Consumption-investment optimization problem in a Lévy financial model with transaction costs and làdlàg strategies

We consider the consumption-investment optimization problem for the financial market model with constant proportional transaction rates and Lévy price process dynamics. Contrarily to the recent work of De Vallière (Financ Stoch 20:705–740, 2016), portfolio process trajectories are only left and right limited. This allows us to identify an optimal làdlàg strategy, e.g. in the two dimensional case, as it is possible to suitably rebalance the portfolio processes when they jump out of the no-trade region in the solvency cone.


Introduction
We study a consumption-investment problem with infinite horizon for financial market models including proportional transaction costs and price's dynamics driven by a Lévy process. This problem originates from the seminal paper of [3]. Davis and Norman [5] rigorously solved the problem and provided the optimal consumption plan in a diffusion model with transaction costs. Although the value function W is in general not smooth, Soner and Shreve [12] show that W is solution to the HJB equation in a weak sense, i.e. is a viscosity solution.
B E. Lepinette emmanuel.lepinette@ceremade.dauphine.fr T. Q. Tran tuan.maths@gmail.com 1 When the risky asset prices follow a one dimensional exponential Lévy processes, Framstad et al. [8] have obtained the same results as those of [12] under some mild conditions. An extension to multidimensional Levy processes is proposed by Kabanov et al. [6] where a general market model with conic constraints is considered. Nevertheless, this paper does not provide an optimal solution.
Traditionally, for general discontinuous price processes such as Levy processes, both the price and the trading strategies are supposed to be càdlàg (right-continuous and leftlimited), see the classical book written by Rama Cont and Peter Tankov. However, it is now accepted that trading strategies have to be modeled by làdlàg trading strategies. This allows to establish the characterization of super-hedging prices and the needed duality for portfolio optimization under transaction costs, see [2,4]. Moreover, it is shown in [4] that, if the price process admits jumps at predictable stopping times, then the optimal strategy has left and right jumps. Moreover, contrarily to the case of predictable jumps, a price process modeled by a Levy process only jumps at totally inaccessible times. Right after a jump time, a re-adjustment is then necessary as a response to the new information given by the jump. Therefore, the traditional way of modeling trading strategies by càdlàg processes is not appropriate. The strategies should be làdlàg processes (left and right limited) or, at least, làdcàg (left-continuous and right-limited) processes.
As claimed in [8], the optimal policy for the jump diffusion case has the same form as in the pure diffusion case. In particular, there is a no transaction cone D for the wealth position such that it is optimal to make no transactions inside and to rebalance the position in the trade region as soon as the wealth position gets out of D, in particular after an unpredictable jump of the price process.
Why do we write a new paper on optimal consumption with transaction costs and a Levy price process? The aim of this paper is two folds: First, it extends [6,8] to the case where the controls (the portfolio strategies and consumption plans) are only làdlàg. This generalization is necessary to identify the optimal control policy of the problem. Actually, Framstad et al. [8] provide such làdlàg optimal solution [8,Theorem 4.3] under additional assumptions but, unfortunately, it does not belong to the set of controls they consider, i.e. right-continuous processes [8,Definition 1.2]. In our paper, we consider the right set of controls, which is coherent with the conjectured solution of [8]. Secondly, we provide a full characterization of the optimal strategy, which completes the analysis of [8]. This issue will be carefully addressed in the two assets case in the last sections of the paper. By carefully analyzing the arguments in [6,8,11], we study the regularity of the Bellman function and give a rigorous construction of the optimal strategy. Comparatively to the previous works, we need to adapt the stochastic calculus to làdlàg processes. Moreover, we need to consider a new concept of viscosity solution, precisely in a weaker sense than the usual one, see Definition 9.10, in order to prove the regularity of the wealth process. One of the most difficult part is to show that the Bellman function is the viscosity solution to the HJB equation with a non-local operator. In order to obtain the optimal strategy, we also extend the Skorohod problem into the case of a Levy process in the context of a market with two assets and transaction costs.
The paper is organized as follows: -Section 2: description of the consumption optimization problem.
-Section 3: elementary properties of the value function (Bellman function) W .
-Section 4: we show that W is a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. -Section 5: we show that the HJB equation admits a unique solution under some mild conditions and, as soon as there exists a Lyapunov function.
-Section 6: we propose a condition under which W is finite. This is in particular the case when there exists a non negative classical supersolution to the HJB equation. -Section 7: we show that W is continuous when finite.
-Sections 8 and 9: for the power utility function, we construct a Lyapunov function and a non negative classical supersolution for Sects. 5 and 6. -Section 10: we apply the general results to a two-dimensional model. Under some conditions, we prove that the value function is continuously twice differentiable and we construct an optimal control. To do so, we solve a Skorokhod problem. Note that we introduce in this part a new definition of viscosity solution in a weaker sense. This allows us to change the global operator by a local one and then deduce that the Bellman function is C 2 . -Appendix: resolution of the Skorokhod problem for Sect. 10.

Notations
In R d , we use standard notations like |x|, the Euclidean norm of x ∈ R d , we define d(x, y) = |x − y|, etc. The Euclidean scalar product between two vectors x, y ∈ R d is denoted by x y ∈ R.
We shall use the notations A + and A − to designate the left (resp. right) limit of a process A and we also denote by A t− and A t+ its left and right limits at time t. If A is a làdlàg predictable process of finite variations, the left and right jump processes are denoted by and we introduce the associated càdlàg processes: The continuous part of A is defined as We denoteȦ c the optional version of the Radon-Nikodym derivative d A c /d A c where A c is the total variation of A c .

Optimal consumption investment problem
We consider the financial market model with jumps adopted in [6]. The price return process is modeled by a d-dimensional Lévy process (Y t ) t≥0 defined on a stochastic basis (Ω, F , (F t ) t≥0 , P) satisfying the usual conditions. We denote by p(dz, dt) its jump measure and q(dz, dt) = Π(dz)dt its compensator. We suppose that Π(dz) is a non negative measure concentrated on (−1, ∞) d and The dynamics of Y is given by where μ ∈ R d , W is a m-dimensional standard Brownian motion and Ξ is a matrix of dimension d × m. In the identification of an optimal strategy, we shall only consider a pure jump Lévy process with finite activity, i.e. R |z|π(dz) < ∞. The general case remains open.
Two constant cones K and C are given. They are supposed to be closed and proper, i.e. K ∩ (−K ) = {0} and C ∩ (−C) = {0}. We assume that C ⊆ int K = ∅. In finance K and C stand respectively for the set of transaction constraints (solvency cone of financial positions with non negative liquidation values, see [9]) and consumption constraints, respectively. The dynamics of a portfolio process is defined by: where the controls π = (B, C) are làdlàg predictable processes of finite variations. The dynamics (2.3) means that the portfolio process V is self-financing. The variations of V are only due to the increments of Y . The transaction costs described by B are withdrawn from the portfolio value. At last, C represents the cumulated sum of consumed wealth. If x ∈ R d is an initial endowment, we assume that π = (B, C) belongs to the class denoted by A x of all admissible controls satisfying the following properties: (1)Ḃ c ∈ −K , dP, d B c a.e.,Ċ c ∈ C, dP, d C c a.e., s. for all stopping times τ , (3) ΔB τ ∈ −K , ΔC τ ∈ C, a.s. for all predictable stopping times τ , The three last conditions mean that the portfolio manager does not deliberately get his position out of the solvency cone. It is also assumed that ΔB + 0 = ΔC 0 = Δ + C 0 = 0 and dC c is absolutely continuous with respect to the Lebesgue measure and we write dC c t = c t dt. Using the monotonicity of the controls B and C with respect to the partial order induced by K (i.e. ∀ x, y ∈ R d , x y ⇔ y − x ∈ K ), we may deduce that B and C are of finite variations. Indeed, since int K = ∅, by an appropriate change of coordinates we may assume w.l.o.g. that all coordinates of B, C are monotonic, hence are of finite variations.
Without loss of generality, we assume that C = C c is continuous as the jumps of C are ignored in the optimization problem we consider. Precisely, we assume that dC t = c t dt almost everywhere w.r.t. the Lesbegue measure dt on R.
For every control π ∈ A x , let us introduce the stopping time where V π is the portfolio process starting from x ∈ R d and satisfying (2.3). We suppose that the strategy π = (B, C) is frozen after the exit time, i.e. Δ + B θ = 0 and d B t = c t = 0 for t > θ. Throughout the paper, we fix a discount coefficient β > 0. For every control π = (B, C) ∈ A x , and x ∈ int K , we define the utility process where U is a non-negative utility function defined on C. We assume that U is concave, U (0) = 0 and U (x)/|x| → 0 as |x| → ∞. The optimal consumption problem consists in optimizing the utility process J π (x) over the set of all admissible strategies. To do so, we define the Bellman function as Showing that the Bellman function is finite is not a trivial task. Indeed, this is based on the existence of classical supersolutions to the associated HJB equation, see Lemma 5.2. Moreover, the continuity of W is proven in Theorem 6.4, Sect. 6.

Some elementary properties of the Bellman function
We denote by the partial order defined by

Proposition 3.1
The function W is increasing with respect to the partial order .
Proof It suffices to adapt the proof of [6] to làdlàg strategies with (2.4).
In the following, we obtain lower bounds for the Bellman function. To do so, let us define the liquidation function associated to the solvency cone K , i.e. for x ∈ K , We have x − l(x)e 1 ∈ K . Then, consider the strategy ΔB 0 := l(x)e 1 − x and B t = B 0 for t ≥ 0. For a given consumption plan c, the corresponding portfolio process is: If the consumption plan is c s = κ X s , then Y t := X t (S 1 t ) −1 satisfies

The HJB equation
In the following, we denote by C p (K ) the set of all continuous functions f on K such that sup x∈K | f (x)|(1 + |x|) − p < ∞. The set of all functions f which are C 2 , i.e. twice continuously differentiable on int K is denoted by C 2 (K ). We use the notation C 2 (x), x ∈ R 2 , for the functions having only these properties on a neighbourhood of x. For each π = (B, C) ∈ A x , x ∈ int K , and every function f ∈ C 1 (K ) ∩ C 2 (x) which are increasing with respect to the order K , we consider the integro-differential operator where I (v, z) = I v+diag (v)z∈intK . Applying a Taylor expansion to f ∈ C 1 (K ), we claim that where C x is a constant depending on x. Therefore, the operator H is well defined. Let us now define and the operators: (4.11) Let us introduce the Dirichlet problem associated to the HJB equation: We consider a possible solution to the HJB equation in the viscosity sense. Observe that the integro-differential operator L is not locally defined because of the operator H . Therefore, we define viscosity solutions in the global sense as follows: is called a viscosity supersolution of (4.12) with operator L on a subsetK ⊆ K if for every x ∈ intK and every f ∈ and v ≥ f on K , the inequality L f (x) ≤ 0 holds.
is called a viscosity subsolution of (4.12) with operator L on a subsetK ⊆ K if for every x ∈ intK and every f ∈ and v ≤ f on K , the inequality L f (x) ≥ 0 holds.
is called a viscosity solution of (4.12) on a subset K ⊆ K if v is simultaneously a viscosity super-and subsolution on the subsetK ⊆ K .
When, the subsetK = K , we just say that v is a viscosity solution (resp. super-or subsolution). At last, a function v ∈ C 1 (K ) ∩ C 2 (int K ) is called classical supersolution of (4.12) with operator L on a subsetK ⊆ K if Lv ≤ 0 on intK . We add the adjective strict when Lv < 0 on the set int K .
is a viscosity supersolution of (4.12) with operator L on a subsetK ⊆ K if and only if, for every point x ∈ intK , the inequality L f (x) ≤ 0 holds for any function φ ∈ C 2 (x), such that the difference v − φ attains its global minimum on K at x.

Proof
The proof is an adaptation of the proof of [11,Lemma 4.2.4] where we replace the notion of local minimum by the global one.

Remark 4.5
In the classical theory developed for differential equations, the notion of viscosity solution admits an equivalent formulation in terms of super-and sub-jets J + , J − (see definition in [6]). But this is not the case in our formulation due to the non-local property of the integro-differential operator. Although, there is a link between the notion of viscosity solutions and super-and sub-jets as stated in [6,Section 7]. Theorem 4.6 is proven in Appendix, Sect. 10.1. It implies that the Bellman function is a viscosity solution to (4.12). We shall see conditions under which W is finite.

Definition 4.7
We say that a positive function ∈ C 1 (K ) ∩ C 2 (int K ) is a Lyapunov function if the following properties are satisfied: In other words, is a classical supersolution of the truncated equation (excluding the term U * ), continuous up to the boundary, and growing to infinity at infinity. Let us introduce the following condition on Π which guaranties the uniqueness of the HJB equation we consider, under the condition that there exists a Lyapunov function as stated in the next theorem.  (4.14) Moreover, W is concave.
The proof of this result is given in Appendix, Sect. 10.2.

Finiteness of the value function
We denote by Φ the set of all continuous functions f : K → R + increasing with respect to the partial ordering K and such that for every x ∈ int K and π = (B, C) the positive process X f t = e −βt∧θ f (V θ t+ ) + J π t , t ≥ 0, is a supermartingale. The following lemma is a consequence of Lemma 10.2. It may be proven as in [6].
be a non negative classical supersolution of (4.12) which vanishes out of int K , then f ∈ Φ.
The following proposition formulates finiteness and continuity up to boundary ∂ K of the Bellman function in term of Φ. This corresponds to [6,Lemma 8.1 ].

Corollary 5.3
Let f ∈ C 2 (R d ) be a non negative classical supersolution of (4.12) which vanishes out of int K , then W is finite.

Continuity of the value function
The following results may be proved as in [6].

Proof
Suppose that V t− ∈ ∂ K for some t < θ. Then, by Lemma 10.1 and Assumption 5 of the model, we have ΔY t = ΔB t = ΔC t = 0. Therefore, V t = V t − ∈ ∂ K hence a contradiction.
Note that we also deduce from Lemma 10.1 that the portfolio process exits K either in a continuous manner (in the case V θ − ∈ ∂ K ) or after a jump (in the case V θ − ∈ int K ).

Lemma 6.3 Let us consider a sequence x n
where θ , θ n are the stopping times defined by V := V π,x , V (n) respectively in (2.4).

Existence of Lyapunov functions
In this subsection, we study the existence of Lyapunov functions. We only focus on the case where the matrix A = (a i j ) is diagonal with a ii = σ i , such that σ 0 = 0, μ 0 = 0 and the first asset is a numéraire and the others are risky assets. We also suppose that the utility function is U (x) = u γ (xe 0 ), where u γ (t) = t γ /γ , γ ∈ (0, 1), and C = R + e 0 . At last, we suppose the following condition Note that the Bellman function is homogeneous of degree γ . Therefore, we choose η ≥ γ so that the Lyapunov function grows faster than the Bellman function. If such a Lyapunov exists, the HJB equation admits a unique concave solution in C 1 by Theorem 4.9.
We have u η (x) = ( px) η−1 p ∈ int K * as required for v to be a Lyapunov function. Moreover, Let us denote the integral expression above by H η (x). Our goal is to choose u such that Let us introduce We have We then choose β such that .
Note that k( p, t x) = k( p, x) for t > 0. Therefore, instead of considering the r.h.s of the equality above on K , we may simply consider it on B 1 : It is easy to prove that this expression is bounded on B 1 . We then define and we chose β such that We may deduce the following: Then, the HJB equation admits a unique solution under the conditions of Theorem 4.9.

Classical supersolution
The hypothesis of this section are those of Sect. 7 and the notations are the same. By Our goal is to choose p and k so that, on intK , we have the following inequality Adapting the reasoning of the previous subsection, we choose β such that (8.17) is a classical supersolution to the HJB equation for some p ∈ K * with p 1 = 1.
(ii) In the two-dimensional model with the power utility function, assume that the Merton parameter satisfies there exists a classical supersolution to the HJB equation and a Lyapunov function with higher growth rate than the Bellman function.

Application to the two-asset model
We investigate the properties satisfied by the Bellman function and we construct an optimal policy in the case where d = 2. We use the notation z = (x, y) = (z 1 , z 2 ) to designate a generic element z of R 2 where is the transpose operator. The canonical basis of R 2 is (e 1 , e 2 ) where e 1 = (1, 0) and e 2 = (0, 1) . The risk-free asset (a bond) is supposed to be constant, and the risky asset follows a geometric Lévy process: where p is the jump measure of S 2 and q(dy, dt) = π(dy)dt is its compensator. We suppose that π(dy) is a positive measure concentrated in (−1, ∞) which does not charge the singletons and satisfies the following condition: The inequality (9.19) ensures that π is a finite measure such that the associated Lévy process has a finite activity. This implies that where N t = ∞ n=1 1 T n ≤t is a Poisson process of intensity λ > 0 and (χ i ) i≥1 is a sequence of i.i.d. π-distributed random variables independent of N . A portfolio process satisfies by definition the following dynamics: where L i j , i, j = 1, 2, are the transfer processes supposed to be làdlàg and (λ i j ) i, j=1,2 are the transaction cost coefficients. We rewrite the dynamics of a portfolio process under the vector form: In the following, we use the notationsμ = (0, μ) ,σ = (0, σ ) and, with z = (x, y) , This means that A = diag(0, σ 2 ) is the diagonal matrix with diagonal elements 0 and σ 2 . Morever, we introduce dC t = (c t dt, 0) and The optimization problem reads as where u : R + → R is a concave utility function. In the sequel, we consider the case of power utility functions, i.e. u(r ) = r γ γ , γ ∈ (0, 1). Therefore, W is homogeneous of degree γ : Note that for other utility functions, the following analysis needs to be reconsidered. For instance, for the logarithmic utility function, a simple adaptation of our work does not seem to be enough.
In our example, the solvency cone K is simply a sector generated by the vectors For the sake of simplicity, we suppose that λ 12 = λ 21 = λ. The consumption region is C = R + e 1 . Therefore, the HJB equation is given by: y) . We now formulate properties satisfied by the operator H (u, z):

u is increasing with respect to the natural order on K ), then the operator H
Moreover, if u y (x, 0 + ) and u yy (x, 0 + ) exist and are finite, then so is H (u, (x, 0 + )).
. We now consider t ≥ 0. Using the homogeneity of u, we obtain and, similarly, We deduce that This implies that H (u, z n ) → H (u, z) when n → ∞. The case y = 0 is proved similarly, using the boundedness of (u y (z n )) n≥1 when y n → 0. (iii) The proof is similar than (ii).
Let us also suppose the following: where the function h * is given in Corollary 8.2. By Corollary 8.2, conditions I and II imply the existence of a Lyapunov function with higher growth rate than the Bellman function and a classical supersolution to the HJB equation. Therefore, using Proposition 5.2, we deduce that the Bellman function is finite on K and continuous up to the boudary ∂ K , see Theorem 6.4. Then, we deduce by Theorem 4.9 that the HJB equation admits a unique concave solution in the class C 1 (K ). Actually, we prove that W ∈ C 2 (K ) in Sect. 9.1. To do so, we first formulate some well known results from the literature on the Bellman function. Recall that these results hold provided that W is continuous, concave and monotone with respect to K . The proofs (for continuous diffusion processes, see [11,12]) may be extended to the processes with jumps, as we shall see.

Theorem 9.2 The cone K can be splitted into three non-empty open disjoint cones
where a 1 , a 2 are some constants such that Proof We adapt the proof of [11,Proposition 4.8.2]. With ϕ(x) := a 1 u( p 1 x), we need to show that for all x ∈ cone (g 1 , e 1 ). By Lemma 9.1 (i), we have H (ϕ, x) ≤ 0. Moreover, as in the proof of [11,Proposition 4.8.2], L 0 ϕ(x) + u * (ϕ x ) ≤ 0 due to the lower bound given by Lemma 3.2. This implies that ϕ is a classical super solution of our HJB equation on the subsetK = cone (g 1 , e 1 ). Moreover, by construction, W = ϕ onK . By Theorem 10.7, we deduce that W ≤ ϕ onK . Similarly, we follow the proof of [11, Proposition 4.8.2, statement b)] using Theorem 10.7. We then define K 1 , K 2 as the largest sectors on which (9.26) and (9.27) hold.

Lemma 9.3 If
then the axis of the abscises is not the common boundary of K 1 and K 2 .
Proof We follow the proof of [11,Lemma 4.8.4 ]. Suppose the opposite. Then, the function ψ(z) = a 2 ( p 2 z) γ coincides with W (z) on the sector cone (g 2 , e 1 ). We may deduce the value of a 2 using the assumption of the lemma. Hence, we deduce with z = (x, y) that With z = (x, y) where x, y > 0 and x is fixed, the first term in the r.h.s. above admits a lower bound c x y provided that y is sufficiently close to 0 and c x > 0 only depends on x, a 2 , γ , μ and λ. On the other hand, we may write where |θ t | ≤ |t| for all t ≥ −1, and P is defined by (9.25). We choose M large enough such that π([M, ∞)) is close to 0. Precisely, as ψ y (z) = a 2 ((1 + λ)x + y) γ −1 is bounded by a constant depending on x when y is small enough, we deduce that ψ(x, ·) is Lipschitz and finally where x is arbitrarily small provided that we choose M large enough. We fix M such that M), the factor ψ (z + yθ t )|t| appearing in the first term of the r.h.s. of (9.28) is also bounded by a constant depending on M but we may choose y small enough such that ψ (z + yθ t )|t|y is as small as we want. We finally deduce that |H (ψ, z)| ≤ (2c x /3)y provided that y is small enough. This implies that L 0 ψ(z) + u * (ψ x (z)) > 0 for some z small enough. This yields a contradiction.

Reduction to one variable and regularity of the value function
Using the homogeneity property of the Bellman function, we reduce our problem to the case of one variable by considering the restriction of the Bellman function on the intersection of the line {z=(x,y) : x+y=1} with the interior of K . Indeed, if we define ψ(t) , we may reconstruct W from ψ by the formula As in [11,12], we may show that ψ is the viscosity solution to the new HJB equation obtained by the change of variables above: with the two first-order operators and the second-order operator Recall that W is concave by Theorem 10.8. Then, the function ψ defined above is also concave on Δ and its derivatives ψ , ψ exist almost everywhere. Therefore, (9.29) holds in the classical sense as stated in [6, Lemma 6.1]. Moreover, ψ being concave, it admits left and right derivatives which are respectively left and right continuous and such that the inequality D + ψ ≤ D − ψ holds and is strict only on a countable set. Moreover, by Theorem 9.2, ψ(z) > 0 when z is sufficiently close to λ −1 or 1 + λ −1 . Since ψ is concave, we deduce that ψ > 0 on int Δ.
We now adapt some results from the literature on the regularity of the Bellman function. We mainly focus on the extra term H (z, ψ, ψ(z), ψ (z)).
A direct consequence of the proposition above is that the Bellman function is C 1 on int K \R + e 1 . More precisely we have: Corollary 9.5 The value function is C 1 on int K \R + e 1 . If ψ is not C 1 on R + e 1 , then (9.30) holds. Furthermore, even if ψ is not C 1 on R + e 1 , the partial derivative W x is defined and continuous, and the one-sided derivatives W y (x, 0±) are also defined and satisfy the onesided continuity conditions Proof The first claim is an immediate consequence of Proposition 9.4. When y = 0, W (x, y) = x γ ψ(0) so that W x (x, y) exists and is given by W x (x, y) = γ x γ −1 ψ(0). Otherwise, when y = 0 or equivalently z = z(x, y) = y/(x + y) = 0, then ψ is differentiable at the point z hence W (x, y) = (x + y) γ ψ(z(x, y)) exists and is given by z(x, y)). (9.33) Since D ± ψ(0) exists, it is natural to take the convention 0 × ψ (0) = 0 even if ψ (0) does not exist. It follows that (9.33) holds everywhere. In particular, W x is continuous. Similarly, when y = 0, i.e. z(x, y) = 0, W y (x, y) exists and is given by The claim follows.

Lemma 9.6 The interior of K 0 is nonempty.
Proof It suffices to micmic the proof of [11, Lemma 4.8.7] by virtue of Proposition 9.4 and Lemma 9.3.

Proposition 9.7
The point e 1 belongs to int K 1 . , we observe that H (z n , ψ, ψ ) → 0 as z n → 0. This is indeed proved in the last part of the proof of Proposition 9.4. We then conclude.
In order to apply the Itô formula and construct an optimal control, we need for the value function W to be C 2 across the boundary of the cone K 0 , except at 0. To do so, we introduce the following local operator deduced from the global operator l 0 by freezing the value function ψ.l Although, by doing so, ψ is no more a viscosity solution to the (new) local operator but it is only a viscosity solution in a weak sense we precise as follows. Precisely, we define the notion of weak viscosity solution for an operator L. The corresponding weak solution for the one dimensional equation is easily deduced. Definition 9.10 A function v ∈ C(K ) is a weak viscosity supersolution (resp. subsolution) of (4.12) with operator L on a subsetK ⊆ K if and only if, for every point x ∈ intK , the inequality Lφ(x) ≤ 0 (resp. Lφ(x) ≥ 0) holds for any function φ ∈ C 2 (x) satisfying (v − φ)(x) < 0 (resp. (v − φ)(x) > 0) such that the difference v − φ attains its global minimum (resp. maximum) on K at x.
As usual, a weak viscosity solution is both a weak viscosity supersolution and weak viscosity subsolution. Adapting the proof of [11, Lemma 4.2.5], we get a similar result for weak viscosity solutions.

Lemma 9.11
Let ψ ∈ C 1 (a, b) be a weak viscosity solution of the (local) equation nonempty subinterval (a , b ) ⊆ (a, b). a , b ) and the equation holds in the classical sense.
Proof We adapt the proof of [11,Lemma 4.2.5]. Indeed, with the same notations, the case of interest is when the minimum of ψ − ψ is negative. In that case, we use the weak supersolution property. Otherwise, ψ − ψ ≥ 0, i.e. ψ ≥ ψ on [z 1 , z 2 ], which is the desired conclusion. Symmetrically, when the maximum of ψ − ψ is positive, we use the weak subsolution property. On the contrary, we have ψ ≤ ψ and the conclusion follows.
In the following proposition, we also need the notion of weak viscosity solution contrarily to [11].
Proof By continuity, we may assume that l 1 ψ(z) < 0 and l 2 ψ(z) < 0 for all z in some interval [a, b] where a < z 0 < b. As z 0 = 1, we may suppose that 1 / ∈ [a, b]. We show that ψ is a weak viscosity solution to the equationl 0 ψ = 0 on [a, b].
Let us now consider any function f ∈ C 2 (z 0 ) where z 0 ∈ (a, b) and suppose that that the difference ψ − f attains its global minimum on K at z 0 such that (ψ − f )(z 0 ) < 0. Let us considerf r := f ξ r + ψ(1 − ξ r ) where, by the one dimensional version of Lemma 10.9, ξ r ∈ [0, 1] is infinitely differentiable, vanishes out of the ball of center z 0 and radius r → 0 and ξ r = 1 on a smaller ball around z 0 . Note that ψ −f r admits a global minimum at z 0 . Therefore, l 0f r (z 0 ) ≤ 0. Notice that by assumption f (z 0 ) > ψ(z 0 ) so that, by continuity of f , we may choose r → 0 and > 0 small enough such thatf r (z) ≥ ψ(z) − h(z) on z ∈ D h where h is a continuous function we may choose arbitrarily. Actually, we choose h on a set D h z 0 such that h(z 0 ) = 1. Precisely, 1] and h = 0 on [(z 0 + 1)/2, 1]. Notice that the range of the mapping δ : t → (z 0 + z 0 t)/(1 + z 0 t) is D h when 1 + z 0 t > 0 and lim t→∞ δ(t) = 1. Therefore, the integral is well defined and finite as h[(z 0 + z 0 t)/(1 + z 0 t)] vanishes when (z 0 + z 0 t)/(1 + z 0 t) is closed to 1, i.e. when t is closed to ∞. From the inequalities l 0f r (z 0 ) ≤ 0 andf ≥ ψ − h on z ∈ D h , we then deduce thatl 0 f (z 0 ) ≤ 0 as → 0, i.e. we have replacedf by ψ in the global operator.
Since ψ is a viscosity subsolution of the equation max i=0,1,2 l i ψ = 0, for any test function dominating ψ, we have max i=0,1,2 l i φ ≥ 0. Moreover, max i=1,2 l i φ = max i=1,2 l i ψ < 0 since l 1 and l 2 are local operators only depending on the values of ψ and its first derivative at the considered point z ∈ (a, b). It follows that l 0 φ = max i=0,1,2 l i φ ≥ 0. As above, we deduce that ψ is a weak viscosity subsolution of the equationl 0 ψ = 0 on [a, b] hence ψ is finally a weak viscosity solution to the equationl 0 ψ = 0 on [a, b].
is continuous. Recall that the operator which appears in the definition of G is local as we have frozen the dependence in the test function ψ before. Therefore, by the weak solution property and Lemma 9.11, The following result corresponds to [12,Proposition 8.5]. Its proof may be easily adapted to the case with jumps. To do so, we use the continuity of the function z → H (z, ψ, ψ ) as stated by Lemma 9.1 (i).

Proposition 9.13
Suppose that π(R) < ∞. The function ψ is C 2 on the set (z 1 , z 2 )\{1} and satisfies the HJB equation ψ = 0 on this set in the classical sense.
The proposition above implies that the value function satisfies the HJB equation L 0 (W ) + U * (W x ) = 0 on K 0 \Re 2 in the classical sense and it is C 2 on this set. It remains to study W on the set Re 2 . To do so, we follow the proof of in [12, Theorem 9.1].
Proposition 9.14 The second derivative W yy is well defined and is continuous accross R + e 2 . Moreover, W satisfies the equation L 0 (W ) + U * (W x ) = 0 on K 0 in the classical sense.
Finally, we deduce the following: Corollary 9.15 Suppose that π(R) < ∞. Then, the value function W is C 2 .

Optimal control
The following important result provides an optimal policy for the optimization problem. The proof is based on the resolution of a Skorohod problem described in Appendix, see (10.61).
where W is the Bellman function.
Proof Existence of a solution to the Skorohod problem holds by Theorem 10.12. Note that θ π = ∞ since V + t ∈ K 0 , ∀t, and W x is positive [hence (9.37) makes sense]. We shall only consider the case where K 0 is included in the first quadrant. Otherwise, we refer to Remark 9.18. By the propositions 9.9 and 9.13, the function ψ coincides on the interval (z 1 , z 2 ) with a C 2 -functionψ defined on (− λ −1 , 1 + λ −1 ). Indeed, it suffices to replace ψ by suitable parabolic functions outside (z 1 , z 2 ). In particular, sinceψ is Lipschitz on [z 1 , z 2 ] and W (z) = (x + y) γψ (z) where z = y/(x + y) on K 0 , we also deduce that W is also Lipschitz on K 0 . Moreover, by [11,Lemma 4.7.1, Corollary 4.7.6], W = 0 implies that W (z) ∈ K * \{0} ⊆ (0, ∞) 2 hence W x (z) > 0 for all z. It follows that σ is locally Lipschitz on the set K 0 . We then deduce that the Skorokhod problem admits a unique solution. We check the second assertion. Applying Lemma 10.2, since the term R of the expansion is negative, we have Moreover, when c is defined by (9.37), we have W . Therefore, the last integral term in the equality above is zero by virtue of Proposition 9.14. It remains to prove that N is a martingale and for a sequence of real numbers t n → ∞. To prove (9.38), we observe that |W (z)| ≤ κ|z| γ and |W x (z)| ≤ κ|z| γ −1 where κ is an upper bound of W and W x on the intersection Δ 0 of the set K 0 with the line x + y = 1. This is deduced from the continuity of W on Δ 0 and the fact that ψ (0+) < ∞. We deduce the existence of a constant κ such that Since W is finite, this implies the existence of a sequence t n ↑ ∞ such that (9.38) holds. Details of this assertion are given in Lemma 9.17. We now prove that N t is a true martingale. Indeed, by a similar argument, we have Hence, we infer that the stochastic process The second process defining N is the integral with respect to the Lévy measure. Observe that, for each fixed s, we have I (V s − , z) = 1 (because V + s ∈K 0 , ∀ s). Moreover, using the finite Taylor expansion, we get where the last inequality is deduced from the inequality |W (η)| ≤ κ|η| γ −1 . We then obtain that Therefore, as the Lévy process is of finite activity and (9.39) holds, By [10, Theorem I. 1.33 b., p. 73], we deduce that the purely discontinuous local martingale N satisfies E var(N ) ∞ < ∞ hence is a martingale.
To complete the proof of the theorem, we need the following lemma Since N is a martingale, we deduce that X u − X s ≤ 0, i.e. X is decreasing. Therefore, the integrablity of

Remark 9.18
The situations where x ∈ K i , i = 1, 2, are easily reduced to the one treated in the theorem above. Indeed, recall that the function W restricted on the set K i is constant along the direction g i , i = 1, 2. Instead of considering the initial position x ∈ K i , i = 1, 2, we consider the pointx lying on the boundary of K 0 by projecting x onto K 0 parallel to g i . This translation does not change the value of the Bellman function, meaning that W (x) = W (x). Therefore, the optimal strategy for x is constructed simply by adding the initial jump ΔB 0 :=x − x to the optimal strategy given by the Skorokhod problem with the initial pointx.

Proof of Theorem 4.6
The proof is based on preliminary results we formulate before. It is an adaptation of [6] to làdlàg strategies by considering right limits of the wealth process. The proof of the following lemma is given in [6].

Lemma 10.1 For every portfolio process
Let us define for each n the compact set Note that (K n ) n≥1 is an increasing sequence whose union is int K . For every control π = (B, C) ∈ A x and x ∈ int K , we define V θ n + as the stopped portfolio process.
where θ n is the first instant when the portfolio exits K n . We also define B θ n + similarly. Note that the value of V π,x θ n may get out of int K due to a possible jump of the Lévy process at θ n but, in this case, V π,x θ n − / ∈ ∂ K by virtue of Lemma 10.1. From the dynamics of V π,x , we deduce that p(dz, ds) − q(dz, ds)) .
We propose to study the quantitȳ X f ,n t where, we recall that We have the following key result: be an increasing function with respect to the order K . Then, we havē where N is a local martingale and R is a decreasing process such that R 0 = 0.
Proof Note that we do not assume any regularity of f on ∂ K . Therefore, we can not directly apply the Ito formula toX f ,n t . To overcome this difficulty, instead of considering V θ n + , we study the process V θ n − defined by We also have a representation for V θ n − : For a sake of simplicity, we write Using (10.42) and the fact that B , ds)) .
Since I ( V s − , ΔY s ) = 1 for s < θ n , we may omit the indicator I within the operator H for s < θ n . We deduce that du)) .

(10.45)
The residual term is Proof We fix a strategy π and omit its symbol in the notations below. Observe that only the behaviour of the processes we consider does matter on [0, τ ]. For n large enough, we have O 2r (x) ⊆ int K n . Therefore, τ π ≤ θ n hence we may apply Lemma 10.2 so that We deduce thatX where R is a decreasing process such that R 0 = 0 and N is a local martingale. As it is shown in [6,Lemma 8.3], the stopped process N τ is a martingale hence EN t∧τ = 0.
By assumption, L f (y) ≤ −ε for all y ∈ O r (x) and so We deduce that the following term above [appearing in the expression (10.49) of R] is bounded as follows: On the other hand, the other terms defining R in (10.49) can be estimated as follows: If s = τ we may assume without loss of generality that the controls ΔB + τ = ΔB τ = 0 hence V τ = V τ + as we consider the supremum given in (10.51). Therefore, we have Therefore, by equality (10.52), we deduce that On the set Γ ∩ {τ > t}, the inequality Z t ≥ t obviously holds. Thus, E Z t ≥ tP(Γ ) if t ∈ [0, t 0 ] and the result is proven.
Observe that, if n is large enough, τ π ≤ θ n hence X f ,n t∧τ π does not depend on n.
Lemma 10.5 Suppose that W is continuous on int K . Let T f be the set of finite stopping times. Then, Proof The proof is an adaptation of the proof of [6, Lemma 9.2] but we replace ρ by Proof of Theorem 4.6 (i) We adapt the proof of [6, Lemma 10.2] since the arguments of the proof are based on a strategy π = (B, C) such that B = 0 and, in our case, we need Lemma 10.6 to replace [6, Lemma 9.2]). In that case, V t+ = V t for all t ≥ 0 and the Ito formula is valid as observed in Remark 10.3 but also in [6]. (ii) Let x ∈ int K and φ ∈ C 1 (K ) ∩ C 2 (K ) be a function with φ(x) = W (x) and W ≤ φ on K . Suppose that φ is not a subsolution, i.e. there exists x ∈ int K such that the required inequality fails. Precisely, by continuity, suppose that Lφ ≤ −ε, ε > 0, on a neighborhood O r (x) ⊆ int K of x. By virtue of Lemma 10.4, there exists a constant η := η(ε) and an interval (0, t 0 ] such that where τ π := τ π r is given by (10.50). We may assume w.l.o.g. that r = r . Fix an arbitrary t ∈ (0, t 0 ]. Applying Lemma 10.5, we deduce that there exists π ∈ A x such that As W ≤ φ and, since I (V x,π t∧τ π r − , ΔY t∧τ π r ) = 0, we get that I t∧τ π r <θ = 0 and we obtain from above that W (x) ≤ φ(x) − 1 2 ηt, in contradiction since W (x) = φ(x).

Proof of Theorem 4.9
Uniqueness is an immediate consequence of Theorem 10.7 below. Moreover, Theorem 10 We also consider the corresponding stopping times θ = θ x,π := inf{t : V x,π t / ∈ int K } for π ∈ A x and we have V π t = 0 for t ≥ θ .Therefore, the consumption strategy c is zero after θ . We deduce that the Bellman function W is a global viscosity solution to the same HJB equation as W . Indeed, we use Lemma 10.5 or equivalently [6, Lemma 9.1] and we adapt [6, Lemma 9.2], where we replace the random variable ρ in the proof of [6, Lemma 9.2] by ρ := inf{ j ≥ 1 : V x,π τ + ∈ O j } and the strategyπ is replaced bỹ = V x,π τ k − = 0 by Lemma 10.1. It follows thatπ ∈ A x and we may conclude as in [6, Lemma 9.2] since u ≥ 0.
By assumption, the global viscosity solution of this HJB equation is unique hence W = W . It is well known that the function W is concave. This is proven by Akian et al. [1] as mentioned by Framstad et al. [8]. Therefore, W is also concave.

Skorokhod problem for continuous diffusion processes
The construction of the optimal control for the two-dimensional optimal consumption problem we consider in Sect. 9 is based on the resolution of the so-called Skorokhod problem.
This problem is about existence and uniqueness of the solution to a S.D.E. with reflection. We first recall some known results for the continuous diffusion case. We provide the proof for the sake of completeness. In the next section, we extend these results to finite activity pure-jumps Lévy processes.
Let γ : ∂ K 0 → R 2 be a vector-valued function with g( where W is a standard Brownian motion. Let σ = R 2 → R 2 × R 2 be a matrix-valued function which is Lipschitzcontinuous. On the closed coneK 0 of Sect. 9, we consider the Skorokhod problem formulated as follows: find a pair of adapted continuous processes V ∈ R 2 and k ∈ R, starting respectively from x ∈K 0 and zero, such that k is non decreasing and The aim of this section is to show that the R.S.D.E (10.57) admits a solution on the set K 0 which is trapped at zero. To do so, we shall prove several intermediate lemmas. The main proof is based on the existence of a solution to a R.S.D.E. on a bounded domain G if the direction of the reflection is given by a function γ ∈ C 2 satisfying the following condition (see [7]): C1: γ ∈ C 2 (R 2 , R 2 ) and there is b ∈ (0, 1) such that The case x = 0 being trivial, we assume that x = 0 hence x ∈ K n 0 if n is large enough.

Skorokhod problem for pure-jumps Lévy processes
The setting of this subsection is given in Sect. 10.3. Let γ : ∂ K 0 → R 2 be a vector-valued function with g(x) = −g i on (∂ K 0 ∩ ∂ K i )\{0} and γ (0) = 0. Recall that, as the Lévy process we consider is of finite activity, it can be represented as the sum of a compound poison process and a Wiener process. So, consider a process Y defined by Y t = (Y 1 0 , Y 2 0 , 0) + (t, W t , N t ), t ≥ 0, where W is a standard Brownian motion and N is a pure jump process of finite activity. This means that where ΔN T k are i.i.d. random variables andÑ t = k 1 T k ≤t is a Poisson process with jump stopping times (T k ) k≥1 . Let σ = R 2 → R 2 × R 2 × R 2 be a matrix-valued function which is Lipschitz-continuous.
We consider the Skorokhod problem on K 0 formulated as follows: find a pair of adapted làdlàg (resp. càglàd) processe V , starting from x ∈ K 0 and real-valued process k, starting at zero and increasing, such that The goal is to show that this R.S.D.E has a solution on the setK 0 . To do so, we shall use the following: Lemma 10.11 (Projection onto K 0 parallel to − K ) Assume that K ⊆ R 2 is a constant cone satisfying the hypothesis of the introduction and K 0 ⊆ K is a closed cone with ∂ K 0 ⊆ int K and int K 0 = ∅. For every x ∈ K , there exists a unique y := P −K K 0 (x) ∈ K 0 such that We omit the proof which is standard. It is easily observable that the direction of x − P −K K 0 (x) is given by g 2 if x ∈ K 2 and g 1 if x ∈ K 1 .

Theorem 10.12
There exists a unique solution to the Skorokhod problem (10.61).
Proof Let (T k ) k≥1 be the jump stopping times of the process Y . Assume that we have already constructed a solution (V , k) to (10.61) on the interval [0, T k ). Define

Let us introduce
where the projection operator P K 0 is defined in Lemma 10.11. We define Δ + k T k by the equality Applying Theorem 10.10 and the strong markov property, there exists a solution (Ṽ ,k) to (10.61) from the starting pointṼ 0 := V T k + with respect toÑ t := N T k +t − N T k and W t := W T k +t − W T k defined on the interval [0, T k+1 − T k ]. We then define V t :=Ṽ t−T k , and k t := k T k + +k t−T k , t ∈ (T k , T k+1 ).
Uniqueness follows from uniqueness on each interval [T k , T k+1 ).