Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint
aa r X i v : . [ q -f i n . M F ] A p r Continuous viscosity solutions to linear-quadratic stochasticcontrol problems with singular terminal state constraint ∗ Ulrich Horst † and Xiaonyu Xia ‡ April 29, 2020
Abstract
This paper establishes the existence of a unique nonnegative continuous viscosity solu-tion to the HJB equation associated with a linear-quadratic stochastic control problem withsingular terminal state constraint and possibly unbounded cost coefficients. The existenceresult is based on a novel comparison principle for semi-continuous viscosity sub- and su-persolutions for PDEs with singular terminal value. Continuity of the viscosity solution isenough to carry out the verification argument.
AMS Subject Classification:
Keywords:
HJB equation, viscosity solution, terminal state constraint
Let T ∈ (0 , ∞ ) and let (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) that satisfies the usual conditions and carries aPoisson process N and an independent ˜ d -dimensional standard Brownian motion W . We analyzethe linear-quadratic stochastic control problemess inf ξ,µ E (cid:20)Z T η ( Y s ) | ξ s | + θγ ( Y s ) | µ s | + λ ( Y s ) | X ξ,µs | ds (cid:21) (1.1)subject to the state dynamics dY t = b ( Y t ) dt + σ ( Y t ) dW t , Y = ydX ξ,µt = − ξ t dt − µ t dN t , X ξ,µ = x (1.2)and the terminal state constraint X ξ,µT = 0 . (1.3) ∗ Financial support by d-fine GmbH is gratefully acknowledged. We thank Paulwin Graewe for many discus-sions and valuable comments. We thank two anonymous referees for valuable comments and suggestions thatgreatly helped to improve the presentation of the results. † Department of Mathematics, and School of Business and Economics, Humboldt-Universit¨at zu Berlin Unterden Linden 6, 10099 Berlin, Germany; email: [email protected] ‡ Department of Mathematics, Humboldt-Universit¨at zu Berlin Unter den Linden 6, 10099 Berlin, Germany;email: [email protected]
1e assume that θ is a positive constant, that the cost coefficients η, λ, γ are continuous and ofpolynomial growth, that η is twice continuously differentiable and that the diffusion coefficients b, σ are Lipschitz continuous. We prove the existence of a unique continuous viscosity solutionto the resulting HJB equation and give a representation of the optimal control in terms of theviscosity solution.Control problems of the form (1.1)-(1.3) arise in models of optimal portfolio liquidation undermarket impact when a trader can simultaneously trade in a primary venue and a dark pool.Dark pools are alternative trading venues that allow investors to reduce market impact andhence trading costs by submitting liquidity that is shielded from public view. Trade executionis uncertain, though, as trades will be settled only if matching liquidity becomes available. Insuch models, X ξ,µ describes the portfolio process when the traders submits orders at rates ξ to the primary venue for immediate execution and orders of sizes µ to the dark pool. Darkpools executions are governed by the Poisson process N with rate θ . The process η describesthe instantaneous market impact; it often describes the so-called market depth. The process γ describes adverse selection costs associated with dark pool trading while λ usually describesmarket risk, e.g. the volatility of a portfolio holding.Starting with the work of Almgren and Chriss [1] portfolio liquidation problems have receivedconsiderable attention in the financial mathematics and stochastic control literature in recentyears; see [3, 9, 10, 12, 15–17, 20, 21, 23] and references therein for details. From a mathematicalperspective one of their main characteristics is the singular terminal condition of the valuefunction induced by the terminal state constraint (1.3). The constraint translates into a singularterminal state constraint on the associated HJB equation and causes significant difficulties inproving the existence and, even more so, the uniqueness of solutions to that equation.Under a continuity and polynomial growth condition on the cost coefficients η, λ, γ it has beenshown in [10] that the HJB equation admits at most one continuous viscosity solution of poly-nomial growth. The proof used a comparison principle for continuous viscosity solutions toPDEs with singular terminal value. Since the comparison principle applies only to continuousfunctions, it can not be used to establish the existence of a viscosity solution. Instead, it wasshown in [10] that a (unique) classical solution to the HJB equation exists under strong bound-edness and regularity assumptions on the model parameters. In this paper we prove a novelcomparison principle for semi-continuous viscosity solutions for PDEs with singular terminalvalue from which we deduce the existence of a continuous viscosity solution to our HJB equationusing Perron’s method. The existence of a continuous viscosity solution is enough to carry outthe verification arguments and to give a representation of the optimal control in feedback form.There are several papers that provide verification arguments without assuming continuity ofviscosity solutions. For instance, a utility optimization problem with delays and state constraintshas been considered in [8]. The authors solved in the viscosity sense the associated HJB equationunder the assumption that the utility function satisfies the Inada condition, a condition that isnot satisfied in our model. In [6], the authors studied the general verification result for stochasticimpulse control problems, assuming that a comparison principle for discontinuous viscositysolutions of the HJB equation holds. This is a very strong hypothesis that can be avoided inour case. The linear-quadratic structure of our control problem allows us to characterize thevalue in terms of a PDE without jumps, and the verification argument can be given in terms of2he associated FBSDE after the existence of the viscosity solution has been established.To the best of our knowledge, existence of continuous solutions to HJB equations associated withcontrol problems of the form (1.1)-(1.3) has so far only been established under L ∞ assumptionson the model parameters. The existence of unique continuous viscosity solution was establishedwhen η is a constant and λ is of polynomial growth in [4]. Existence and uniqueness of solutionsin suitable Sobolev spaces for bounded stochastic cost and diffusion coefficients was provedin [9, 13]; classical solutions were considered in [10].The restriction to constant market impact terms and/or bounded impact functions and diffusioncoefficients is unsatisfactory. In a portfolio liquidation framework, it is natural to choose atwo-dimensional driving factor where the first component is a mean-reverting process, e.g. anOrnstein-Uhlenbeck process that describes a liquidity index and the second component is ageometric Brownian motion with zero drift that describes the dynamics of the unaffected stockprice process. It is then natural to chose η to be a strictly monotone unbounded function of theliquidity index and λ to be the square of the geometric Brownian motion so that market risk ismeasured by the volatility of the portfolio value. Our results apply to such setting.The papers [3, 17, 21] allow for unbounded coefficients. They characterize the value function asthe minimal solution to some BSDE with singular terminal value. BSDEs with singular terminalvalue were first studied in [20]. In [21] the same author showed that the minimal solution toa certain singular BSDE yields a probabilistic representations of a (possibly discontinuous)viscosity solution to the associated PDE. Our comparison result yields sufficient conditionsfor this minimal viscosity solution to be the unique (and hence continuous) solution. Thiscomplements the analysis is [3, 17]. The existence (and uniqueness) of minimal solutions toBSDEs with singular terminal values for more general drivers has recently been establishedin [11] under (suitable regularity and) boundedness assumptions on the model parameters. Theframework in [23] allows for unbounded coefficients but requires strong a priori estimates on themarket impact term that are not satisfied in our main example. Complementing the analysisin [23] our results show when value function derived in terms of Dawson-Watson superprocessestherein solves the HJB equation in the viscosity sense.The remainder of this paper is organized as follows. In Section 2, we summarize our mainresults. The existence of viscosity solution is proved in Section 3.1; the verification argumentis carried out in Section 3.2. Section 4 is devoted to an extension of our uniqueness result to anon-Markovian model with unbounded coefficients. Notation.
We denote by C b ( R d ) the set of all functions φ : R d → R which are continuous andbounded on R d . For a given m ≥ , we define C m ( R d ) to be set of continuous functions thathave at most polynomial growth of order m , i.e. the set of functions φ ∈ C ( R d ) such that ψ := φ ( y )1 + | y | m ∈ C b ( R d ) . This space is a Banach space when endowed with the norm k φ k m := sup y ∈ R d | φ ( y ) | | y | m . Let I be a compact subset of R . A function φ belongs to U SC m ( I × R d ) (or LSC m ( I × R d )) ifit has at most polynomial growth of order m in the second variable uniformly with respect to3 ∈ I and is upper (lower) semi-continuous on I × R d . Whenever the notation T − appears inthe definition of a function space we mean the set of all functions whose restrictions satisfy therespective property when T − is replaced by any s < T , e.g., C m ([0 , T − ] × R d ) = { u : [0 , T ) × R d → R : u | [0 ,s ] × R d ∈ C m ([0 , s ] × R d ) for all s ∈ [0 , T ) } . Throughout, all equations and inequalities are to be understood in the a.s. sense. We adopt theconvention that C is a constant that may vary from line to line. For each initial state ( t, y, x ) ∈ [0 , T ) × R d × R we define by V ( t, y, x ) := inf ( ξ,µ ) ∈A ( t,x ) E (cid:20)Z Tt η ( Y t,ys ) | ξ s | + θγ ( Y t,ys ) | µ s | + λ ( Y t,ys ) | X ξ,µs | ds (cid:21) (2.1)the value function of the control problem (1.1) subject to the state dynamics dY t,ys = b ( Y t,ys ) ds + σ ( Y t,ys ) dW s , Y t,yt = ydX ξ,µs = − ξ s ds − µ s dN s , X t = x. (2.2)Here, ξ = ( ξ s ) s ∈ [ t,T ] describes the rates at which the agent trades in the primary market, while µ = ( µ s ) s ∈ [ t,T ] describes the orders submitted to the dark pool. The infimum is taken overthe set A ( t, x ) of all admissible controls , that is, over all pairs of controls ( ξ, µ ) such that ξ isprogressively measurable, such that µ is predictable and such that the resulting state process X ξ,µs = x − Z st ξ r dr − Z st µ r dN r , t ≤ s ≤ T, satisfies the terminal state constraint X ξ,µT = 0 . (2.3)The expected costs associated with an admissible liquidation strategy ( ξ, µ ) are given by J ( t, y, x ; ξ, µ ) := E (cid:20)Z Tt c ( Y t,ys , X ξ,µs , ξ s , µ s ) ds (cid:21) , where the running cost function c ( y, x, ξ, µ ) is given by c ( y, x, ξ, µ ) := η ( y ) | ξ | + θγ ( y ) | µ | + λ ( y ) | x | . Remark . We assume that the cost function is quadratic in the controls and the state variable.A generalization to general powers p > − ∂ t V ( t, y, x ) − L V ( t, y, x ) − inf ξ,µ ∈ R H ( t, y, x, ξ, µ, V ) = 0 , ( t, y, x ) ∈ [0 , T ) × R d × R , (2.4) We show later that we restrict ourselves to monotone portfolio processes so we could just as well assume that µ is bounded. L := 12 tr( σσ ∗ D y ) + h b, D y i denotes the infinitesimal generator of the factor process and the Hamiltonian H is given by H ( t, y, x, ξ, µ, V ) := − ξ∂ x V ( t, y, x ) + θ ( V ( t, y, x − µ ) − V ( t, y, x )) + c ( y, x, ξ, µ ) . The quadratic cost function suggests an ansatz of the form V ( t, y, x ) = v ( t, y ) | x | . The followingresult confirms this intuition. Its proof can be found in [10, Section 2.2]. Lemma 2.2.
A nonnegative function v : [0 , T ) × R d → [0 , ∞ ) is a (sub/super) solution to thePDE − ∂ t v ( t, y ) − L v ( t, y ) − F ( y, v ( t, y )) = 0 , (2.5) where F ( y, v ) := λ ( y ) − | v | η ( y ) + θγ ( y ) vγ ( y ) + | v | − θv, (2.6) if and only if v ( t, y ) | x | is a (sub/super) solution to the HJB equation (2.4) . In this case theinfimum in (2.4) is attained at ξ ∗ ( t, y, x ) = v ( t, y ) η ( y ) x and µ ∗ ( t, y, x ) = v ( t, y ) γ ( y ) + v ( t, y ) x (2.7) and H ( t, y, x, ξ ∗ ( t, y, x ) , µ ∗ ( t, y, x ) , v ( · , · ) | · | ) = F ( y, v ( t, y )) | x | . (2.8) In order to prove the existence of a unique non-negative continuous viscosity solution of poly-nomial growth to our HJB equation we assume throughout that the factor process Y t,ys = y + Z st b ( Y t,yr ) dr + Z st σ ( Y t,yr ) dW r , t ≤ s ≤ T. (2.9)satisfies the following condition. Assumption 2.3.
The coefficients b : R d → R d and σ : R d → R d × ˜ d are Lipschitz continuous.The preceding assumption guarantees that the SDE (2.9) has a unique strong solution ( Y t,ys ) s ∈ [ t,T ] for every initial state ( t, y ) ∈ [0 , T ] × R d and that the mapping ( s, t, y ) Y t,ys is a.s. continuous.We repeatedly use the following well known estimates; cf. [18, Corollary 2.5.12]. For all m ≥ , there exists a constant C > y ∈ R d , ≤ t ≤ s ≤ T, E sup t ≤ s ≤ T | Y t,ys | m ≤ C (1 + | y | m ) . (2.10)Furthermore, we assume that the cost coefficients are continuous and of polynomial growth andthat η is twice continuously differentiable and satisfies a mild boundedness condition. Assumption 2.4.
The cost coefficients satisfy the following conditions:5i) The coefficients η, γ, λ, /η : R d → [0 , ∞ ) are continuous and of polynomial growth.(ii) η ∈ C and k L ηη k is bounded. Remark . The preceding assumption is satisfied if, for instance Y is a geometric Brownianmotion or an Ornstein-Uhlenbeck (OU) process and η ( y ) = 1 + | y | . In both cases, condition (2.13) in [23] is violated. Our assumptions are also weaker than thosein [10]. For instance, OU processes do not generate analytic semigroups, they do not satisfy theassumptions therein.
Before stating our first main result, we recall the notion of viscosity solutions for parabolicequations that will be used in this paper. The following definition can be found in [7, Section8].
Definition 2.6.
For semicontinuous functions v : [0 , T ) × R d → R we use the following solutionconcepts for the parabolic PDE: − ∂ t v ( t, y ) − G ( t, y, v ( t, y ) , D y v ( t, y ) , D y v ( t, y )) = 0 , (2.11)where G : [0 , T ) × R d × R × R d × S d → R and S d denotes the set of symmetric d × d matrices.(i) v ∈ U SC m ([0 , T − ] × R d ) is a (strict) viscosity subsolution if for every ϕ ∈ C , loc ([0 , T ) × R d )such that ϕ ≥ v and ϕ ( t, y ) = v ( t, y ) at a point ( t, y ) ∈ [0 , T ) × R d it holds − ∂ t ϕ ( t, y ) − G ( t, y, v ( t, y ) , D y ϕ ( t, y ) , D y ϕ ( t, y ))( < ) ≤ . (ii) v ∈ LSC m ([0 , T − ] × R d ) is a (strict) viscosity supersolution if for every ϕ ∈ C , loc ([0 , T ) × R d )such that ϕ ≤ v and ϕ ( t, y ) = v ( t, y ) at a point ( t, y ) ∈ [0 , T ) × R d it holds − ∂ t ϕ ( t, y ) − G ( t, y, v ( t, y ) , D y ϕ ( t, y ) , D y ϕ ( t, y ))( > ) ≥ . (iii) v is a viscosity solution if v is both viscosity sub- and supersolution.We are now ready to state the main result of this paper. Its proof is given in Section 3 below. Theorem 2.7.
Under Assumptions 2.3, 2.4, the singular terminal value problem ( − ∂ t v ( t, y ) − L v ( t, y ) − F ( y, v ( t, y )) = 0 , ( t, y ) ∈ [0 , T ) × R d , lim t → T v ( t, y ) = + ∞ locally uniformly on R d , (2.12) with the nonlinearity F given in (2.6) admits a unique nonnegative viscosity solution in C m ([0 , T − ] × R d ) for some m ≥ . Proposition 2.8.
Under Assumptions 2.3 ,2.4, let v be the unique nonnegative viscosity solu-tion to the singular terminal value problem (2.12) . Then, the value function (2.1) is given by V ( t, y, x ) = v ( t, y ) | x | , and the optimal control ( ξ ∗ , µ ∗ ) is given in feedback form by ξ ∗ s = v ( s, Y t,ys ) η ( Y t,ys ) X ∗ s and µ ∗ s = v ( s, Y t,ys ) γ ( Y t,ys ) + v ( s, Y t,ys ) X ∗ s − . (2.13) In particular, the resulting optimal portfolio process ( X ∗ s ) s ∈ [ t,T ] is given by X ∗ s = x exp − Z st v ( r, Y t,yr ) η ( Y t,yr ) dr ! ∆ N r =0 Y t 11 + | Y | , if Y ≥ , γ ( Y ) = 1 , and λ ( Y ) = σ | Y | . The process Y specifies a liquidity indicator that fluctuates around a stationary level (nor-malized to zero) with the market impact increasing when below average liquidity is availableand decreasing when above average liquidity is available. Instantaneous market risk, on theother hand is captured by the volatility of the portfolio value assuming that asset prices followa geometric Brownian motion. For the above choice of model parameters all assumptions onthe cost and diffusion coefficients are satisfied. Hence, there exists a unique optimal liquidationstrategy. Remark . To the best of our knowledge, numerical methods for simulating solutions to generalPDEs with singular terminal values are still to be developed. At least two problems arise whensimulating solutions to HJB equations with singular terminal state constraint. The most obviousproblem is the singular terminal condition. This problem can potentially be overcome by notingthat the function w ( t, y ) := ( T − t ) v ( t, y ) , ( t, y ) ∈ [0 , T ) × R satisfies the following PDE with finite terminal value, yet singular driver (see [10,11] and Section3 for details) − ∂ t w ( t, y ) − L w ( t, y ) − w ( t, y ) T − t − ( T − t ) F ( y, w ( t, y ) T − t ) = 0 , ( t, y ) ∈ [0 , T ) × R d , lim t → T w ( t, y ) = η ( y ) on R d . σ = 0), w ( t, y ) ≤ Cη ( y ) , ( t, y ) ∈ [0 , T ) × R for some C > η ( y ) → | y | → ∞ . Ingeneral we can not expect the above inequality to be an equality, though, not even asymptoticallywhen | y | → ∞ . If we choose σ = 0 and the dynamics dY t = − tanh( Y t − Y t ) dt + dW t for the liquidity index, then the index is mean-reverting to the levels ± 1, the “regimes of averageliquidity”. Choosing η ( y ) = y all our assumptions on the model parameters are satisfied.In this case we may regard the interval ( − , +1) as the low and the set [ − , c as the highliquidity regime. Since w ( t, y ) → | y | → ∞ , the boundary problem can be dealt with. In this section, we prove Theorem 2.7. In a first step, we establish a comparison principle forsemicontinuous viscosity solutions to (2.12). In view of the singular terminal state constraint wecan not follow the usual approach of showing that if a l.s.c. supersolution dominates an u.s.c.subsolution at the boundary, then it also dominates the subsolution on the entire domain.Instead, we prove that if some form of asymptotic dominance holds at the terminal time, thendominance holds near the terminal time.In a second step, we construct smooth sub- and supersolutions to (2.12) that satisfy the requiredasymptotic dominance condition. Subsequently, we apply Perron’s method to establish an u.s.c.subsolution and a l.s.c. supersolution that are bounded from above/below by the smooth solu-tions. From this, we infer that the semi-continuous solutions can be applied to the comparisonprinciple, which then implies the existence of the desired continuous viscosity solution. Throughout this section, we fix δ ∈ (0 , T ] and for some m ≥ , let u ∈ LSC m ([ T − δ, T − ] × R d )and u ∈ U SC m ([ T − δ, T − ] × R d ) be a viscosity super- and a viscosity subsolution to (2.12).8 roposition 3.1. Under Assumptions 2.3, 2.4, if, uniformly on R d , lim sup t → T u ( t, y )( T − t ) − η ( y )1 + | y | m ≤ ≤ lim inf t → T u ( t, y )( T − t ) − η ( y )1 + | y | m , (3.1) and u ( t, y )( T − t ) , u ( t, y )( T − t ) ≥ η ( y ) , t ∈ [ T − δ, T ) , (3.2) then u ≤ u on [ T − δ, T ) × R d . Assumptions (3.1), (3.2) are uncommon in the viscosity literature. However, we shall only usethe comparison result to establish the existence of a solution, not the uniqueness. As a result, weonly need to guarantee that the semi-continuous solutions established through Perron’s methodsatisfy both assumptions.The proof of the comparison principle is based on three auxiliary results. The first lemma istaken from [10, Lemma A.2]. It is a modification of [5, Lemma 3.7]. Lemma 3.2. The difference w := u − u ∈ U SC m ([ T − δ, T − ] × R d ) is a viscosity subsolution to − ∂ t w ( t, y ) − L w ( t, y ) − l ( t, y ) w ( t, y ) = 0 , ( t, y ) ∈ [ T − δ, T ) × R d , (3.3) where l ( t, y ) := F ( y, u ( t, y )) − F ( y, u ( t, y )) u ( t, y ) − u ( t, y ) I u ( t,y ) = u ( t,y ) . The next lemma constructs a smooth strict supersolution to (3.3) of polynomial growth. Lemma 3.3. For every n ∈ N , there exists K n large enough such that χ ( t, y ) := e K n ( T − t ) (1 + | y | ) n T − t satisfies − ∂ t χ ( t, y ) − L χ ( t, y ) + χ ( t, y ) T − t > , ( t, y ) ∈ [ T − δ, T ) × R d . Proof. Direct calculations verify that h ( t, y ) := e K n ( T − t ) (1+ | y | ) n satisfies − ∂ t h ( t, y ) −L h ( t, y ) > T − δ, T ) × R d when K n is chosen sufficiently large; see also [2, Proposition 5]. Here it isused that b and σ are Lipschitz and thus are of linear growth. Hence, − ∂ t χ ( t, y ) − L χ ( t, y ) + χ ( t, y ) T − t = − ∂ t h ( t, y ) − L h ( t, y ) T − t > . The following lemma is key to the proof of the comparison principle. Lemma 3.4. If n ∈ N in Lemma 3.3 is chosen large enough, then independent of α > , thefunction Φ α ( t, y ) := w ( t, y ) − αχ ( t, y ) is either nonpositive or attains its supremum at some point ( t α , y α ) in [ T − δ, T ) × R d . roof. Suppose that the supremum of Φ α on [ T − δ, T ) × R d is positive and denote by ( t k , y k )a sequence in [ T − δ, T ) × R d approaching the supremum point. The representationΦ α ( t, y ) = h u ( t,y )( T − t ) − η ( y )1+ | y | m − u ( t,y )( T − t ) − η ( y )1+ | y | m i (1 + | y | m ) − αe K n ( T − t ) (1 + | y | ) n T − t , along with condition (3.1) shows that for any n > m, lim sup t → T Φ α ( t, y ) = −∞ , uniformly on R d . Hence lim k t k < T. Furthermore, w ∈ U SC m ([ T − δ, T − ] × R d ) is bounded by a function ofpolynomial growth uniformly away from the terminal time. Choosing n large enough this showsthat lim k | y k | < ∞ . As a result, the supremum is attained at some point ( t α , y α ) because Φ α isupper semicontinuous. This proves the assertion.We are now ready to prove the comparison principle. Proof of Proposition 3.1. Let us fix α > . By letting α → α is nonpositive.In view of Lemma 3.4, we just need to consider the case where there exists a point ( t α , y α ) ∈ [ T − δ, T ) × R d such that w ( t, y ) − αχ ( t, y ) ≤ w ( t α , y α ) − αχ ( t α , y α ) , ( t, y ) ∈ [ T − δ, T ) × R d . This inequality can be interpreted as w − ψ α having a global maximum at ( t α , y α ), where ψ α := αχ ( t, y ) + ( w − αχ )( t α , y α ) . Since ψ α is smooth and w is a viscosity subsolution to (3.3), − ∂ t ψ α ( t α , y α ) − L ψ α ( t α , y α ) − l ( t α , y α ) w ( t α , y α ) ≤ . By the mean value theorem along with the monotonicity of ∂ u F , condition (3.2) and the factthat ∂ v F ( y, v ) ≤ − vη ( y ) we get that l ( t, y ) = F ( y, u ( t, y )) − F ( y, u ( t, y )) u ( t, y ) − u ( t, y ) I u ( t,y ) = u ( t,y ) ≤ ∂ v F ( y, η ( y )2( T − t ) ) ≤ − T − t . (3.4)Thus, Lemma 3.3 implies0 ≥ − ∂ t ψ α ( t α , y α ) − L ψ α ( t α , y α ) − l ( t α , y α ) w ( t α , y α )= α [ − ∂ t χ ( t α , y α ) − L χ ( t α , y α ) − l ( t α , y α ) w ( t α , y α )] > − α χ ( t α , y α ) T − t α − l ( t α , y α ) w ( t α , y α ) ≥ αl ( t α , y α ) χ ( t α , y α ) − l ( t α , y α ) w ( t α , y α )= − l ( t α , y α )Φ α ( t α , y α ) . (3.5)Since l ≤ , we can conclude that Φ α ( t α , y α ) ≤ , thus Φ α ≤ . .1.2 Existence via Perron’s method Armed with our comparison principle, the existence of a viscosity solution to our HJB equationcan be established using Perron’s method as soon as suitable sub- and supersolutions can beidentified. In view of Assumption 2.4, η, λ ∈ C m ( R d ) for some m ≥ k L ηη k is well-definedand finite. Hence δ := 1 / k L ηη k ∧ T > . (3.6)By a direct computation, we can find a constant K ′ large enough such that the function:ˆ h ( t, y ) := e K ′ ( T − t ) (1 + | y | ) m/ satisfying − ∂ t ˆ h ( t, y ) − L ˆ h ( t, y ) − λ ( y ) ≥ . Let us then defineˇ v ( t, y ) := η ( y ) − η ( y ) k L ηη k ( T − t ) e θ ( T − t ) ( T − t ) and ˆ v ( t, y ) := η ( y ) + η ( y ) k L ηη k ( T − t )( T − t ) + ˆ h ( t, y ) . Proposition 3.5. Under Assumption 2.3, 2.4 the functions ˇ v, ˆ v are a nonnegative classicalsub- and supersolution to (2.12) on [ T − δ, T ) × R d , respectively.Proof. To verify the supersolution property of ˆ v , we first verify that − ∂ t ˆ v ( t, y ) − L ˆ v ( t, y )= − η ( y ) + L η ( y )( T − t ) + L η ( y ) k L ηη k ( T − t ) ( T − t ) − ∂ t ˆ h ( t, y ) − L ˆ h ( t, y ) (3.7)Recalling the definition (2.6) of F , we have since ˆ v ≥ − F ( y, ˆ v ( t, y )) ≥ − λ ( y ) + ˆ v ( t, y ) η ( y ) . Next, we apply the inequality ( u + v + w ) ≥ u + 2 uv for u, v, w ≥ v ( t, y ) toobtain − F ( y, ˆ v ( t, y )) ≥ − λ ( y ) + η ( y ) + 2 η ( y ) k L ηη k ( T − t ) η ( y )( T − t ) . (3.8)Adding (3.7) and (3.8) yields − ∂ t ˆ v ( t, y ) − L ˆ v ( t, y ) − F ( y, ˆ v ( t, y )) ≥ η ( y ) k L ηη k − L η ( y ) − L η ( y ) k L ηη k ( T − t )( T − t ) − ∂ t ˆ h ( t, y ) − L ˆ h ( t, y ) − λ ( y ) . The definition of δ yields 1 ≥ k L ηη k ( T − t ) for t ∈ [ T − δ, T ) and so,2 η ( y ) k L ηη k − L η ( y ) − L η ( y ) k L ηη k ( T − t ) ≥ η ( y ) k L ηη k · (cid:20) k L ηη k ( T − t ) (cid:21) − L η ( y ) − L η ( y ) k L ηη k ( T − t )= (cid:20) k L ηη k ( T − t ) (cid:21) · (cid:20) η ( y ) k L ηη k − L η ( y ) (cid:21) ≥ . We use the convention 1 / ∞ . 11e conclude that − ∂ t ˆ v ( t, y ) − L ˆ v ( t, y ) − F ( y, ˆ v ( t, y )) ≥ . Next, we verify the subsolution property of ˇ v . By direct computation, − ∂ t ˇ v ( t, y ) − L ˇ v ( t, y ) = − η ( y ) + L η ( y )( T − t ) − L η ( y ) k L ηη k ( T − t ) e θ ( T − t ) ( T − t ) − θ ˇ v ( t, y ) . (3.9)On the other hand, since λ, γ ≥ 0, and ˇ v ≥ T − δ, T ) × R d , − F ( y, ˇ v ( t, y )) ≤ ˇ v ( t, y ) η ( y ) + θ ˇ v ( t, y ) . We estimate ˇ v ( t, y ) using the inequality ( u − v ) ≤ u − uv for u ≥ v ≥ − F ( y, ˇ v ( t, y )) ≤ η ( y ) − η ( y ) k L ηη k ( T − t ) e θ ( T − t ) ( T − t ) + θ ˇ v ( t, y ) . (3.10)Since e − θ ( T − t ) ≤ e − θ ( T − t ) , adding (3.9) and (3.10) yields − ∂ t ˇ v ( t, y ) − L ˇ v ( t, y ) − F ( t, ˇ v ( t, y )) ≤ − η ( y ) k L ηη k + L η ( y ) − L η ( y ) k L ηη k ( T − t ) e θ ( T − t ) ( T − t ) . Using again that 1 ≥ k L ηη k ( T − t ) we obtain, η ( y ) k L ηη k + L η ( y ) − L η ( y ) k L ηη k ( T − t ) ≥ η ( y ) k L ηη k · (cid:20) − k L ηη k ( T − t ) (cid:21) + L η ( y ) − L η ( y ) k L ηη k ( T − t )= (cid:20) − k L ηη k ( T − t ) (cid:21) · (cid:20) η ( y ) k L ηη k + L η ( y ) (cid:21) ≥ . Thus, − ∂ t ˇ v ( t, y ) − L ˇ v ( t, y ) − F ( t, ˇ v ( t, y )) ≤ . Proof of Theorem 2.7. From the definition of ˇ v, ˆ v we have( T − t )ˇ v ( t, y ) = η ( y ) + η ( y ) O ( T − t ) uniformly in y as t → T .( T − t )ˆ v ( t, y ) = η ( y ) + (1 + | y | m ) O ( T − t ) uniformly in y as t → T . (3.11)Then for ε = , there exists δ ∈ (0 , δ ] such that for all t ∈ [ T − δ , T ) , ˇ v ( t, y )( T − t ) > η ( y ) − η ( y ) = 12 η ( y ) uniformly on R d . Since η ∈ C m ( R d ) , we obtain from (3.11) thatlim t → T ˇ v ( t, y )( T − t ) − η ( y )1 + | y | m = lim t → T ˆ v ( t, y )( T − t ) − η ( y )1 + | y | m = 0 , uniformly on R d . (3.12)12n order to apply Perron’s method, we set S = { u | u is a subsolution of (2.12) on [ T − δ , T ) × R d and u ≤ ˆ v } . From Proposition 3.5 we know that ˇ v ∈ S , so S is non-empty. Thus, the function v ( t, y ) = sup { u ( t, y ) : u ∈ S} is well-defined. Classical arguments show that the upper semi-continuous envelope v ∗ is aviscosity subsolution to (2.12). From [24, Lemma A.2], the lower semi-continuous envelope v ∗ of v is also a viscosity supersolution to (2.12). Since ˇ v ≤ v ∗ ≤ v ∗ ≤ ˆ v, we have that for all t ∈ [ T − δ , T ) , v ∗ ( t, y )( T − t ) , v ∗ ( t, y )( T − t ) ≥ η ( y ) , uniformly on R d . and ˇ v ( t, y )( T − t ) − η ( y )1 + | y | m ≤ v ∗ ( t, y )( T − t ) − η ( y )1 + | y | m ≤ v ∗ ( t, y )( T − t ) − η ( y )1 + | y | m ≤ ˆ v ( t, y )( T − t ) − η ( y )1 + | y | m . Hence, it follows from (3.12) that,lim t → T v ∗ ( t, y )( T − t ) − η ( y )1 + | y | m = lim t → T v ∗ ( t, y )( T − t ) − η ( y )1 + | y | m = 0 , uniformly on R d . (3.13)From our comparison principle [Proposition 3.1] we conclude that v ∗ ≤ v ∗ on [ T − δ , T ) × R d , which shows that v is the desired viscosity solution to (2.5) that belongs to C m ([ T − δ , T − ] × R d ).By [2, Remark 6], there exists a unique viscosity solution v ∈ C m ([0 , T − δ ] × R d ) to (2.5) whenimposed at t = T − δ with a terminal value in C m ( R d ) . Hence from the comparison principlefor continuous viscosity solutions [10, Lemma 3.1], we get a unique global viscosity solution v ∈ C m ([0 , T − ] × R d ) . Remark . If all the coefficients of the generator F and the SDE (2.9) are bounded, then onecan show that twice differentiability of η is not needed; only a uniform continuity is requiredto choose continuous solutions which satisfying the conditions (3.1) and (3.2). Thus a uniqueviscosity solution can be obtained by the same argument above. This section is devoted to the verification argument. Throughout, v ∈ C m ([0 , T − ] × R d ) denotesthe unique nonnegative viscosity solution to the singular terminal value problem (2.12). We willprove that the viscosity solution is indeed the value function to our stochastic control problem.In a first step we are now going to show that the feedback control given in (2.13) is indeedadmissible. The standard Perron method of finding viscosity solutions for elliptic PDEs can be found in [7]. We referto [24, Appendix A] for the proof of this method for parabolic equations. emma 3.7. The pair of feedback controls ( ξ ∗ , µ ∗ ) given by (2.13) is admissible.Proof. Given the feedback form in (2.13), one can easily obtain that the pair of controls ( ξ ∗ , µ ∗ )is admissible and the resulting portfolio process ( X ∗ s ) s ∈ [ t,T ] is monotone. It remains to verifythe liquidation constraint. Since ˇ v ≤ v ≤ ˆ v on [ T − δ, T ) where δ is defined in (3.6), it holds forany r ∈ [ T − δ, T ) that,1 − k L ηη k ( T − r ) e θ ( T − r ) ( T − r ) η ( Y t,yr ) ≤ v ( r, Y t,yr ) ≤ k L ηη k ( T − r ) T − r η ( Y t,yr ) + ˆ h ( r, Y t,yr ) . For s ∈ [ T − δ, T ) , | X ∗ s | ≤ | x | exp − Z st v ( r, Y t,yr ) η ( Y t,yr ) dr ! ≤ | x | exp − Z sT − δ v ( r, Y t,yr ) η ( Y t,yr ) dr ! ≤ | x | exp − Z sT − δ − k L ηη k ( T − r ) e θ ( T − r ) ( T − r ) dr ! ≤ | x | exp Z sT − δ e θ ( T − r ) − [1 − k L ηη k ( T − r )] e θ ( T − r ) ( T − r ) dr ! exp (cid:18) − Z sT − δ T − r dr (cid:19) ≤ | x | exp Z sT − δ " e θ ( T − r ) − e θ ( T − r ) ( T − r ) + k L ηη k e θ ( T − r ) dr ! · T − sδ ≤ C | x | T − sδ . (3.14)The last inequality holds because lim r → T e θ ( T − r ) − e θ ( T − r ) ( T − r ) = θ. As a result, X ∗ T − = 0 and hence X ∗ T = 0 . It has been shown in [10, Lemma 5.2] that we may w.l.o.g restrict ourselves to admissible controlsthat result in a monotone portfolio process. We denote by ¯ A ( t, x ) the set of all admissiblecontrols under which the portfolio process is monotone.Next, we give a probabilistic representation of the viscosity solution to (2.12). In [21], theauthor showed that the possibly discontinuous minimal solution of a certain backward stochasticdifferential equation with singular terminal condition gives a probabilistic representation of theminimal viscosity solution of an associated partial differential equation; continuity of the solutionwas not established. However, continuity is necessary to carry out the verification argument.We obtain a solution to the corresponding FBSDE in a different way since the existence of the(continuous) viscosity solution has already been proved. Proposition 3.8. Under Assumptions 2.3, 2.4, for any fixed ǫ ∈ (0 , T ) and ( t, y ) ∈ [ ǫ, T ) × R d ,there exists a pair of processes ( U t,y , Z t,y ) ∈ S F ( t, T ; R ) × L F ( t, T ; R × ˜ d ) satisfying that U t,yt = v ( t − ǫ, y ) and for any ǫ ≤ t ≤ r ≤ s ≤ T,U t,yr = U t,ys + Z sr F ( Y t,yρ , U t,yρ ) dρ − Z sr Z t,yρ dW ρ . roof. We consider the forward-backward system dY s = b ( Y s ) ds + σ ( Y s ) dW s , s ∈ [ t, T ] ,dU s = − f ( s, Y s ) ds + Z s dW s , s ∈ [ t, T ] ,Y t = y, U T = v ( T − ǫ, Y T ) , (3.15)and the corresponding PDE ( − w t ( t, y ) − L w ( t, y ) − f ( t, y ) = 0 , ( t, y ) ∈ [ ǫ, T ) × R d ,w ( T, y ) = v ( T − ǫ, y ) , y ∈ R d (3.16)where f ( t, y ) := F ( y, v ( t − ǫ, y )) and F is defined in (2.6). Recalling the polynomial growthcondition on the cost coefficients in Assumption 2.4 and the polynomial growth property of thesolution v established in Theorem 2.7, we know that f ∈ C m ′ ([ ǫ, T ] × R d ), for some m ′ ≥ m. Together with Assumption 2.3 and the fact that v ( T − ǫ, · ) ∈ C m ( R d ), we conclude from [14,Theorem 2.1] that the system admits a unique solution( Y t,y , U t,y , Z t,y ) ∈ S F ( t, T ; R d ) × S F ( t, T ; R ) × L F ( t, T ; R × ˜ d ) . Let w ( t, y ) := U t,yt . By the Feynman-Kac formula [19, Theorem 3.2], w is the unique viscositysolution of (3.16) with driver f . Due to the time-homogeneity of the PDE in (2.12), viscositysolutions stay viscosity solutions when shifted in time. Let ˜ v ( t, y ) := v ( t − ǫ, y ) on [ ǫ, T ].By the definition of f, we see that ˜ v is also a viscosity solution of (3.16) with driver f on[ ǫ, T ]. Hence it follows that w = ˜ v . By the Markov property, we have for any r ∈ [ t, T ] that0 ≤ U t,yr = v ( r − ǫ, Y t,yr ) . Thus U t,y is also a solution to the following FBSDE: dY s = b ( Y s ) ds + σ ( Y s ) dW s , s ∈ [ t, T ] ,dU s = − F ( Y s , U s ) ds + Z s dW s , s ∈ [ t, T ] ,Y t = y, U T = v ( T − ǫ, Y T ) . For any ǫ ∈ (0 , T ) , we can restrict our interval on [ t, T − ǫ ] and repeat the arguments abovewithout shifting in time. This yields a solution ( ˜ U t,y , ˜ Z t,y ) ∈ S F ( t, T − ǫ ; R ) × L F ( t, T − ǫ ; R × ˜ d )satisfying that ˜ U t,yt = v ( t, y ) and for any 0 ≤ t ≤ r ≤ s < T − ǫ, dY s = b ( Y s ) ds + σ ( Y s ) dW s , s ∈ [ t, T − ǫ ] ,d ˜ U s = − F ( Y s , ˜ U s ) ds + ˜ Z s dW s , s ∈ [ t, T − ǫ ] ,Y t = y, ˜ U T − ǫ = v ( T − ǫ, Y T − ǫ ) . Since ǫ is arbitrary, a global solution on [0 , T ) can be obtained. Corollary 3.9. Under Assumptions 2.3, 2.4, there exists processes ( ˜ U t,y , ˜ Z t,y ) ∈ S F ( t, T − ; R ) × L F ( t, T − ; R × ˜ d ) satisfying that ˜ U t,yt = v ( t, y ) and for any ≤ t ≤ r ≤ s < T, ˜ U t,yr = ˜ U t,ys + Z sr F ( Y t,yρ , ˜ U t,yρ ) dρ − Z sr ˜ Z t,yρ dW ρ . (3.17)15he following lemma is key to the verification argument. Lemma 3.10. Fix ǫ ∈ (0 , T ) and ( t, y ) ∈ [ ǫ, T ) × R d . For every ( ξ, µ ) ∈ ¯ A ( t, x ) and s ∈ [ t, T ) , v ( t − ǫ, y ) | x | ≤ E h v ( s − ǫ, Y t,ys ) | X ξ,µs | i + E (cid:20)Z st c ( Y t,yr , X ξ,µr , ξ r , µ r ) dr (cid:21) . Proof. By Proposition 3.8, we know that ( U t,y , Z t,y ) solves the following BSDE: U t,yt = U t,ys + Z st F ( Y t,yr , U t,yr ) dr − Z st Z t,yr dW r . This allows us to apply to U t,ys | X ξ,µs | the classical integration by parts formula for semimartin-gales in order to obtain U t,yt | x | = U t,ys | X ξ,µs | + Z st (cid:8) F ( Y t,yr , U t,yr ) | X ξ,µr | + 2 ξ r U t,yr sgn( X ξ,µr ) | X ξ,µr | − θU t,yr ( | X ξ,µr − µ r | − | X ξ,µr | ) (cid:9) dr − Z st σ ( Y t,yr ) Z t,yr | X ξ,µr | dW r − Z st U t,yr ( | X ξ,µr − − µ r | − | X ξ,µr − | ) d e N r , where e N r = N r − θr denotes the compensated Poisson process. Moreover, | X ξ,µ | ≤ | x | and | µ | ≤ | x | , due to the monotonicity of the portfolio process. Furthermore, Z st σ ( Y t,yr ) Z t,yr | X ξ,µr | dW r is a uniformly integrable martingale because2 E "(cid:18)Z st | σ ( Y t,yr ) | · | Z t,yr | | X ξ,µr | dr (cid:19) / ≤ E (cid:18) sup t ≤ r ≤ s | σ ( Y t,yr ) | + | x | Z st | Z t,yr | dr (cid:19) < ∞ . As a consequence, the above stochastic integrals are true martingales. Hence, recalling (2.8), U t,yt | x | = E h U t,ys | X ξ,µs | i + E (cid:20)Z st c ( Y t,yr , X ξ,µr , ξ r , µ r ) dr (cid:21) + E (cid:20)Z st (cid:8) F ( Y t,yr , U t,yr ) | X ξ,µr | − H ( r, Y t,yr , X ξ,µr , ξ r , µ r , U t,yr | X ξ,µr | ) (cid:9) dr (cid:21) ≤ E h U t,ys | X ξ,µs | i + E (cid:20)Z st c ( Y t,yr , X ξ,µr , ξ r , µ r ) dr (cid:21) . (3.18)Since U t,yt = v ( t − ǫ, y ) , U t,yr = v ( r − ǫ, Y t,yr ) , we have v ( t − ǫ, y ) | x | ≤ E h v ( s − ǫ, Y t,ys ) | X ξ,µs | i + E (cid:20)Z st c ( Y t,yr , X ξ,µr , ξ r , µ r ) dr (cid:21) . We are now ready to carry out the verification argument.16 roof of Proposition 2.8. Let ( ξ, µ ) ∈ ¯ A ( t, x ). By the liquidation constraint of X ξ,µ , letting s → T yields E h v ( s − ǫ, Y t,ys ) | X ξ,µs | i → . Hence, v ( t − ǫ, y ) | x | ≤ J ( t, y, x ; ξ, µ ) . Finally, by letting ǫ → 0, we conclude v ( t, y ) | x | ≤ J ( t, y, x ; ξ, µ ) . on [0 , T ) × R d by the continuity of v and the nonnegativity of J .Using similar arguments to the proof of Proposition 3.10 on the BSDE (3.17), we can obtainthat v ( t, y ) | x | ≤ E h v ( s, Y t,ys ) | X ξ,µs | i + E (cid:20)Z st c ( Y t,yr , X ξ,µr , ξ r , µ r ) dr (cid:21) . By Lemma 2.2 equality holds in the preceding inequality if ξ = ξ ∗ and µ = µ ∗ . Thus, v ( t, y ) | x | = E h v ( s, Y t,ys ) | X ξ ∗ ,µ ∗ s | i + E (cid:20)Z st c ( Y t,yr , X ξ ∗ ,µ ∗ r , ξ ∗ r , µ ∗ r ) dr (cid:21) ≥ E (cid:20)Z st c ( Y t,yr , X ξ ∗ ,µ ∗ r , ξ ∗ r , µ ∗ r ) dr (cid:21) from which we conclude that v ( t, y ) | x | ≥ J ( t, y, x ; ξ ∗ , µ ∗ ) . This shows that the strategy ( ξ ∗ , µ ∗ ) is indeed optimal. In this section we assume that the filtration is solely generated by the Brownian motion. Theexistence of a minimal nonnegative solution( Y , Z ) ∈ L F (Ω; C ([0 , T − ]; R + )) × L F (0 , T − ; R × ˜ d )to the BSDE − d Y t = (cid:26) λ t − |Y t | η t (cid:27) dt − Z t dW t , ≤ t < T ; lim t → T Y t = + ∞ (4.1)has been established in [3] under the assumption that η ∈ L F (0 , T ; R + ), η − ∈ L F (0 , T ; R + ), λ ∈ L F (0 , T − ; R + ), and E [ R T ( T − t ) λ t dt ] < ∞ . In this section we extend our uniqueness result to non-Markovian models and prove the existenceof a unique nonnegative solution under the following conditions; they correspond to those inthe Markovian setting. 17 ssumption 4.1. (i) The process η is a positive Itˆo diffusion satisfying that dη t = α t dt + β t dW t with ( α, β ) ∈ L F (0 , T ; R × R × ˜ d ).(ii) The processes η, η − ∈ L F (Ω; C ([0 , T ]; R )) and η − α ∈ L ∞F (0 , T ; R ).(iii) There exists a positive Itˆo diffusion h t such that dh t = α ′ t dt + β ′ t dW t with ( α ′ , β ′ ) ∈ L F (0 , T ; R × R × ˜ d ) and h − λ, h − α ′ ∈ L ∞F (0 , T ; R ). Proposition 4.2. Let Asssumption 4.1 hold. Set τ := 1 / k η − α k L ∞ ∧ T and ˜ K := k h − α ′ k L ∞ + k h − λ k L ∞ . For any solution ( Y , Z ) ∈ L F (Ω; C ([0 , T − ]; R + )) × L F (0 , T − ; R × ˜ d ) to (4.1) the following estimates hold for T − τ ≤ t < T : η t (cid:18) T − t − k η − α k L ∞ (cid:19) ≤ Y t ≤ η t (cid:18) T − t + k η − α k L ∞ (cid:19) + e ˜ K ( T − t ) h t . (4.2) Proof. For 0 < ǫ < τ we define ( Y tǫ ) t ∈ [ T − τ,T − ǫ ) by Y tǫ = η t (cid:18) T − ǫ − t + k η − α k L ∞ (cid:19) + e ˜ K ( T − ǫ − t ) h t . We will show that these processes are supersolutions to (4.1) but with the singularity at t = T − ǫ ,lim t → T − ǫ Y tǫ = + ∞ . Precisely, − d Y tǫ = g ǫ ( t, Y tǫ ) dt − Z tǫ dW t , T − τ ≤ t < T − ǫ, where g ǫ ( t, Y tǫ ) := − η t ( T − ǫ − t ) − α t (cid:18) T − ǫ − t + k η − α k L ∞ (cid:19) + ˜ Ke ˜ K ( T − ǫ − t ) h t − e ˜ K ( T − ǫ − t ) α ′ t and Z ǫ ∈ T t ∈ [ T − τ,T − ǫ ) L F ( T − τ, t ; R × ˜ d ). A calculation as in the proof of Proposition 3.5verifies that for all T − τ ≤ t < T − ǫ , g ǫ ( t, Y tǫ ) ≥ λ t − |Y tǫ | η t =: f ( t, Y tǫ ) . Indeed, applying the inequality ( u + v + w ) ≥ u + 2 uv for u, v, w ≥ |Y tǫ | , weobtain that |Y tǫ | η t ≥ η t + 2 η t k η − α k L ∞ ( T − ǫ − t ) η t ( T − ǫ − t ) . τ := 1 / k η − α k L ∞ ∧ T , k η − α k L ∞ ( T − ǫ − t ) ≤ t ∈ [ T − τ, T − ǫ ) , we have that |Y tǫ | η t − η t ( T − ǫ − t ) − α t (cid:18) T − ǫ − t + k η − α k L ∞ (cid:19) ≥ η t k η − α k L ∞ − α t (1 + k η − α k L ∞ ( T − ǫ − t ))( T − ǫ − t ) ≥ η t k η − α k L ∞ (1 + k η − α k L ∞ ( T − ǫ − t )) − α t (1 + k η − α k L ∞ ( T − ǫ − t ))( T − ǫ − t )= ( η t k η − α k L ∞ − α t )(1 + k η − α k L ∞ ( T − ǫ − t ))( T − ǫ − t ) ≥ K := k h − α ′ k L ∞ + k h − λ k L ∞ , we have that ˜ Ke ˜ K ( T − ǫ − t ) h t − e ˜ K ( T − ǫ − t ) α ′ t ≥ e ˜ K ( T − ǫ − t ) λ t ≥ λ t . Therefore, we can conclude that g ǫ ( t, Y tǫ ) ≥ λ t − |Y tǫ | η t . We now consider the difference of Y and Y ǫ for T − τ ≤ t ≤ s < T − ǫ : Y tǫ − Y t = E (cid:20) Y sǫ − Y s + Z st g ǫ ( r, Y rǫ ) dr − Z st f ( r, Y r ) dr (cid:12)(cid:12)(cid:12) F t (cid:21) ≥ E (cid:20) Y sǫ − Y s + Z st f ( r, Y rǫ ) − f ( r, Y r ) dr (cid:12)(cid:12)(cid:12) F t (cid:21) = E (cid:20) Y sǫ − Y s + Z st ( Y rǫ − Y r )∆ r dr (cid:12)(cid:12)(cid:12) F t (cid:21) where ∆ r = f ( r, Y rǫ ) − f ( r, Y r ) Y rǫ − Y r , if Y rǫ − Y r = 0 , , else. Note that ∆ ≤ . By the explicit representation of the solution to linear BSDEs, Y tǫ − Y t ≥ E " ( Y sǫ − sup t ≤ s ≤ T − ǫ Y s ) exp (cid:18)Z st ∆ r dr (cid:19) . Since Y sǫ ≥ , E [sup t ≤ s ≤ T − ǫ Y s ] < + ∞ due to Y ∈ L F (Ω; C ([0 , T − ]; R + )) , we can apply Fatou’slemma to the expectation above as s → T − ǫ to obtain that Y tǫ − Y t ≥ . Taking ǫ → Lemma 4.3. Suppose that Asssumption 4.1 holds. Let ( Y , Z ) be a solution of (4.1) in thespace L F (Ω; C ([0 , T − ]; R + )) × L F (0 , T − ; R × ˜ d ) . Let X ∗ t = exp( − R t Y s η s ds ) denote the associatedportfolio process. Then X ∗ Z ∈ L F (0 , T ; R ) . Proof. Let M t = Y t X ∗ t + R t λ s X ∗ s ds. Integration by parts yields dM t = X ∗ t Z t dW t . (4.3)Hence, M is a nonnegative local martingale on [0 , T ) and in particular a nonnegative super-martingale. Thus, it converges almost surely in R as t goes to T . Similarly to (3.14), we usethe lower estimate in (4.2) to obtain that for s ∈ [ T − τ, T ) | X ∗ s | ≤ C ( T − s ) . 19n view of the upper estimate in (4.2), we have that E " sup T − τ ≤ t ≤ s |Y t X ∗ t | ≤ C E " sup T − τ ≤ t ≤ T ( | η t | + | h t | ) , where the constant C is independent of s. Thus, applying the dominated convergence theoremimplies E " sup ≤ t ≤ T | M t | ≤ C E " sup ≤ t ≤ T − τ |Y t | + E " sup T − τ ≤ t ≤ T ( | η t | + | h t | ) + E (cid:20)Z T | λ s | ds (cid:21)! < + ∞ . Recalling the equation (4.3), we have that X ∗ Z ∈ L F [0 , T ; R ) and that M is indeed a nonneg-ative martingale on [0 , T ] . It follows from [3, Proposition 4.4] that Y is the minimal solution of (4.1). Therefore, we canobtain the uniqueness result. Theorem 4.4. Under Asssumption 4.1, there exists a unique solution to the BSDE (4.1) in L F (Ω; C ([0 , T − ]; R + )) × L F (0 , T − ; R × ˜ d ) . In this paper we established a novel comparison principle for viscosity solutions to HJB equationswith singular terminal conditions arising in models of optimal portfolio liquidation under marketimpact. Our method is flexible enough to allow for possibly unbounded coefficients. Thecomparison principle allowed us to prove the existence of a unique continuous viscosity solutionto the HJB equation and hence the existence of a unique optimal trading strategy. 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