∇ g (0)) z ≥ 0, multiplying both sides by H yields |H| = − M H + ψ Y H≤ a | M | |H| a + ψ Y H≤ a | M | |H| a + k ψ k ∞ | Y | a |H| a , with Y ∈ ∇ g ( t, 0) and a > 0. Choosing a fixed and sufficently large constant a yields thelemma. Example 8. Also if g is not positively homogeneous, H can be computed explicitly for specialcases. For example assume that d = 1 . • If g ( t, z ) = a t z + b t for some deterministic functions a, b and U is a CARA function,meaning that U ( x ) = − e − xγ , with a risk aversion coefficient γ ∈ (0 , ∞ ) then we have H ( t, X, ζ, M ) = − a t /γ − M t which is independent of ( X, ζ ) . • Similarly, if g ( t, z ) = a t | z | / and U is a CARA function we get H ( t, X, ζ, M ) = − γM/ ( γ + a t ) . Now, the optimal strategy can be characterized by a solution of a fully-coupled forward-backward system. Theorem 3.2. Under the assumptions of Theorem 3.1 and Assumption 3.1, the optimalstrategy for Problem (3.1) is characterized by Z θ ∗ t = H ( t, X t , ζ t , M t ) , (3.17) where ( X, ζ, M ) is a triple of adapted processes which solves the FBSDE X t = x − R t g ( s, H ( s, X s , ζ s , M s ))d s + R t H ( s, X s , ζ s , M s )d W s ,ζ t = 0 − R Tt M s d W s + R Tt U (3) U ′′ ( X s + ζ s ) |H ( s, X s , ζ s , M s ) + M s | d s − R Tt g ( s, H ( t, X s , ζ s , M s ))d s. (3.18)15 roof. From Theorem 3.1, ( ζ t , M ζ ) is a solution of the BSDE (3.7) with driver12 |Z θ ∗ t + M t | U (3) U ′′ ( X θ ∗ t + ζ t ) − g ( t, Z θ ∗ t ) . Furthermore, by Theorem 3.1, Z θ ∗ satisfies (3.4). Using Assumption 3.1 we obtain thenimmediately Z θ ∗ t = H ( t, X t , ζ t , M t ). The proof is completed by plugging this identity backinto the BSDE and into the dynamics of X θ ∗ t . Remark 3.1. Using a measurable selection theorem (see Aumann [5]) there exists a P ⊗B ( R d ) -measurable function e Z : Ω × [0 , T ] × R d → R such that Z ( ω, t, e Z ( ω, t, z )) = z for z ∈ Image ( Z ( t, ω, · )) . (3.19) If θ ∗ ∈ Image ( Z ( t, ω, · )) we can therefore compute the optimal strategy θ ∗ explicitly by Theorem3.1 as θ ∗ t = e Z ( t, H ( t, X θ ∗ t , ζ t , M ζt )) . (3.20) We further remark that in complete markets Z is onto so that in this case θ ∗ is always welldefined through (3.20). See also Example 3. Next, let us give the other direction of Theorem 3.2. Define ψ ( x ) := U ′ U ′′ ( x ) and ψ ( x ) := U (3) U ′′ ( x ) . Lemma 3.2. Let ( X, ζ, M ) be a triple of adapted processes that solves the FBSDE (3.18) andlet Q be the measure defined by d Q d P := U ′ ( X T + ζ T ) E [ U ′ ( X T + ζ T )] . Assume furthermore ψ and ψ are bounded and Lipschitz continuous. Then, W Q t := W t − R t Y s d s for any progressively measurable process Y s ∈ ∇ g ( s, H ( s, X s , ζ s , M s )) is a standardBrownian motion under Q . Proof. We will treat X, ζ and M in the sequel as variables and suppress the time index. Notefirst that the conclusion is obvious under Assumption (HL) when g is Lipschitz (as ∇ g is thenbounded). So assume condition (Hg) . Below C denotes a universal positive constant whichmay change from line to line. Under Condition (Hg) , for all ( t, z ), | g z ( t, z ) | ≤ C (1 + | z | )for some universal positive constant C uniformly in t . By Assumption 3.1 we have that |H ( t, X, ζ, M ) | ≤ C (1 + | M | ) uniformly in t and X .We remark in addition that as ψ and ψ are bounded and Lipschitz continuous, g is ofquadratic growth, and the terminal condition of the BSDE (3.18) is obviously bounded so that R t M s d W s is a BM O ( P ) martingale; see the Appendix E and [9, 36]. Hence, g z ( t, H ( t, X t , ζ t , M t ))integrated with respect to a Brownian motion gives rise to a BM O ( P ) martingale as well andthe Girsanov transformation is consequently well defined.16 heorem 3.3. Let ( X, ζ, M ) be a triple of adapted processes which solves the FBSDE (3.18) and satisfies E [ | U ′ ( X T ) | ] < ∞ ; E [ U ( X T )] < ∞ , E [ Z T |H ( t, X t , ζ t , M t ) | d t ] < ∞ . Assume that ψ and ψ are bounded and Lipschitz continuous, that U ′ ( X t + ζ t ) is a positivemartingale and that Assumption 3.1 holds. Assume furthermore that θ ∗ ∈ Image ( Z ( t, ω, · )) .Then θ ∗ t = e Z ( t, H ( t, X t , ζ t , M t )) (3.21) is an optimal strategy of problem (3.2) . Proof. From the FBSDE (3.18) and Assumption 3.1 we haved U ′ ( X t + ζ t ) = U ′′ ( X t + ζ t )( Z θ ∗ t + M t )d W t = U ′ ( X + ζ ) Y t d W t (3.22)with Z θ ∗ t = H ( t, X t , ζ t , M t ) and a progressively measurable process Y t ∈ ∇ g ( t, H ( t, X t , ζ t , M t )).This implies that the positive martingale U ′ ( X t + ζ t ) can be expressed as a Dolans-Dadeexponential U ′ ( X t + ζ t ) = U ′ ( x + ζ ) E ( R t Y s d W s ). Now let us introduce the measure Q defined by d Q d P := U ′ ( X T + ζ T ) E [ U ′ ( X T + ζ T )] . By Lemma 3.2, W Q t := W t − R t Y s d s is a standard Brownian motion under Q . Let b X be anarbitrary admissible portfolio corresponding to a given b Z which satisfies E [ R T | b Z t | d t ] < ∞ .Due to the concavity of U we obtain U ( b X T ) − U ( X T ) ≤ U ′ ( X T )( b X T − X T ) . Observe that b X T − X T = Z T ( b Z t − H ( t, X t , ζ t , M t ))d W t − Z T ( g ( t, b Z t ) − g ( t, H ( t, X t , ζ t , M t )))d t = Z T ( b Z t − H ( t, X t , ζ t , M t ))d W Q t − Z T (cid:18) g ( t, b Z t ) − g ( t, H ( t, X t , ζ t , M t )) − ( b Z t − H ( t, X t , ζ t , M t )) Y t (cid:19) d t. The last integral is always non-negative due to the convexity of g . Therefore, E [ U ( b X T ) − U ( X T )] ≤ E [ U ′ ( X T )( b X T − X T )]= 1 E [ U ′ ( X T )] E Q [ b X T − X T ] ≤ E [ U ′ ( X T )] E Q (cid:20) Z T ( b Z t − H ( t, X t , ζ t , M t ))d W Q t (cid:21) . 17y the BDG inequality and the Cauchy-Schwarz inequality we have E Q (cid:20) sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t ( b Z t − H ( t, X t , ζ t , M t ))d W Q t (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) ≤ C E Q (cid:20)(cid:18) Z T ( b Z t − H ( t, X t , ζ t , M t )) d t (cid:19) / (cid:21) = C (cid:18) E | U ′ ( X T ) | ( E [ U ′ ( X T )]) (cid:19) / (cid:18) E Z T ( b Z t − H ( t, X t , ζ t , M t )) d t (cid:19) / < ∞ . Let ( τ n ) be an localizing sequence of the local martingale R t ( b Z s − H ( s, X s , ζ s , M s ))d W Q s .Using stopping time localization and Lebesgue’s dominated convergence theorem we obtain E Q (cid:20) Z T ( b Z t − H ( t, X t , ζ t , M t ))d W Q t (cid:21) = lim n →∞ E Q (cid:20) Z T ∧ τ n ( b Z t − H ( t, X t , ζ t , M t ))d W Q t (cid:21) = 0 , which shows that X is indeed an optimal solution. Remark 3.2. We deduce from the proof above that for every arbitrary admissible portfolio b X ,the product U ′ ( X + ζ )( b X − X ) is a local supermartingale. It becomes a local martingale when g is linear in z , which is well-known as the first order condition of optimality in case of nomarket impact. In this section we show that the FBSDEs (3.18) admits a solution under appropriate assump-tions on g and the utility function U . To keep the exposition simple we assume that d = 1throughout this section. Assume that U is a CARA utility function and that Assumption 3.1 holds. Then, from (3.14)we deduce that H ( t, X, ζ, M ) = H ( t, M ) is of at most linear growth in M by Lemma 3.1 andgiven by 0 ∈ ∇ g ( t, H ( t, M )) + γ H ( t, M ) + γM. (4.1)The backward equation in (3.18) can be decoupled as ζ t = 0 − Z Tt M s d W s − Z Tt f ( s, M s )d s, ζ T = 0 , (4.2)where f ( t, M ) = γ |H ( t, M ) + M | + g ( t, H ( t, M t )) . The following theorem gives the existence of a solution under Condition (HL) or (Hg) on g , which is a direct consequence of [37, 42] for BSDEs with quadratic or Lipschitz generator.18 heorem 4.1. Suppose that either g is twice differentiable with bounded second derivative orthat g is positively homogeneous. Then the FBSDE (3.18) admits a solution given by the triple ( ξ t , M t , X t ) , where X t = x − Z t g ( s, H ( s, M s ))d s + Z t H ( s, M s )d W s , and ( ζ t , M t ) is the unique solution of BSDE (4.2) . Proof. In the case that g is twice differentiable and the second derivative is bounded theunique solution of the ODE (4.1), H , has a uniformly bounded derivative. On the other hand,in the case that g is positively homogeneous H is given explicitly through (3.15)-(3.16). Hence,in both cases we can conclude that there exist two deterministic bounded functions k t , k ′ t anda constant C > | f ( t, z ) | ≤ k t + C | z | and |∇ f ( t, z ) | ≤ k ′ t + C | z | , ∀ ( t, z ) ∈ [0 , T ] × R , (4.3)where the second inequality holds for every element of the subgradient. The theorem followsnow directly from [37, 42] for BSDEs with Lipschitz or quadratic generator. g Assume that g is linear and of the form g ( t, z ) = a t z + b t for some bounded deterministicfunctions a, b . Then our FBSDE system can be decoupled as in [55, 30] by studying theforward SDE of Y = X + ξ . In particular, define ψ ( x ) = U ′ U ′′ ( x ) , ψ ( x ) = U (3) U ′′ ( x ) , ψ ( x ) := ψ ( x ) ψ ( x ) . Then we have that H ( t, X, ζ, M ) = a t ψ ( X t + ζ t ) − M t . Now, the FBSDE (3.18) can be writtenas X t = x − R t (cid:0) a s ( a s ψ ( X s + ζ s ) − M s ) + b s (cid:1) d s + R t (cid:0) a s ψ ( X s + ζ s ) − M s (cid:1) d W s ,ζ t = − R Tt M s d W s + R Tt (cid:0) | a s | ψ ( X s + ζ s ) − | a s | ψ ( X s + ζ s ) + a s M s − b s (cid:1) d s. (4.4)It can be observed that if ( X, ζ, M ) is an adapted solution of (4.4) then Y := X + ζ is asolution of the following forward SDE Y t = x + ζ − Z t | a s | ψ ( Y s )d s + Z t a s ψ ( Y s )d W s . (4.5)Thus, a necessary condition for (4.4) to have a solution is that the SDE (4.5) admits a solution,which is guaranteed by the Lipschitz continuity of ψ and ψ . Proposition 4.1. Assume that ψ and ψ are bounded and Lipschitz continuous. Then, thereexists a solution to system (4.4) . Proof. It is sufficient to remark that for a linear g our setting reduces to the classical utilitymaximation problem so that the result follows for instance from [30].19 Explicit examples in complete markets In this section we show that the optimal investment strategy and its FBSDE characterizationcan be explicitly determined in settings where the market is complete and g is linear orquadratic. To this end, we assume that the utility U in addition satisfies the Inada conditionlim x →−∞ U ′ ( x ) = ∞ and lim x → + ∞ U ′ ( x ) = 0 . g Let us assume that g ( t, z ) = γ t z , where γ is some deterministic function satisfying R T | γ t | d t < ∞ . Then we have − Π t ( − X T ) = E Q γ [ X T |F t ] , (5.1)where Q γ is a measure equivalent to P , defined byd Q γ d P = E (cid:18) Z T γ t d W t (cid:19) := e R T γ t d W t − R T | γ t | d t . Therefore, the problem of utility maximization (3.1) is equivalent to the following static opti-mization problem max X E [ U ( X )] , s.t. E [ Xξ γT ] ≤ x , (5.2)where the pricing kernel ξ γT is defined by ξ γT := d Q γ d P = E ( R T γ s d W s ) . In this case the optimalwealth process is given by the well known Merton-like solution X ∗ T := I ( λξ γT ) , where I isthe inverse of the marginal utility U , and λ is determined such that the budget constraint E [ X ∗ T ξ γT ] = x is met. We summarize the connection between the solution to (5.2) with theFBDES (3.18) in the following proposition. The verification step is straightforward and willbe omitted. Proposition 5.1. Assume that E [ U ( I ( λξ γT ))] < ∞ and E [ ξ γT I ( λξ γT )] < ∞ for any λ > .Moreover, define ζ ∗ t := I ( λξ γt ) − X ∗ t , M ∗ t := − λξ γt γ t U ′′ ( X ∗ t + ζ ∗ t ) and X ∗ t = E Q γ [ I ( λξ γT ) |F t ] = x + Z t Z θ ∗ s d W Q γ s , (5.3) where Z θ ∗ s is a progressively measurable process resulting from the martingale representationof the conditional expectation process E Q γ [ I ( λξ γT ) |F t ] . Then the triple ( X ∗ t , ζ ∗ t , M ∗ t ) solves theFBSDE (3.18) and the optimality condition (3.14) holds. g In this section, we assume that the Market Maker evaluates the market risks in terms of anexponential utility certainty equivalent principle under an equivalent measure Q ∼ P . Moreprecisely, for any X ∈ D T e − γ Π t ( X ) = E Q [ e − γX |F t ] , γ > Q d P = exp (cid:26) − Z T | η t | d t − Z T η t d W t (cid:27) := ξ T , where η is a deterministic and bounded process. By Girsanov’s Theorem we have that W Q t = W t + R t η s d s is a standard Brownian motion under Q . Using Itˆo’s lemma (see e.g. [24]) weobserve that for any X ∈ D T X = Π t ( X ) + Z Tt g ( s, Z s ( X ))d s − Z Tt Z s ( X )d W s , where g ( t, z ) = γ | z | − η t z . We show that the expected utility maximization (3.1) can besolved in a martingale approach. By Theorem 1 in [24], a terminal portfolio value X T ∈ D T can be hedged perfectly with an initial endowment x satisfying e γx ≥ E Q [ e γX T ] = E [ e γX T ξ T ] . Therefore, the problem of utility maximization (3.1) is equivalent to the following static opti-mization problem max X E [ U ( X )] , s.t. E [ e γX ξ T ] ≤ e γx . Using that by the concavity of U lim x → + ∞ U ′ ( x ) e − γx = 0 and lim x →−∞ U ′ ( x ) e − γx = + ∞ we can conclude that U ′ ( x ) e − γx γ − is a decreasing function on R → R + whose inverse wedenote by f . Proposition 5.2. Assume that for any λ > , E [ e γf ( λξ T ) ξ T ] < ∞ . The optimal terminalwealth of Problem (3.1) is then given by X ∗ T := f ( λξ T ) , where λ is determined such that the budget constraint E [ e γX ∗ T ξ T ] = e γx is met. The optimalstrategy can be characterized as the strategy θ ∗ that perfectly replicates the terminal optimalwealth X ∗ T , i.e., X ∗ t = x − γ Z t |Z θ ∗ s | d s + Z t Z θ ∗ s d W Q s , t ∈ [0 , T ] , (5.4) where Z θ ∗ t := 1 γ (cid:18) β ∗ t R ∗ t + η t (cid:19) (5.5) with β ∗ t being the progressively measurable process resulting from the martingale representation R ∗ t := E [ U ′ ( X ∗ T ) |F t ] = U ′ ( X ∗ T ) − Z Tt β ∗ s d W s . (5.6)21 urthermore, define ζ ∗ t := I ( R ∗ t ) − γ log (cid:18) R ∗ t γλξ t (cid:19) , (5.7) and M ∗ t := β ∗ t U ′′ ( X ∗ t + ζ ∗ t ) − γ (cid:18) β ∗ t R ∗ t + η t (cid:19) . (5.8) Then the triple ( X ∗ t , ζ ∗ t , M ∗ t ) solves the FBSDE (3.18) and the optimality condition (3.14) holds. Proof. Observe first that the function ϑ : (0 , ∞ ) → (0 , ∞ ), λ ϑ ( λ ) = E [ e γf ( λξ T ) ξ T ] byassumption is well-defined. Since f is continuous, surjective and decreasing we observe that ϑ is continuous and surjective as well by the monotone convergence theorem. Hence, for any x ∈ R , there exists λ > E [ e γX ∗ T ξ T ] = e γx . Noting from the definition of f that U ′ ( X ∗ T ) = U ′ ( f ( λξ T )) = γλξ T e γf ( λξ T ) , (5.9)we derive X ∗ t = 1 γ log (cid:20) E Q [ e γf ( λξ T ) |F t ] (cid:21) = 1 γ log (cid:20) E [ ξ − t ξ T e γf ( λξ T ) |F t ] (cid:21) = 1 γ log (cid:20) λγξ t E [ U ′ ( X ∗ T ) |F t ] (cid:21) , which implies that X ∗ t = γ log( R ∗ t /λγξ t ). Recall that d ξ t = − η t ξ t d W t . By Itˆo’s lemma we getd X ∗ t = 12 γ (cid:18) | η t | − | β ∗ t | | R ∗ t | (cid:19) d t + 1 γ (cid:18) β ∗ t R ∗ t + η t (cid:19) d W t . Identifying the last SDE with (5.4) we obtain (5.5). It remains to verify that ( X ∗ t , ζ ∗ t , M ∗ t )solves the FBSDE system (3.18). First, it follows from (5.9) that ζ ∗ T = I ( R ∗ T ) − γ log (cid:18) R ∗ T γλξ T (cid:19) = I ( U ′ ( X ∗ T )) − γ log (cid:18) U ′ ( X ∗ T ) γλξ T (cid:19) = X ∗ T − γ log( e γX ∗ T ) = 0 . Set H ( t, X ∗ t , ζ ∗ t ) := 1 γ (cid:18) β ∗ t R ∗ t + η t (cid:19) . (5.10)It follows that X ∗ t = x − Z t g ( s, H ( s, X ∗ s , ζ ∗ s ))d s + Z t H ( s, X ∗ s , ζ ∗ s )d W s . (5.11)Furthermore, applying Itˆo’s lemma for (5.7) we obtain that ζ ∗ t = − Z Tt M ∗ s d W s + 12 Z Tt U (3) U ′′ ( X ∗ s + ζ ∗ s ) |H ( s, X ∗ s , ζ ∗ s ) + M ∗ s | d s − Z Tt g ( s, H ( s, X ∗ s , ζ ∗ s ) (cid:1) d s, (5.12)where M ∗ t is defined by (5.8). From (5.11) and (5.12) we conclude that the triple ( X ∗ t , ζ ∗ t , M ∗ t )solves the FBSDE system (3.18). To finish the proof we recall that R ∗ t = U ′ ( X ∗ t + ζ ∗ t ) and g z ( t, z ) = γz − η t . It then follows from (5.8) and (5.10) that − U ′ ( X ∗ t + ζ ∗ t ) g z ( t, H ( t, X ∗ t , ζ ∗ t )) + U ′′ ( X ∗ t + ζ ∗ t )( H ( t, X ∗ t , ζ ∗ t ) + M ∗ t ) = 0 , which means that the optimality condition (3.14) is fulfilled.A more explicit solution can be given when the Large Trader uses an exponential utilityfunction. This case is also considered in [4] without using FBSDEs.22 roposition 5.3. Assume U ( x ) = − e − γ A x , for some constant γ A > . The optimal wealthand optimal investment strategy are then given by X ∗ t = x − γ Z t |Z θ ∗ s | d s + Z t Z θ ∗ s d W Q s , t ∈ [0 , T ] , with Z θ ∗ t := η t γ + γ A . (5.13) Furthermore, the triple (cid:18) X ∗ t , ζ ∗ t = − γ + γ A ) R Tt | η s | d s, M ∗ t = 0 (cid:19) is a solution of the FBSDEsystem (3.18) . Proof. For U ( x ) = − e − γ A x we easily compute the inverse marginal utility as f ( x ) = − γ + γ A log (cid:0) γγ A x (cid:1) . Therefore, the optimal terminal wealth is given by X ∗ T = − γ + γ A log (cid:0) γλξ T γ A (cid:1) ,where the multiplier λ is defined by x = 1 γ log (cid:18) E Q [ e γX ∗ T ] (cid:19) = 1 γ log (cid:18) E Q (cid:20) e − γγ + γA log (cid:0) γλξTγA (cid:1)(cid:21)(cid:19) = − γ + γ A log (cid:18) γλγ A (cid:19) + − γ A γ + γ A ) Z T | η s | d s. (5.14)In the same way we obtain X ∗ t = 1 γ log (cid:18) E Q [ e γX ∗ T |F t ] (cid:19) = − γ + γ A log (cid:18) γλγ A (cid:19) + − γ + γ A (cid:18) − Z t η s d W Q s + 12 Z t | η s | d s (cid:19) + − γ A γ + γ A ) Z Tt | η s | d s. (5.15)Plugging (5.14) into the last equation we get X ∗ t = x − γ Z t | η s | ( γ + γ A ) d s + Z t η s d W Q s γ + γ A , (5.16)which implies (5.13). Now, consider the martingale process R ∗ t := E [ U ′ ( X ∗ T ) |F t ]. A directcalculation using (5.15) shows that R ∗ t = U ′ ( X ∗ T ) + R Tt R ∗ s γ A γ + γ A η s d W s , which means that β ∗ s = − γ A γ + γ A η s and (5.10) holds true. Furthermore, we get from (5.7) that ζ ∗ t = − γ + γ A ) R Tt | η s | d s ,which means that M ∗ t = 0. Finally, it is straightforward to check that the optimality condition(3.14) is fulfilled. Proposition 5.4. Suppose that g ( t, z ) is convex with quadratic growth and satisfies g z ( t, 0) = g ( t, 0) = 0 . Then the optimal terminal wealth is given by X ∗ T = x . This means that it isoptimal for the Large Trader to invest nothing, i.e., H ( t, X ∗ t , ζ ∗ t ) = 0 . In addition, the triple ( X ∗ t = x , ζ ∗ t = 0 , M ∗ t = 0) is a solution of the FBSDE system (3.18) . Proof. Suppose that g is convex and satisfies g z ( t, 0) = g ( t, 0) = 0 and | g ( t, z ) | ≤ C (1 + | z | ).It is straightforward to see that the triple ( X ∗ t = x , ζ ∗ t = 0 , M ∗ t = 0) solves the FBSDE system(3.18) and the optimality condition (3.14) is fulfilled with H ≡ 0. Clearly, the constant process U ′ ( x ) is a martingale so that we do not need the assumption that ψ and ψ are bounded,and Theorem 3.3 applies. 23 Connection with BSPDEs In this section we characterise the value function of our expected utility maximization problem(3.1) by a BSPDE which resutls from a direct application of the It’s-Ventzel formula (seeAppendix D) for regular families of semimartingales. We remark that SPDEs have been alsostudied in the context of progressively forward utility; see e.g. [23, 49]. We first introduce I s,t ( θ ) := − Z ts g ( u, Z θu )d u + Z ts Z θu d W u , ≤ s ≤ t ≤ T, which represents the total gain/loss of the strategy θ in [ s, t ]. For any t ∈ [0 , T ] and x ∈ R wedefine V ( t, x ) := ess sup θ ∈ Θ ,θ s ,s ∈ [ t,T ] E (cid:2) U ( x + I t,T ( θ )) (cid:12)(cid:12) F t (cid:3) . (6.1)The following conditions are assumed throughout this section. (CV) For any t ∈ [0 , T ] and x ∈ R , the supremium in (6.1) is attained, i.e., there exists anadmissible strategy θ ∗ ( x ) s , s ∈ [ t, T ] such that V ( t, x ) = E (cid:2) U ( x + I t,T ( θ ∗ ( x ))) (cid:12)(cid:12) F t (cid:3) . (C Θ ) Either the driver function g is positively homogeneous with Z t ( − S ) = 0 , d P × d t a.s.,or the market is complete and g is continuously differentiable. Note that Condition (C Θ ) is fulfilled in Example 1 and Example 3. We additionally assumethat g satisfies either Condition ( HL ) or ( Hg ) specified in Section 2. Lemma 6.1. Under Conditions (CV) and (C Θ ) , the value function V ( t, x ) is strictly concavewith respect to x . Proof. Let θ , θ be the corresponding optimal strategy starting with initial wealth x , x respectively, i.e., V ( t, x ) = E (cid:2) U ( x + I t,T ( θ )) (cid:12)(cid:12) F t (cid:3) , and V ( t, x ) = E (cid:2) U ( x + I t,T ( θ )) (cid:12)(cid:12) F t (cid:3) . Let λ ∈ [0 , 1] arbitrary. Let us first consider the case that g is positively homogeneous. FromExample 1 we have Z θ = θZ ( − S ) so that I t,T ( θ i ) = − R Tt g ( s, θ is Z s ( S ))d s + R Tt θ is Z s ( S )d W s for i = 1 , 2. By the convexity of g this yields I t,T ( λθ + (1 − λ ) θ ) = Z Tt − g ( s, ( λθ s + (1 − λ ) θ s ) Z s ( − S ))d s + Z Tt ( λθ s + (1 − λ ) θ s ) Z s ( − S )d W s ≥ λ (cid:26) Z Tt − g ( s, θ s Z s ( − S ))d s + Z Tt θ s Z s ( − S )d W s (cid:27) + (1 − λ ) (cid:26) Z Tt − g ( s, θ s Z s ( − S ))d s + Z Tt θ s Z s ( − S )d W s (cid:27) = λI t,T ( θ ) + (1 − λ ) I t,T ( θ ) . Hence, λV ( t, x ) + (1 − λ ) V ( t, x ) ≤ E (cid:2) U ( λx + (1 − λ ) x + λI t,T ( θ ) + (1 − λ ) I t,T ( θ )) (cid:3) ≤ E (cid:2) U ( λx + (1 − λ ) x + I t,T ( λθ + (1 − λ ) θ )) (cid:12)(cid:12) F t (cid:3) ≤ ess sup θ ∈ Θ , ( θ s ) ,s ∈ [ t,T ) E (cid:2) U ( λx + (1 − λ ) x + I t,T ( θ )) (cid:12)(cid:12) F t (cid:3) = V ( t, λx + (1 − λ ) x ) , V . Next, let us consider the case that the market is complete.This means every X T ∈ D T is attainable for a θ ∈ Θ as long as − E g ( − X T ) ≤ x . Since the g -expextation is concave, see [34], the set of attainable portfolios is convex, from which theconcavity of V immediately follows.To show the strict concavity let us assume that λV ( t, x ) + (1 − λ ) V ( t, x ) = V (cid:0) t, λx + (1 − λ ) x (cid:1) for some λ ∈ [0 , U we deduce that x + I t,T ( θ ) = x + I t,T ( θ ) , which leads to x − x = − Z Tt ( g ( s, Z θ s ) − g ( s, Z θ s ))d s + Z Tt ( Z θ s − Z θ s )d W s = Z Tt ( Z θ s − Z θ s )d W Gs , (6.2)where W Gt := W t − R t P dj =1 G j ( s, Z θ s , Z θ s )d s and G j ( s, z, ˜ z ) := g ( s, z , . . . , z j , ˜ z j +1 , . . . , ˜ z d ) − g ( s, z , . . . , z j − , ˜ z j , . . . , ˜ z d ) z j − ˜ z j . In case of Condition ( HL ), G is bounded. In case of Condition ( Hg ) we observe that | G ( s, Z θ s , Z θ s ) | ≤ K (1 + |Z θ s | + |Z θ s | ) . It follows then directly from (6.2) that Z θ and Z θ integrated with respect to the Brown-ian motion are BMO( P )’s (see Appendix E). Thus, we conclude by our growth conditionsthat R t G ( s, Z θ s , Z θ s )d W s ∈ BMO( P ) as well. Hence, by Kazamaki’s Theorem (see e.g.[9] [Th. 3.24, Chapter 3]), W G is a Brownian motion under P G defined by d P G / d P = E T (cid:0) − R t G ( s, Z θ s , Z θ s )d W s (cid:1) . Moreover, since R t ( Z θ s − Z θ s )d W s is a BMO( P ) martingaleit follows from general properties of BMOs, see [9], that R t ( Z θ s − Z θ s )d W Gs is a BMO( P G )martingale. Taking expectations with respect to P G on both sides in (6.2) we obtain x = x and the proof is completed. Lemma 6.2. Let θ be admissible and s ∈ [0 , T ] , then the process { V ( t, x + I s,t ( θ )) , t ≥ s } isa supermartingale for all x ∈ R . Furthermore, V ( s, x ) := ess sup θ ∈ Θ ,θ u ,u ∈ [ s,T ] E (cid:20) V ( t, x + I s,t ( θ )) (cid:12)(cid:12)(cid:12)(cid:12) F s (cid:21) (6.3) and a strategy θ ∗ is optimal if and only if V ( t, x + I s,t ( θ ∗ )) is a martingale process. Proof. Let θ ∈ Θ. Denote by Θ( θ , t, T ) the set of all admissible strategies being equal to θ until time t , i.e., θ ∈ Θ( θ , t, T ) if θ ∈ Θ and θ s ≤ s ≤ t = θ s ≤ s ≤ t . Let us show that thefamily (cid:8) Υ θt := E [ U ( x + I ,T ( θ )) |F t ] , θ ∈ Θ( θ , t, T ) (cid:9) admits the lattice property. Indeed, for θ , θ ∈ Θ( θ , t, T ) we define θ := θ s ≤ s The functions b ( t, x ) , α ( t, x ) and α x ( t, x ) in Definition 6.1 are continuous with respectto x and satisfy, for any constant c > , E (cid:20) Z T max | x |≤ c ( | b ( t, x ) | , | α ( t, x ) | , | α x ( t, x ) | )d s (cid:21) < ∞ . Below we denote by V , the class of all regular families semimartingales V defined byDefinition 6.1 whose coefficients b , and α satisfy Condition ( CR ). Recall also that a process V belongs to the class D if the family of processes V τ τ ≤ T for all stopping times τ is uniformlyintegrable. The following lemma is a straightforward consequence of the global convexity of g . Lemma 6.3. There exists a unique progressively measurable process denoted by υ ( t, x ) suchthat the supremum of L V ( t, x ) := ess sup Z∈ Image ( Z θt ) (cid:18) − g ( t, Z ) V x ( t, x ) + 12 |Z| V xx ( t, x ) + Z α x ( t, x ) (cid:19) , (6.4) is attained. In particular, when g is continuously differentiable and the market is complete, υ ( t, x ) is given by the first order condition U V ( t, υ ( t, x ) , x ) = 0 , where U V ( t, z, x ) = − g z ( t, z ) V x ( t, x ) + zV xx ( t, x ) + α x ( t, x ) . On the other hand, when g is positively homogeneous, υ ( t, x ) = b θ ( t, x ) Z t ( − S ) , where b θ ( t, x ) = b θ ( t, x ) b θ ( t,x ) > + b θ ( t, x ) b θ ( t,x ) < + 0 b θ ( t,x ) ≥ ≥ b θ ( t,x ) , (6.5) and b θ ( t, x ) ≤ b θ ( t, x ) a.s. which are defined by b θ ( t, x ) = − Z t ( − S ) α x ( t, x ) + g ( t, Z t ( − S )) V x ( t, x ) | Z t ( − S ) | V xx ( t, x ) , (6.6) b θ ( t, x ) = − Z t ( − S ) α x ( t, x ) − g ( t, − Z t ( − S )) V x ( t, x ) | Z t ( − S ) | V xx ( t, x ) . (6.7) Proof. When g is continuously differentiable and the market is complete so that Image( Z θt ) = R d , optimizing over Z yields that υ is determined by the unique solution of − g z ( t, Z ) V x ( t, x ) + V xx ( t, x ) Z + α x ( t, x ) = 0. On the other hand, when g is positively homogeneous, we recallfrom Example 1 that Z θt = θ t Z ( − S ). Therefore, the formula inside the essential supremum(6.4) becomes −| θ t | g (cid:0) t, Z t ( − S )sgn( θ t ) (cid:1) V x ( t, x ) + 12 | θ t | | Z t ( − S ) | V xx ( t, x ) + θ t Z t ( − S ) α x ( t, x )and the supremum may be taken over θ t ∈ R . Taking the derivative with respect to θ t givesthe necessary and sufficient optimality condition0 ∈ − sgn( b θ ( t, x ) g ( t, sgn( b θ ( t, x )) Z t ( − S )) V x ( t, x ) + b θ ( t, x ) | Z t ( − S ) | V xx ( t, x ) + Z t ( − S ) α x ( t, x ) . (6.8)A direct calculation then leads to (6.5)We show in the following that the value function can be characterized as solution of aBSPDE where g satisfies (Hg) or (HL) . 27 heorem 6.1. Let V ( t, x ) ∈ V , . Then the value function is a solution of the BSPDE V ( t, x ) = V (0 , x ) + Z t α ( s, x )d W s + Z t L V ( s, x )d s, (6.9) where the operator L is defined by (6.4) . Moreover, a strategy θ ∗ ∈ Θ with V ( t, X θ ∗ t ) belongingto class D is optimal if and only if Z θ ∗ t = υ ( t, X θ ∗ t ) , i.e., U V ( t, υ ( t, X θ ∗ t ) , X θ ∗ t ) = − g z ( t, υ ( t, X θ ∗ t )) V x ( t, X θ ∗ t ) + Z θ ∗ t V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) = 0 , (6.10) when g is continuously differentiable and the market is complete, and Z θ ∗ t = b θ ( t, X θ ∗ t ) Z t ( − S ) (see (6.5) ) when g is positively homogeneous. The optimal wealth process X θ ∗ t in each case isthen characterized by the forward SDE X θ ∗ t = x − Z t g ( s, Z θ ∗ s )d s + Z t Z θ ∗ s d W s . (6.11) Proof. Using Itˆo-Ventzel’s formula (see Appendix D) we can represent V ( t, x + I s,t ( θ )) = V ( s, x ) + Z ts (cid:16) G ( u, x + I s,u ( θ ) , Z θu ) − b ( u, x + I s,u ( θ )) (cid:17) d u + Z ts (cid:16) V x ( u, x + I s,u ( θ )) Z θu + α ( u, x + I s,u ( θ )) (cid:17) d W u , (6.12)where G ( u, z ) := − g ( u, z ) V x ( u, p ) + 12 | z | V xx ( u, p ) + α x ( u, p ) z. (6.13)We recall from Lemma 6.2 that for any x , the value process V ( t, x + I s,t ( θ )) is a supermartingale.Hence, the finite variation part in (6.12) is decreasing. Therefore, for any ǫ > Z s + ǫs b ( u, x + I s,u ( θ ))d u ≥ Z s + ǫs G ( u, x + I s,u ( θ ) , Z θu )d u. Dividing both sides by ǫ and letting ǫ → b ( s, x ) ≥ G ( s, x, Z θs ) d P × d t a.s . Noting that V xx < V x > xb ( t, x ) ≥ ess sup Z∈ Image ( Z θt ) G ( t, x, Z ) = G ( t, x, υ ( t, x )) = L V ( t, x ) , (6.14)where υ ( t, x ) and L V ( t, x ) are defined by (6.4). Now assume that θ ∗ is an optimal strategy,i.e., V ( t, x + I s,t ( θ ∗ )) is a martingale. Let X θ ∗ s,t ( x ) = x + I s,t ( θ ∗ ) and note that X θ ∗ ,t ( x ) = x + I ,t ( θ ∗ ) = X θ ∗ t . Using the Itˆo-Ventzel formula we get for any s ∈ [0 , t ], G ( t, X θ ∗ s,t ( x ) , Z θ ∗ t ) − b ( t, X θ ∗ s,t ( x )) = 0 . (6.15)It follows from (6.14) that G ( t, X θ ∗ s,t ( x ) , Z θ ∗ t ) − L V ( t, X θ ∗ s,t ( x )) ≥ . L V ( t, x ) = ess sup Z∈ Image ( Z θt ) G ( t, x, Z ). Therefore, Z θ ∗ t must be the maximizerof G ( t, X θ ∗ s,t ( x ) , Z ) and therefore Z θ ∗ t = υ ( t, X θ ∗ s,t ( x )). Hence, for any s ∈ [0 , t ] we have U V ( t, Z θ ∗ t , X θ ∗ s,t ( x )) = 0 if the market is complete and g is continuously differentiable, or Z θ ∗ t = b θ ( t, X θ ∗ s,t ( x )) Z t ( − S ) if g is positively homogeneous. Consequently, taking s = 0 and s = t we obtain (6.10) and b ( t, x ) = L V ( t, x ), which leads to the BSPDE (6.9).If e θ is a strategy satisfying (6.9)-(6.11), it is straightforward to verify that the value process V ( t, x + I s,t ( e θ )) is a local martingale. Since it by assumption belongs to the class D , V ( t, x + I s,t ( e θ )) is a martingale and e θ is optimal by Lemma 6.2.We obtain the following explicit BSPDE characterization for the value function in the casewhere g is quadratic. Corollary 6.1 (quadratic g ) . Let V ( t, x ) ∈ V , and assume that the market is complete andthat g ( t, z ) = γ t | z | − η t z where η t is a vector of bounded deterministic functions of t and γ is a positive bounded deterministic function. Then the value function is a solution of theBSPDE V ( t, x ) = V (0 , x ) + Z t α ( s, x )d W s + 12 Z t | η s V x ( s, x ) + α x ( s, x ) | V xx ( s, x ) − γ s V x ( s, x ) d s. (6.16) Moreover, a strategy θ ∗ with V ( t, X θ ∗ t ) belonging to class D is optimal if and only if Z θ ∗ t = − η t V x ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) V xx ( t, X θ ∗ t ) − γ t V x ( t, X θ ∗ t ) (6.17) is admissible. The optimal wealth process X θ ∗ t is characterized by following forward SDE X θ ∗ t = x − Z t (cid:18) γ s |Z θ ∗ s | − η s Z θ ∗ s (cid:19) d s + Z t Z θ ∗ s d W s , (6.18) where Z θ ∗ is defined by (6.17) . Proof. For g ( t, z ) = γ t | z | − η t z which is continuously differentiable we have g z ( t, z ) = γ t z − η t . Hence, we obtain (6.17) directly from (6.10) and L V ( t, x ) = − | η t V x ( t, x ) + α x ( t, x ) | V xx ( t, x ) − γ t V x ( t, x ) , which leads to the BSPDE (6.16). Finally, if e θ is a strategy satisfying (6.17) and (6.18) thenthe value process V ( t, x + I s,t ( e θ )) is a local martingale and belongs to the class D . Therefore, V ( t, x + I s,t ( e θ )) is a martingale and e θ is optimal by Lemma 6.2.Since γ > 0, (6.16) gives a hightly nonlinear SPDE which, as far as we know, has not beenderived before. Hence, further investigation on the uniqueness and regularity of the solutionof the class of SPDEs (6.16) would be of interest but very challenging due to the possibledegeneracy and full nonlinearity of the equation. In the case where U is exponential, a moreexplicit analysis is possible but is omitted. 29e have seen that the value function of an optimal strategy can be characterized by aBSPDE (6.9)-(6.11). Differentiating this BSDPE (assuming all derivatives below exist) weobtain V x ( t, x ) = V x (0 , x ) + R t α x ( s, x )d W s + R t L Vx ( s, x )d s, V x ( T, x ) = U ′ ( x ) ,X θ ∗ t = x − R t g ( s, Z θ ∗ s )d s + R t Z θ ∗ s d W s . (6.19)Below we will show that the BSPDE (6.19) is equivalent to the FBSDE (3.18). We remarkthat [45] obtains a similar result for the case without price impact. Theorem 6.2. Assume the conditions of Theorem 6.1, that ( V x ( t, x ) , α x ( t, x ) , L Vx ( t, x ) , X θ ∗ t ) is a solution of the BSPDE (6.19) and that V x ( t, x ) is a regular family of semimartingales.Let υ ( t, X θ ∗ t ) be the unique adapted maximizer process in (6.4) . Then, the triple ( X θ ∗ t , ζ t , M t ) defined by ζ t = I ( V x ( t, X θ ∗ t )) − X θ ∗ t , M t = υ ( t, X θ ∗ t ) V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) U ′′ ( X θ ∗ t + ζ t ) − υ ( t, X θ ∗ t ) , is a solution of the FBSDE (3.18) . Furthermore, the optimality condition (3.4) holds. Proof. First, by the Itˆo-Ventzel formula, it can be seen directly that V x ( t, X θ ∗ t ) = V x (0 , x ) + Z t (cid:0) Z θ ∗ s V xx ( s, X θ ∗ s ) + α x ( s, X θ ∗ s ) (cid:1) d W s + Z t (cid:0) L Vx ( s, X θ ∗ s ) − |Z θ ∗ s | V xxx ( s, X θ ∗ s ) − g ( s, Z θ ∗ s ) V xx ( s, X θ ∗ s ) + Z θ ∗ s α xx ( s, X θ ∗ s ) (cid:1) d s. From (6.4) we observe that the finite variation term is zero, which implies that V x ( t, X θ ∗ t ) isa local martingale whose decomposition is given by V x ( t, X θ ∗ t ) = V x (0 , x ) + Z t (cid:0) Z θ ∗ s V xx ( s, X θ ∗ s ) + α x ( s, X θ ∗ s ) (cid:1) d W s . (6.20)Let ζ t = I ( V x ( t, X θ ∗ t )) − X θ ∗ t . Clearly, V x ( t, X θ ∗ t ) = U ′ ( X θ ∗ t + ζ t ) and therefore I ′ ( V x ( t, X θ ∗ t )) = 1 U ′′ ( X θ ∗ t + ζ t ) , I ′′ ( V x ( t, X θ ∗ t )) = − U (3) ( X θ ∗ t + ζ t )( U ′′ ) ( X θ ∗ t + ζ t ) . Note that ζ T = I ( U ′ ( X θ ∗ T )) − X θ ∗ T = 0. By Itˆo’s formula we getd ζ t = Z θ ∗ t V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) U ′′ ( X θ ∗ t + ζ t ) d W t − Z θ ∗ t d W t + 12 I ′′ ( V x ( t, X θ ∗ t )) |Z θ ∗ t V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) | d t + g ( t, Z θ ∗ t )d t. (6.21)Set M t = υ ( t, X θ ∗ t ) V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) U ′′ ( X θ ∗ t + ζ t ) − υ ( t, X θ ∗ t ) . (6.22)30onsider first the case where the market is complete with g continuously differentiable. By(6.10) we have υ ( t, X θ ∗ t ) V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) = g z ( t, υ ( t, X θ ∗ t )) V x ( t, X θ ∗ t ). Hence, it followsfrom (6.22) that − g z ( t, υ ( t, X θ ∗ t )) U ′ ( X θ ∗ t + ζ t ) + U ′′ ( X θ ∗ t + ζ t )( M t + υ ( t, X θ ∗ t )) = 0 , and we can define a function H ( t, X θ ∗ t , ζ t , M t ) := υ ( t, X θ ∗ t ) = Z θ ∗ t , where the last equationholds by Theorem 6.1. It is then straightforward to see using (6.21) that the triple ( X θ ∗ t , ζ t , M t )satisfies the FBSDE (3.18). We remark that the last identity (since the partial derivative ∂ y Z does not appear) defines a slightly stronger optimality condition than (3.4). In the casethat g is positively homogeneous, we note that we can define a function H ( t, X θ ∗ t , ζ t , M t ) = υ ( t, X θ ∗ t ) = ˆ θ ( t, X θ ∗ t ) Z t ( − S ), see (6.5), and the triple ( X θ ∗ t , ζ t , M t ) again satisfies the FBSDE(3.18). Finally, using (6.22) we observe that U ′′ ( X θ ∗ t + ζ t ) Z t ( − S ) M t = U ′′ ( X θ ∗ t + ζ t ) Z t ( − S ) (cid:18) ˆ θ ( t, X θ ∗ t ) Z t ( − S ) V xx ( t, X θ ∗ t ) + α x ( t, X θ ∗ t ) U ′′ ( X θ ∗ t + ζ t ) − ˆ θ ( t, X θ ∗ t ) Z t ( − S ) (cid:19) = ˆ θ ( t, X θ ∗ t ) | Z t ( − S ) | V xx ( t, X θ ∗ t ) + Z t ( − S ) α x ( t, X θ ∗ t ) − ˆ θ ( t, X θ ∗ t ) U ′′ ( X θ ∗ t + ζ t ) | Z t ( − S ) | = Z t ( − S ) α x ( t, X θ ∗ t ) + ˆ θ ( t, X θ ∗ t ) | Z t ( − S ) | ( V xx ( t, X θ ∗ t ) − U ′′ ( X θ ∗ t + ζ t )) . Hence, − U ′ ( X θ ∗ t + ζ t )sgn( b θ ( t, X θ ∗ t )) g ( t, sgn( b θ ( t, X θ ∗ t )) Z t ( − S ))+ Z t ( − S ) U ′′ ( X θ ∗ t + ζ t )( b θ ( t, X θ ∗ t ) Z t ( − S ) + M t )= − V x ( t, X θ ∗ t )sgn( b θ ( t, X θ ∗ t )) g ( t, sgn( b θ ( t, X θ ∗ t )) Z ( − S )) + b θ ( t, X θ ∗ t ) | Z t ( − S ) | V xx ( t, X θ ∗ t )+ Z t ( − S ) α x ( t, X θ ∗ t ) ∋ , where the last relationship is true as by Lemma 6.3, b θ ( t, X θ ∗ t ) satisfies (6.8). Since the laststatement corresponds to (3.4) in the case that g is positively homogeneous (see Example 7)the proof is completed. AppendixA Proof for Proposition 2.2 Proof. Recall that Z t,r ( yS ) is defined through the BSDEΠ t,rs ( yS ) = h M ( R t,rT ) − ys ( R t,rT ) − Z Ts g ( u, Z t,ru ( yS )) d u + Z Ts Z t,ru ( yS ) d W u , s ∈ [0 , T ] . (A.1)Denote by ( F t,r,y , V t,r,y ) the solution to the BSDE F t,r,ys = ( h M r ( R t,rT ) − ys r ( R t,rT )) ∇ r R t,rT − Z Ts g z ( u, Z t,ru ( yS )) V t,r,yu d u + Z Ts V t,r,yu d W u = ( h M r ( R t,rT ) − ys r ( R t,rT )) ∇ r R t,rT − Z Ts g z ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) V t,r,yu d u + Z Ts V t,r,yu d W u , (A.2)31here R Ts V u d W u is defined by P ≤ i ≤ d R Ts V iu d W iu , with V i denoting the i -th line of the d × d matrix process V . In the second equation we used that Z t,r,yu = − F t,r,yu ( ∇ r R t,ru ) − σ, (A.3)see [21] or [51]. We remark that ∇ r R t,rs and ( ∇ r R t,rs ) − are bounded and bounded awayfrom zero. Note that the index y in F and V refers to the dependence of these process in y through the terminal condition. We remark further that ( F t,r,y , V t,r,y ) is again the solution ofa BSDE. Hence, the d P × d t a.s. differentiability of y → F t,r,yt (and therefore using (A.3) alsothe differentiability y → Z t,r,yt ) follows now from Theorem 2.1 in [3]. Now in case a) usingclassical truncation arguments, see [21], we may assume that the driver in (A.2) has compactsupport in the argument V t,r,y meaning that it can be assumed to be Lipschitz continuous. Forconvenience, let us briefly repeat their argument. For an integer N , define a smooth function ρ N : R × d → R + such that ∀| z | ≤ N , ρ N ( z ) = 1; and ∀| z | ≥ N + 1, ρ N ( z ) = 0. Then ρ N g is abounded Lipschitz function. Hence, for any N , there exists a unique solution (Π N ; t,r , Z N ; t,r,y )to the BSDEΠ N ; t,rs ( yS ) = h M ( R t,rT ) − ys ( R t,rT ) − Z Ts ( ρ N g )( u, Z N ; t,r,yu ) du + Z Ts Z N ; t,r,yu dW u . On the other hand, denote by ( F N ; t,r,y , V N ; t,r,y ) the unique solution to the following BSDE F N ; t,r,ys = ( h M r ( R t,rT ) − ys r ( R t,rT )) ∇ r R t,rT − Z Ts ( ρ N g ) z ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) V N ; t,r,yu du + Z Ts V N ; t,r,yu dW u . Using a Girsanov transformation it follows that F N ; t,r,y is bounded by a constant C . Inparticular, for N ≥ C , F N ; t,r,y is independent of N and ρ N g ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) agreeswith g ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) . Hence, F N ; t,r,y satisfies the BSDE (A.2) and is bounded by C. In particular, Π t,r ( yS ) = Π N ; t,r ( yS ), and F t,r,y = F N ; t,r,y is the solution of a BSDE with aLipschitz continuous driver. By well known stability results for Lipschitz-continuous BSDEs,see for instance [42], we can conclude that F t,r,y is continuous in y and grows at most linearly.The continuity of Z in y and its linear growth follow then directly from (A.3).We further remark that in case b) existence, comparison principle and L ∞ -continuity ofsolutions of the BSDE (A.2) in the terminal conditions follow by [37]. Hence, F t,r,y is boundedand continuous in y , and therefore by (A.3), Z is bounded and continuous in y as well. B Proof of Proposition 2.3 Proof. The case of g being quadratic will be omitted. In the sequel, assume that g is Lipschitz.Choose s n , σ n , µ n , g n uniformly Lipschitz continuous with the first three derivatives boundedconverging uniformly to s, σ, µ and g . Then S n = s n ( R nT ) converges in L to S = s ( R T ),where R n is the Markov process corresponding to the solution of the SDE with functions µ n and σ n . Denote the solution of the BSDE with terminal condition yS n and driver function g n by ( Y n,y , Z n,y ). By the proof of Theorem 4.1 in [24], y → Z n,y ( t, ω ) is continuous, ontoand strictly increasing, with derivative with respect to y greater than ǫ > L (d P × d t )almost surely. By standard results on BSDEs see for instance [37], Z n,y → Z y in L (d P × d t ).32y passing to subsequence if necessary we may assume that convergence holds almost surely.Since the derivative of Z n,y is always greater than ǫ , by the mean-value theorem the samemust be true for all differential quotients of Z n,y . By passing to the limit also all differentialquotients of Z y must thus be greater than ǫ which entails that Z y is strictly increasing andonto. That Z y is also continuous follows then from the monotonicity. That the market iscomplete follows by Theorem 3.8 in [24]. C Proof of Proposition 3.1 Proof. Using similar truncation arguments as in the proof of Proposition 2.2 we can see that V t,r,y in (A.2) for fixed y is uniformly bounded as s rr is bounded. Taking the derivative in(A.2) with respect to y (see [3]) and denoting the derivative of F by F y we obtain for t ≤ s ≤ T the linear BSDE F t,r,yy,s = ( h M r ( R t,rT ) − s r ( R t,rT )) ∇ r R t,rT − Z Ts (cid:18) g z ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) ˜ V t,r,yu − g zz ( u, − F t,r,yu ( ∇ r R t,ru ) − σ ) F t,r,yy,u ( ∇ r R t,ru ) − σV t,r,yu (cid:19) du + Z Ts ˜ V t,r,yu dW u . (C.1)By our assumptions the coefficients and the terminal condition of this linear BSDE are bounded(since either g zz or F t,r,y is bounded.) Hence, the solution F t,r,yy,s is bounded as well. From ∂ Z t,ry,s = − F t,r,yy,s ( ∇ r R t,rs ) − σ , it follows then that ∂ Z y is bounded. D Generalized Itˆo-Ventzel formula For convenience we recall a short version of the generalized Itˆo-Ventzel formula. Detaileddiscussions and proofs can be found in e.g. [40, 39]. Let V ( t, x ), x ∈ R be a family ofone-dimensional C -continuous processes and C -semimartingales whose local characteristicssatisfies appropriate boundedness conditions. The generalized Itˆo integral of V with respectto the kernel (d s, X s ) is denoted by R t V (d s, X s ), see e.g. [40, 39]. The differential rule for V ( t, X t ) is given by the following result. Theorem D.1 ([40, 39]) . Let V ( t, x ) , x ∈ R be a family of one-dimensional C -continuousprocesses and C -semimartingale whose local characteristics satisfies appropriate boundednessconditions, see Theorem 3.3.1 in [39]. Let X t be a continuous semimartingale. 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