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Consumption-Investment Problem with Transaction Costs for L\'evy-Driven Price Processes

We consider an optimal control problem for a linear stochastic integro-diffe\-rential equation with conic constraints on the phase variable and the control of singular-regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs where the price of the assets are given by a geometric L\'evy process and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of the HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the Dynamic Programming Principle.


Introduction
In this paper we study the classical consumption-investment model with infinite horizon in the presence of transaction costs. Our aim is to extend the results of [18] to the case where the price evolution is given by a geometric Lévy process. Namely, we show that the Bellman function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We also prove a uniqueness theorem for the latter.
Mathematically, the consumption-investment problem with transaction costs is a regular-singular control problem for a linear stochastic equation in a cone. Its specificity is that the Bellman function may not be smooth and, therefore, one cannot use the verification theorem (at least, in its traditional form) because the Itô formula cannot be applied. Nevertheless, one can show that the Bellman function is a solution of the HJB equation in viscosity sense. Though the general line of arguments is common, one needs to re-examine each step of the proof. In particular, for the considered jump-diffusion model, the HJB equation contains an integro-differential operator and the test functions involved in the definition of the viscosity solution must be "globally" defined. It seems that already in 1986 H.M. Soner noticed that the control problems with jump parts can be considered in the framework of the theory of viscosity solutions, [26], [27].
There is a growing literature on extension of the concept of viscosity solutions to equations with integro-differential operators, see, e.g., [23], [2], [22], [9], [8], [3], [4]. There are several variants of the definition of viscosity solution. Our choice is intended to serve the model with a positive utility function. The definition can be viewed as a simplified version of that adopted in [15].
A rather detailed study of consumption-investment problems under transaction costs when the prices follow exponential Lévy processes and the investor is constrained to keep long positions in all assets, money included, was undertaken in papers by Benth et al. [10] and [11]. Our geometric approach seems to be more general than that of the mentioned papers where the authors consider a "parametric" version of the stock market, with transactions always passing through money (i.e. either "buy stock" or "sell stock"). A more important difference is that in our setting the investor may take short positions as was always assumed in the classical papers [21], [13], [25]. If short positions are admitted, the ruin may happen due to a jump of the price process. That is why the natural, "classical", setting considered here leads to a different HJB equation of a more complicated structure. Following the ideas from the paper [18] we derive the Dynamic Programming Principle splitted into two separate assertions. Though it is the principal tool which allows to check that the Bellman function is a viscosity solution of the HJB equation, it is rarely discussed in the literature (and even taken for granted, see, e.g., in [1], [25], [10]).
The main results of the paper is Theorem 10.1 claiming that if the Bellman function is continuous up to the boundary then it is a viscosity solution of the HJB equation and the uniqueness theorem for the Dirichlet problem arising in the model, Th. 11.2. We formulate the latter in terms of the Lyapunov function, an object that is defined in terms of the truncated operator, in which the utility function is not involved. Its introduction allows us to disconnect the problems of the uniqueness of a solution and the existence of a classical supersolution.
Probably, the most important result of the paper is the uniqueness theorem for the HJB equation with a non-local operator. In contrast to the methods developed in [?] which are based on (very technical) extensions of the Ishii lemma we use the latter in its original and (very transparent) formulation.
The structure of the problem is the following. In Sections 2 and 3 we introduce the model dynamics and describe the goal functional providing comments on the concavity of the Bellman function W . In Section 4 we show that the Bellman function, if finite, is continuous in the interior of the solvency cone. In Section 5 we give a formal description of the HJB equation. Sections 6 and 7 contain a short account of basic facts on viscosity solutions for integro-differential operators. In Section 8 we explain the role of classical supersolutions to the HJB equations. Section 9 is devoted to the Dynamic Programming Principle. In Section 10 we use it to show that the Bellman function is the solution of our HJB equation. Section 11 contains a uniqueness theorem formulated in terms of a Lyapunov function. In Section 12 we provide examples of Lyapunov functions and classical supersolutions.

The Model
Our setting is more general than that of the standard model of financial market under constant proportional transaction costs. In particular, the cone K is not supposed to be polyhedral. We assume that the asset prices are geometric Lévy processes. Our framework appeals to a theory of viscosity solutions for non-local integro-differential operators.
Let Y = (Y t ) be an R d -valued semimartingale on a stochastic basis (Ω, F, F, P ) with the trivial initial σ-algebra. Let K and C be proper closed cones in R d such that C ⊆ int K = ∅. Define the set A of controls π = (B, C) as the set of predictable R d -valued càdlàg processes of bounded variation such that, up to an evanescent set,Ḃ ∈ −K,Ċ ∈ C. (1) HereḂ denotes a (measurable version) of the Radon-Nikodym derivative of B with respect to the total variation process ||B||. The notationĊ has a similar sense. Though models with arbitrary C is of interest, we restrict ourselves in the present paper by considering consumption processes admitting intensity. To this end, we define Aa as the set of controls π with absolutely continuous components C such that the increment C 0 = 0. For the elements of Aa we have c := dC/dt ∈ C.
The controlled process V = V x,π is the solution of the linear system In general, ∆V 0 = ∆B 0 is not equal to zero: the investor may revise the portfolio when entering the market at time zero. The solution of (2) can be expressed explicitly using the Doléans-Dade expo- Namely, We introduce the stopping time For x ∈ int K we consider the subsets A x and A x a of "admissible" controls for which π = I [0,θ x,π ] π and {V − + ∆B ∈ int K} = {V − ∈ int K}. In financial context, θ is the time of ruin. When V x,π leaves the interior of the solvency cone the control of the portfolio and the consumption stops. The process V given by (2), continues to evolve after the time θ but for us the relevance has only the stopped process V x,π,θ .
It is natural to assume that the process V does not leave the interior of K due to a jump of B: the investor is reasonable enough not to ruin himself by making a too expensive portfolio revision.
The important hypothesis that the cone K is proper, i.e. K ∩ (−K) = {0}, or equivalently, int K * = ∅, corresponds to the model of financial market with efficient friction. In a financial context K (usually containing R d + ) is interpreted as the solvency region and C = (C t ) as the consumption process; the process B = (B t ) describes accumulated fund transfers. In the "standard" model with proportional transaction costs (sometimes referred to as the model of currency market) where λ ij ≥ 0 are transaction costs coefficients, see Section 3.1 in the book [19] for details and other examples.
The process Y represents the relative price movements. If S i is the price process of the ith asset, then Without loss of generality we assume that S i 0 = 1 for all i. In this case Y i is the so-called stochastic logarithm of S i . The formula (4) can be re-written as follows: We shall work assuming that where µ ∈ R d , w is a m-dimensional standard Wiener process and p(dz, dt) is a Poisson random measure with the compensator q(dz, dt) = Π(dz)dt such that Π(dz) is a measure concentrated on ] − 1, ∞[ d . Note that the latter property of the Lévy measure corresponds to the financially meaningful case where S i > 0. For the m × d-dimensional matrix Ξ we put A = ΞΞ * . We assume that and this assumption validates the formula (6): by definition, where the first integral is defined as a stochastic one, while the second and the third are the usual Lebesque integrals, both finite (a.s.).
Notation. By typographical reasons we shall use the notation Dx instead of common diag x for the diagonal operator (or matrix) generated by the vector The system (2) can be written in the integral vector form as follows: It is important to note that the jumps of Y and B cannot occur simultaneously. More precisely, the process |∆B||∆Y | is indistinguishable of zero. Indeed, for any ε > 0 we have, using the predictability of the process because for each ω the set {s : ∆Bs(ω) = 0} is at most countable and its Lebesgue measure is equal to zero. Thus, the process |∆B||∆Y |I {|∆Y |>ε} is indistinguishable of zero and so is the process |∆B||∆Y |.
It follows that ∆B θ = 0. Since the predictable process I {V−∈∂K} I [0,θ] has at most countable number of jumps, the same reasoning as above leads to the conclusion that I {V−∈∂K} |∆Y |I [0,θ] is indistinguishable of zero. This means that θ is the first moment when either V or V − leaves int K. This property will be used in the proof that W is lower semicontinuous on int K.
In our proof of the Dynamic Programming Principle (needed to derive the HJB equation) we shall assume that the stochastic basis is a canonical one, that is the space of càdlàg functions and P is a measure under which the coordinate mapping is the Lévy process.

Goal Functionals and Concavity of the Bellman Function
Let U : C → R + be a concave function such that U (0) = 0 and U (x)/|x| → 0 as |x| → ∞. With every π = (B, C) ∈ A x a we associate the "utility process" where β > 0. We consider the infinite horizon maximization problem with the goal functional EJ π ∞ and define its Bellman function W by a when x 2 − x 1 ∈ K, the function W is increasing with respect to the partial ordering ≥ K generated by the cone K.
If π i , i = 1, 2, are admissible strategies for the initial points x i , then the strategy λπ 1 +(1−λ)π 2 is an admissible strategy for the initial point λx 1 +(1−λ)x 2 , λ ∈ [0, 1], laying on the interval connecting x 1 and x 2 . In the case where the relative price process Y is continuous, the corresponding ruin time for the process dominates the maximum of the ruin times for processes V xi,πi . The concavity of u implies that and, hence, the function W is concave on int K. Unfortunately, in our main case of interest, where Y has jumps, the ruin times are not related in such a simple way since the short positions are allowed. It is easy to give examples of trajectories such that θ x1,π1 = θ λx1+(1−λ)x2,λπ1+(1−λ)π2 < ∞ while θ x2,π2 = ∞ and the relations (10) and (11) do not hold. Therefore, we cannot guarantee, by the above argument, that the Bellman function is concave. Of course, these considerations show only that the concavity of W cannot be obtained in a straightforward way as for a model based on continuous price process but it is not excluded.
The concavity of the Bellman function W is not a property just interesting per se. The classical definition of viscosity solution, as was given by the famous "User's guide" [12], requires the continuity of W . On the other hand, a concave function is continuous in the interior of its domain (and even locally Lipschitz), see, e.g., [5]. Of course, the model must contains a provision which ensures that W is finite. But the latter property, in the case of continuous price processes implies, that W is continuous on int K. In the case of processes with jumps one needs to analyze the continuity of W using other arguments.
In the next section we show that the finiteness of W still guarantees its continuity in the interior of K. We do this using the following assertion. Proof. Let λ > 0. Then λπ ∈ A λx a if and only if π ∈ A x a . For a concave function U with U (0) = 0 we have, for any ε > 0 the inequality U (c) ≥ (1 + ε) −1 U ((1 + ε)c). Hence, for an arbitrary strategy π ∈ A x a we have, for θ = θ x,π = θ (1+ε)x,(1+ε)π , that we infer from here that λ → W (λx) is right-continuous at the point λ = 1. Replacing x by λx we obtain the claim. ✷ If U is a homogeneous function of order γ with γ ∈]0, 1[, i.e. U (λx) = λ γ U (x) for all λ > 0, x ∈ K, then W (λx) = λ γ W (x). Thus, the function λ → W (λx) is concave and, therefore, continuous if finite. Remark 1. In financial models usually C = R + e 1 and σ 0 = 0, i.e. the only first (non-risky) asset is consumed. Correspondingly, U (c) = u(e 1 c) = u(c 1 ) where u is a utility function of a scalar argument. Our presentation is oriented to the power utility function uγ (x) = x γ /γ with γ ∈]0, 1[. The case of γ ≤ 0, where, by convention, u 0 (x) = ln x, is of interest but it is not covered by the present study. Remark 2. We consider here a model with mixed "regular-singular" controls. In fact, the assumption that the consumption process has an intensity c = (c t ) and the agent's utility depends on this intensity is not very satisfactory from the economical point of view. One can consider models with an intertemporal substitution and the consumption by "gulps", i.e. dealing with "singular" controls of the class A x and the goal functionals like with a suitable kernel K(s, r) (the exponential kernel e −γ(s−r) is the common choice).

Continuity of the Bellman Function
Proof. First, we show that the function W is upper semicontinuous on int K. Suppose that this is not the case and there is a sequence xn converging to some x 0 ∈ int K such that lim sup n W (xn) > W (x 0 ). Without loss of generality we way assume that the sequence W (xn) converges. The pointsx k = (1 + 1/k)x 0 , k ≥ 1, belong to the ray R + x 0 and converges to x 0 . We find a subsequence xn k such that x k ≥ K xn k for all k ≥ 1. Indeed, sincẽ and, therefore,x k ∈ xn + int K for all n such that |xn − x 0 | < ε k . Any strictly increasing sequence of indices n k with |xn k − x 0 | < ε k gives us in a subsequence of points xn k having the needed property. The function W is increasing with respect to the partial ordering ≥ K . Thus, On the other hand, the function λ → W (λx 0 ) is right-continuous at λ = 1 and, hence, lim k W (x k ) = W (x 0 ). This contradiction shows that W is upper semicontinuous on int K.

Let us show now that lim infn
Fix ε > 0. Due to the finiteness of the Bellman function there are a strategy π and T ∈ R + such that for θ = θ x0,π we have the bound It remains to show that where we use the abbreviation θn := θ xn,π . Indeed, with this bound we get, using the Fatou lemma, that and the claim follows since ε is arbitrarily small. To prove (12), we observe that by virtue of (5) on the interval [0, θn ∧ θ ∧ T ] we have the representation where S * T := sup t≤T |S t |. Fix arbitrary, "small", δ > 0. For almost all ω the distance ρ(ω) of the trajectory V x0,π (ω) from the boundary ∂K on the interval [0, θ(ω) ∧ T − δ] is strictly positive. The above bound shows that for sufficiently large n the trajectory V xn,π (ω) does not deviate from V x0,π (ω) more than on ρ(ω)/2 on the and (12) holds. ✷

The Hamilton-Jacobi-Bellman Equation
We denote by C 1 (K) the subspace of the space of continuous functions f on K such that sup x∈K |f (x)|/(1 + |x|) < ∞. In other words, C 1 (K) is the space of continuous functions on K of sublinear growth. The notation f ∈ C 2 (x) means that f is smooth in some neighborhood of x.
we introduce the function It is well-defined and continuous in x. Indeed, fix Using the Taylor formula for such value of z and the sublinear growth of f for z with |z| > δ we obtain the following uniform bound for x ∈ Oε(x 0 ):

It implies the needed integrability and the continuity of the integral in x.
We introduce a function of five variables by putting , and the function F 0 is given by In a more detailed form we have that Note that F 0 is increasing in the argument f in the same sense as I. If φ is a smooth function, we put In a similar way, L 0 corresponds to the function F 0 . We show, under mild hypotheses, that W is a viscosity solution of the Dirichlet problem for the HJB equation with the boundary condition understood in the usual classical sense and establish a uniqueness result for this problem.

Viscosity Solutions for Integro-Differential Operators
Since, in general, W may have no derivatives at some points x ∈ intK (and this is, indeed, the case for the model considered here), the notation (13) needs to be interpreted. The idea of viscosity solutions is to substitute W in F by suitable test functions. Formal definitions (adapted to the case we are interested in) are as follows. (13) if v is simultaneously a viscosity super-and subsolution.
At last, a function v ∈ C 1 (K) ∩ C 2 (int K) is called classical supersolution of (13) if Lv ≤ 0 on int K. We add the adjective strict when Lv < 0 on the set int K.
For the sake of simplicity and having in mind the specific case we shall work on, we incorporated in the definitions the requirement that the viscosity superand subsolutions are continuous on K including the boundary. For other cases this might be too restrictive and more general and flexible formulations can be used. Lemma 6.1 Suppose that the function v is a viscosity solution of (13). If v is twice differentiable at x 0 ∈ int K, then it satisfies (13) at this point in the classical sense.
Proof. One needs to be more precise with definitions since it is not assumed that v ′ is defined at every point of a neighborhood of x 0 . "Twice differentiable" means here that the Taylor formula at x 0 holds: and h(r) → 0 as r ↓ 0. We introduce the notation Γr : We define the function fε ∈ C 1 (K) ∩ C 2 (x 0 ) by the formula where M = 1 + sup y∈O1(0) |P 2 (y)|. It follows that Letting ε tend to zero, we obtain that Lv(x 0 ) ≥ 0. Arguing similarly with ε < 0, we get the opposite inequality. ✷

Jets
Let f and g be functions defined in a neighborhood of zero. We shall write have the obvious meaning. For p ∈ R d and X ∈ S d we consider the quadratic function and define the super-and subjets of a function v at the point x: In other words, J + v(x) (resp. J − v(x)) is the family of coefficients of quadratic functions v(x) + Q p,X (y − .) dominating the function v(.) (resp., dominated by this function) in a neighborhood of the point x with precision up to the second order included and coinciding with v(.) at this point.
In the classical theory developed for differential equations the notion of viscosity solutions admits an equivalent formulation in terms of super-and subjets. Since the latter are "local" concepts, such a characterization is not possible for integro-differential operators. Nevertheless, one can construct from semijets test functions with useful properties.
The following lemma claims for v ∈ C 1 (K) with any element (p, X) ∈ J + v(x) x ∈ int K one can relate a test function dominating v, arbitrary close to v in the uniform metric, touching v at the point x, smooth at a neighborhood of x and having at this point the first and the second derivatives coinciding with p and X.
Lemma 7.1 Let v ∈ C 1 (K) and let α > 0. Let x ∈ int K and let (p, X) ∈ J + v(x). Then there exist a number a 0 ∈]0, 1[ and a C 2 -function r : R d → R with compact support such that lim |h|→0 |u| −2 r(h) = 0, (15) and the function f 0 : K → R given by the formula has the following properties: Obviously, δ is continuous, increasing and δ(u) → 0 as u ↓ 0. We extend δ to a continuous function on R + with δ(u) = 0 for u ≥ 1.
The function vanishes at zero with its two right derivatives; u 2 δ(u) ≤ ∆(u) ≤ u 2 δ(4u). It follows that the function r : h → ∆(|h|) has a compact support, belongs to C 2 (Oa 0 (0)), its Hessian vanishes at zero, and Thus, the function y → v(x) + Q p,X (y − x) + r(y − x) dominates v on the ball Oa 0 (x). Without loss of generality, diminishing a 0 if necessary, we may assume that it is dominated by v + α on this ball. Now the assertion of the lemma is obvious. ✷ The corresponding assertion for J − v(x) also holds -with obvious changes in the formulation.
For the proof of the uniqueness theorem we need specific families of test functions coinciding with sub-and supersolutions outside a neighborhood of x. To this end we introduce the following definitions.
Let 0 < a < a ′ . We say that a continuous mapping ξ a,a ′ : It is clear that in the notation of the above lemma for any a ′ ∈]0, a 0 ] the functions f : K → R given by the formulae will satisfy all the properties claimed for f 0 .
The following lemma will be used in the case when D = Dx = diag x,D = Dy = diag y, and x, y has no zero components. Lemma 7.2 Let D,D be two invertible linear operators on R d . Let ξ a,a ′ be a (a, a ′ )cutoff function. Then there is a (ã,ã ′ )-cutoff functionξã ,ã ′ with arbitrary smallã ≤ a||DD −1 || −1 and arbitraryã ′ ≥ a ′ ||DD −1 || such that First, we present sufficient conditions ensuring that the Bellman function W of the considered maximization problem is finite.
Functions we are interested in are defined in the solvency cone K while the process V may jump out of the latter. In order to be able to apply later the Itô formula we stop V = V x,π at the moment immediately preceding the ruin and define the processṼ where θ is the exit time of V from the interior of the solvency cone K. This process coincides with V on [0, θ[ but, in contrast to the latter, either always remains in K (due to the stopping at θ if V θ− ∈ int K) or exits to the boundary in a continuous way and stays on it at the exit point.
, we obtain from the equation (8) the representatioñ Let Φ be the set of continuous functions f : K → R + increasing with respect to the partial ordering ≥ K and such that for every x ∈ int K and π ∈ A x a the positive process X f = X f,x,π given by the formula is a supermartingale.
The set Φ of f with this property is convex and stable under the operation ∧ (recall that the minimum of two supermartingales is a supermartingale). Any continuous function which is a monotone limit (increasing or decreasing) of functions from Φ also belongs to Φ.
The interest to the processes X f with f ∈ Φ is explained by the following: Suppose that for every ε > 0 there exists fε ∈ Φ such that fε(y) ≤ ε. Then W is continuous at y and W (y) = 0.
Proof. (a) On the boundary ∂K the inequality is trivial. Using the positivity of f , the supermartingale property of X f , and, finally, the monotonicity of f we get, for ∈ int K, the following chain of inequalities leading to the required property: The continuity of the function W at the point y ∈ ∂K follows from the inequalities 0 ≤ W ≤ fε. ✷ Remark. Recall that Proposition 4.1 assets that the function W , if finite, is continuous on the interior of K. Thus, the above lemma implies that W is continuous on int K if Φ is not empty. If Φ is reach enough to apply (b) at every point of the boundary, then W is continuous on K and vanishes on the boundary.
supersolution of (13), then f ∈ Φ, i.e. f is increasing with respect to the partial ordering ≥ K and X f is a supermartingale.
Proof. First, notice that a classical supersolution is increasing with respect to the partial ordering ≥ K . Indeed, by the finite increments formula we have that for any The right-hand side is greater or equal to zero because for the supersolution f we have the inequality Σ G (f ′ (y)) ≤ 0 whatever is y ∈ int K, or, equivalently, f ′ (y)h ≥ 0 for every h ∈ K, just by the definition of the support function Σ G and the choice of G as a generator of the cone −K. By continuity, Let θn := inf {t : dist(Ṽ t , ∂K) ≤ 1/n}. The stopped processesṼ θn evolves in int K. Thus, we can apply the "standard" Itô formula to e −βt f (Ṽ t ) and obtain, for t ≤ θ, that Taking into account that the processes Y and B do not jump simultaneously and the ruin cannot happen due to a jump of B we get that where we replace in the integrals by dots the lengthy expression Noting that and using the equation (8) and the above formulae we obtain, after regrouping terms, the following representation where and m is the local martingale By definition of a supersolution, for any x ∈ int K, Thus, the integral in (20) is a decreasing process. The process R is also decreasing. Indeed, the terms of the sum in (21) are less or equal to zero in virtue of the monotonicity of f and where f ′ (V s− )Ḃs ≤ 0 sinceḂ takes values in −K. Let σn be a localizing sequence for m. Taking into account that X f ≥ 0, we obtain from (20) that for each n the negative decreasing process R t∧σn dominates an integrable process and so it is integrable. The same conclusion holds for the stopped integral. Being a sum of an integrable decreasing process and a martingale, the process X f t∧σn is a positive supermartingale and, hence, by the Fatou lemma, X f is a supermartingale as well. ✷ Lemma 8.2 implies that the existence of a smooth positive supersolution f of (13) ensures the finiteness of W on K. We discuss a method how to construct supersolutions in Section 12.
Remark. LetŌ be the closure of an open subset O of K and let f :Ō → R + be a classical supersolution inŌ increasing with respect to the partial ordering ≥ K . Let x ∈ O and let τ be the exit time of the process V x,π fromŌ. The above arguments imply that the process X f t∧τ is a supermartingale and, therefore,

Strict Local Supersolutions
For the strict supersolution we can get a more precise result which will play the crucial role in deducing from the Dynamic Programming Principle the property of W to be a subsolution of the HJB equation. Fix x ∈ int K and a ballŌr(x) ⊆ int K such that the larger ballŌ 2r (x) ⊆ int K. We define τ π = τ π r as the exit time of V π,x from Or(x), i.e.
. Then there exist a constant η = ηε > 0 and an interval ]0, t 0 ] such that Proof. We fix a strategy π and omit its symbol in the notations below. In what follows, only the behavior of the processes on [0, τ ] does matter. Note that |Vτ −x| ≥ r on the set {τ < ∞} and τ ≤ θ. As in the proof of Lemma 8.2, we apply the Itô formula and obtain, with the same notations, (21) and (22), the representation Due to the monotonicity of f we may assume without loss of generality that on the interval [0, τ ] the increment ∆B t does not exceed the distance of V s− to the boundary of Or(x). In other words, if the exit from the ball is due to an action (and not because of a jump of the price process), we can replace this action by a less expensive one, with the jump of the processṼ in the same direction but a smaller one, ending on the boundary of the ball. So, |∆B t | ≤ 2r for t ≤ τ . By assumption, for y ∈Ōr(x) we have the bounds (L 0 f +U * )(y) ≤ −ε (implying that the first integral in the right-hand side above is dominated by −ε (t ∧ τ )) and Σ G (f ′ (y)) ≤ −ε. The latter inequality means that kf ′ (y) ≤ −ε|k| for every k ∈ −K (therefore, we have the inclusion f ′ (Ōr(x)) ⊂ int K * ). In particular, for s ∈ [0, τ ] Since |Ṽ s− − x| ≤ r for s ∈ [0, τ ], we obtain, using the finite increment formula and the linear growth of f , the bounds and, as I(Ṽ s− , z) = 1 when |z| < r/(|x| + r), for some constant κ independent on the strategy. Thus, the integrand in the stochastic integral with respect to the centered Poisson measure in (22) for t ≤ τ is bounded by the function |z| 2 ∧ |z| multiplied by a constant while the integrand in the integral with respect to the Wiener process is bounded. It follows that the local martingale (m t∧τ ) t≥0 is a martingale and Em t∧τ = 0. The above observations imply the inequality where N t := ε (t ∧ τ ) + The process N t looks a bit complicated but we can replace it by another one of a simpler structure. To this end, note that there is a constant κ ("large", for convenience, κ ≥ 1) such that Indeed, being the image of a closed ball under continuous mapping, the set f ′ (Ōr(x)) is a compact in int K * . The lower bound of the continuous function U * on f ′ (Ōr(x)) is finite. For any p from f ′ (Ōr(x)) and c ∈ C ⊆ K we have the inequality pc/|c| ≥ ε. At last, U (c)/|c| → 0 as c → ∞. Combining these facts we infer the claimed inequality. Thus, for the first integral in the definition of N t we have the bound The second integral in the definition dominates κ 1 ||B|| t∧τ for some κ 1 > 0. To see this, let us consider the absolute norm |.| 1 in R d . In contrast with the total variation ||B|| which is calculated with respect to the Euclidean norm |.|, the total variation of B with respect to the absolute norm admits a simpler expression i Var B i where Var B i is the total variation of the scalar process B i . Obviously, But all norms in R d are equivalent, i.e.κ −1 |.| ≤ |.| 1 ≤κ|.| for some strictly positive constantκ. The same inequalities relate the corresponding total variation processes. The claimed property follows from here with the constant κ 1 =κ −2 . Summarizing, we conclude that it is sufficient to check the domination property for EÑ t withÑ These processesÑ =Ñ π have a transparent dependence on the control. The idea of the concluding reasoning is very simple: on a certain set of strictly positive probability, where one may neglect the random fluctuations, either τ is "large", or the total variation of the control is "large": one can accelerate exit only by an intensive trading or consumption. The formal arguments are as follows. Using the stochastic Cauchy formula (4) and the fact that E 0+ (Y i ) = E 0 (Y i ) = 1, we get immediately that there exist a number t 0 > 0 and a measurable set Γ with P (Γ ) > 0 on which whatever is the control π = (B, C). Of course, diminishing t 0 , we may assume without loss of generality that κt 0 ≤ r/8. For any t ≤ t 0 we have on the set Γ ∩ {τ ≤ t} the inequality ||B||τ + ||C||τ ≥ r/4 and, hence, On the set Γ ∩ {τ > t} the inequalityÑ t ≥ t is obvious. Thus, EÑ t ≥ tP (Γ ) on [0, t 0 ] and the result is proven. ✷

Dynamic Programming Principle
The aim of this section is to establish the following two assertions which will serve to derive the HJB equation for the Bellman function. For the considered model, they constitute an analog of the classical Dynamic Programming Principle. The latter is usually written in the form of a single identity (see the remark at the end of the section), but for our purpose this form, more precise, is needed.
1 Let T f be the sets of finite stopping times. Then Lemma 9.2 Suppose that W is continuous on int K. Then for any τ ∈ T f We work on the canonical filtered space of càdlàg functions equipped with the measure P which is the distribution of the driving Lévy process. The generic point ω = ω. of this space is a d-dimensional càdlàg function on R + , zero at the origin. Let F • t := σ{ωs, s ≤ t} and F t := ∩ ε>0 F • t+ε . We add the superscript P to denote σ-algebras augmented by all P -null sets from Ω. Recall that F •,P t coincides with F P t (this assertion follows easily from the predictable representation theorem). The Skorohod metric makes Ω a Polish space and its Borel σ-algebra coincides with F∞, for details see [?].
Let π be a fixed strategy from A x a and let θ = θ x,π be the exit time from int K for the process V x,π .
Recall the following general fact on regular conditional distributions. Let ξ and η be two random variables taking values in Polish spaces X and Y equipped with their Borel σ-algebras X and Y. Then ξ admits a regular conditional distribution given η = y which we shall denote by p ξ|η (Γ, y). This means that p ξ|η (., y) is a probability measure on X , p ξ|η (Γ, .) is a Y-measurable function, and for any X × Y-measurable function f (x, y) ≥ 0.
At last, for fixed s and w s , the shifted control π .+s (g(s, ω s . ,ω.)) is admissible for the initial condition V x,π s (ω) when s ≤ θ(ω). Here we denote byω. a generic point of the canonical space. Proof of Lemma 9.1. For arbitrary π ∈ A x a and T f we have that According to the above discussion we can rewrite the second term of the right-hand side as ∞ 0 e −βr U (c r+τ (g(τ, ω τ ,ω)))dr P (dω) and dominate it by Ee −βτ I {τ <θ} W (V x,π τ ). Thus, This bound leads directly to the announced inequality. ✷ Proof of Lemma 9.2. Fix ε > 0. By hypothesis, the function W is continuous on int K. For each x ∈ int K we can find an open ball Or(x) = x + Or(0) with r = r(ε, x) < ε contained in the open set {y ∈ int K : |W (y) − W (x)| < ε}. Moreover, we can find a smaller ball Or(x) contained in the set y(x) + K with some y(x) ∈ Or(x). Indeed, take an arbitrary x 0 ∈ int K. Then, for some δ > 0, the ball x 0 + O δ (0) ⊂ K. Since K is a cone, λx 0 + O λδ (0) ⊂ K for every λ > 0 and this inclusion implies that Clearly, the requirement is met for y(x) = x − λx 0 andr = λδ when λ|x 0 | < r and λδ < r. The family of sets Or (x)/2 (x), x ∈ int K, is an open covering of int K. But any open covering of a separable metric space contains a countable subcovering (this is the Lindelöf property; in our case, where int K is a countable union of compacts, it is obvious). Take a countable subcovering indexed by points xn. For notational simplicity, we shall denote the open balls Or (xn)/2 (xn) by On and y(xn) by yn. Let π n = (B n , C n ) ∈ A yn a be an ε-optimal strategy for the initial point yn, i.e. such that Let π ∈ A x a be an arbitrary strategy. Put Let us introduce the strategy π ′ := πI [0,τ ] + (0, 0)I ]τ,∞[ and the predictable stopping times τ k := τ + 1/k. Finally, put whereπ n,k is the translation of the strategy π n : namely, for a point ω. with τ (ω.) = s < ∞ we haveπ In other words, the strategyπ coincides with π on [0, τ [, is zero on the interval [τ, τ k [ and with the shift of π n on [τ k , ∞[ when V x,π τ is in On and V x,π τ k − yn ∈ K; the correction term guarantees that in the latter case the trajectory of the control system corresponding to the controlπ passes at time τ k through the point yn.
One can check thatπI [0,θ x,π ] ∈ A x a . Now, using the same considerations as in the previous lemma, we have: Since π and ε are arbitrary, the result follows. ✷ Remark. The previous lemmas imply that for any τ ∈ T f the following identity holds: It can be considered as a form of the dynamic programming principle but, seemingly, it is not sufficient for our derivation of the HJB equation. 10 The Bellman Function and the HJB Equation Theorem 10.1 Assume that the Bellman function W is in C(K). Then W is a viscosity solution of (13).
Proof. The claim follows from the two lemmas below. ✷ Lemma 10.2 If W is in C(int K) then W ≥ 0 is a viscosity supersolution of (13).
Proof. Let x ∈ int K and let φ ∈ C 1 (K)∩C 2 (x) be a function such that φ(x) = W (x) and W ≥ φ on K.
Take now π with B t = 0 and c t = c ∈ C for all t. Let τr = τ π r ≤ θ be the exit time of the process V = V x,π from the ball Or(x); obviously, τr ≤ θ. The properties of the test function and the inequality (26) imply that We get from here, using the Itô formula (20), that Dividing the resulting inequality by t and taking successively the limits as t and r converge to zero we infer that L 0 φ(x) − cφ ′ (x) + U (c) ≤ 0. Maximizing over c ∈ C yields the bound L 0 φ(x) + U * (φ ′ (x)) ≤ 0 and, therefore, W is a supersolution of the HJB equation. ✷ Lemma 10.3 If (25) holds then W ≥ 0 is a viscosity subsolution of (13).
Proof. Let x ∈ int K and let φ ∈ C 1 (K)∩C 2 (x) be a function such that φ(x) = W (x) and W ≤ φ on K. Suppose that the subsolution inequality for φ fails at x. Thus, there exists ε > 0 such that Lφ ≤ −ε on some ballŌr(x) ⊂ int K. By virtue of Lemma 8.3 (applied to the function φ) there are t 0 > 0 and η > 0 such that on the interval ]0, t 0 ] for any strategy π ∈ A x a E J π where τ π is the exit time of the process V x,π from the ball Or(x). Fix arbitrary t ∈]0, t 0 ]. By the second claim of Lemma 9.1 there exists π ∈ A x a such that for every stopping time τ , in particular for τ π . Using the inequality W ≤ φ and applying Lemma 8.3 we obtain from the above relations that W (x) ≤ φ(x) − (1/2)ηt. This is a contradiction because at the point x the values of W and φ are the same. ✷

Uniqueness Theorem
Before formulating the uniqueness theorem we recall the Ishii lemma.
In this statement I is the identity matrix andJ + v andJ −ṽ are values of the set-valued mappings whose graphs are closures of graphs of the set-value mappings J + v and J −ṽ , respectively.
The inequality (27) means that for any vectors x and y from R d Of course, if v is smooth, the claim follows directly from the necessary conditions of a local maximum (with X = v ′′ ( x), Y =ṽ ′′ ( y) and the constant 1 instead of 3 in inequality (27)).
The inequality (27) implies the bound which will be used in the sequel (for the proof see, e.g., Section 4.2 in [19]).
The following concept plays a crucial role in the proof of the purely analytic result on the uniqueness of the viscosity solution which we establish by a classical method of doubling variables using the Ishii lemma.
Definition. We say that a positive function ℓ ∈ C 1 (K) ∩ C 2 (int K) is the Lyapunov function if the following properties are satisfied: In other words, ℓ is a classical strict supersolution of the truncated equation (without the term U * ), continuous up to the boundary, and increasing to infinity at infinity. Theorem 11.2 Suppose that there exists a Lyapunov function ℓ and the Lévy measure Π is such that Then the Dirichlet problem (13), (14) has at most one viscosity solution in the class of continuous functions satisfying the growth condition Proof. Let W andW be two viscosity solutions of (13) coinciding on the boundary ∂K. Suppose that W (z) >W (z) for some z ∈ K. Take ε > 0 such that We introduce a family of continuous functions ∆n : K × K → R by putting Note that ∆n(x, x) = ∆ 0 (x, x) for all x ∈ K and ∆ 0 (x, x) ≤ 0 when x ∈ ∂K. From the assumption that the function ℓ has a higher growth rate than W we deduce that ∆n(x, y) → −∞ as |x| + |y| → ∞. It follows that the level sets {∆n ≥ a} are compacts and the function ∆n attains its maximum on a compact subset of K × K which does not depend on n. That is, there exists (xn, yn) ∈ K × K such that ∆n(xn, yn) =∆n := sup (x,y)∈K×K ∆n(x, y) ≥∆ := sup x∈K ∆ 0 (x, x) > 0.
All (xn, yn) belong to the compact set {(x, y) : ∆ 0 (x, y) ≥ 0}. It follows that the sequence n|xn − yn| 2 is bounded. We continue to argue (without introducing new notations) with a subsequence along which (xn, yn) converge to some limit ( x, x). Necessarily, n|xn − yn| 2 → 0 (otherwise we would have ∆ 0 ( x, x) >∆). It is easily seen that∆n → ∆ 0 ( x, x) =∆. Thus, x is an interior point of K and so are xn and yn for sufficiently large n. Let j ≥ 1 be the number of nonzero components x. Without loss of generality we assume that x j+1 , . . . , xd = 0 and for sufficiently large n the first j components of xn and yn are strictly positive.
If p ∈ int K * , then for any x, y ∈ K \{0} we have the strict inequality yf ′ (x) > 0 implying that f ′ (x) ∈ int K * . Thus, for p ∈ int K * the function f is a Lyapunov function provided that the inequality L 0 f (x) ≤ 0 is satisfied. We show that under some mild conditions this inequality holds for sufficiently large β.
Put Recall that A(x) is the matrix with A ij (x) = a ij x i x j and the vector µ(x) has the components µ i x i .
Taking into account that Combining these two cases, we get the result. ✷ Remark. An inspection of the above arguments shows that one can get that the fp is a Lyapunov function for Of course, such a bound is less tractable than that given above.

Construction of classical supersolutions.
Similar argments are useful in the search of classical supersolutions for the operator L. Since Lf = L 0 f + U * (f ′ ), it is natural to choose u related to U . For a particular case, where C = R d + and U (c) = u(e 1 c), with u satisfying the postulated properties (except, maybe, unboundedness) and assuming, moreover, that the inequality u * (au ′ (z)) ≤ g(a)u(z) (36) holds, we get, using the homogeneity of L 0 , the following result. Proposition 12.2 Let p ∈ int K. Suppose that (36) holds for every a, z > 0 with g(a) = o(a) as a → ∞. If κp < ∞ and β <ηpR + ηp + max i |µ i |κp, then there exists a 0 such that for every a ≥ a 0 the function afp is a classical strict supersolution of (13).