E Lepinette
email: emmanuel.lepinette@ceremade.dauphine.fr.

T Q Tran

E Lépinette

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

Keywords: Consumption-investment Problem, Transaction costs, Hamilton-Jacobi-Bellman equation, Viscosity solution, Lévy process. Mathematics Subject Classification (2000) 60G44 JEL Classification G11, G13

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 in [4], 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 outside the no trade region of the solvency cone.

under some mild conditions. An extension to the jumping diffusion case is studied by Kabanov et al. [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF] where a general market model with conic constraints is considered.

The present paper extends [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF] to the case where the controls (the portfolio strategies and consumption plans) are only left and right limited (làdlàg). This generalization is important as we need it to identify an optimal strategy, as conjectured in the twodimensional case by Framstad et al. [START_REF] Framstad | Optimal consumption and portfolio in a jump diffusion market[END_REF]. Indeed, in [START_REF] Framstad | Optimal consumption and portfolio in a jump diffusion market[END_REF], the set of controls is the family of all right continuous and left limited strategies but the conjectured optimal scontrol is only làdlàg. Actually, despite this contradiction, considering làdlàg strategies is the good intuition as the negative effects induced by price process jumps on portfolio processes may be corrected by an immediate re-balancement. The economic intuition is also straightforward. When asset prices move in response to unpredictable events, which is typically the case at jumping times of a Lévy process, a trading response should be immediately be made after the event i.e. the strategy or control should only be right limited. However, if a market information is announced at a predictable time, e.g. in the presence of a predictable labor income stream, a trade may be executed immediately before the announcement, i.e. the strategy is right-continuous at that predictable time.

Although the main results in this paper extend the results of [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF] and [START_REF] Framstad | Optimal consumption and portfolio in a jump diffusion market[END_REF], one needs to re-examine carefully each step of the proofs. One of the most difficult part is to show that the Bellman function is the unique viscosity solution to the HJB equation.

In the presence of one bond and one risky asset, we study the regularity of the Bellman function and give a construction of the optimal strategy which is conjectured in [START_REF] Framstad | Optimal consumption and portfolio in a jump diffusion market[END_REF].

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 8,9: For the power utility function, we construct a Lyapunov function and a non negative classical supersolution for Sections 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 Section 10.

Notations:

In R d , we use standard notations like |x|, the Euclidean norm of x ∈ R d , d(x, y) = |x -y|, etc. The Euclidean scalar product between two vectors x, y ∈ R d is denotes by xy ∈ 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 limit at time t. If A is a làdlàg predictable process of finite variations, the left and right jump processes are denoted by ∆A := A -A -, ∆ + A := A + -A, and we introduce the associated càdlàg processes:

A d t := s≤t ∆As, A d+ t := s<t ∆ + As, t ≥ 0.
The continuous part of A is defined as

A c := A -A d -A d,+ -.
We denote Ȧc the optional version of the Radon-Nikodym derivative dA 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 [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]. 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

R d |z| 2 ∧ |z| Π(dz) < ∞.
(

The dynamics of Y is given by

dY t = µt + ΞdW t + R d
z (p(dz, dt) -q(dz, dt)) ,

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) and consumption constraints, respectively. The dynamics of a portfolio process is defined by:

dV i t = V i t-dY i t + dB i t -dC i t , i = 1, . . . , d, V 0 = V 0-= x, (2.3)
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 Ax of all admissible controls satisfying the following properties:

1) Ḃc ∈ -K, dP, d B c a.e., Ċc ∈ C, dP, d C c a.e.,

2) ∆ + Bτ ∈ -K, ∆ + Cτ ∈ C, a.s. for all stopping times τ ,

3) ∆Bτ ∈ -K, ∆Cτ ∈ C, a.s. for all predictable stopping times τ , 4) If V = V π , π = (B, C) ∈ Ax, is such that Vτ ∈ int K for some stopping time τ , then

V + τ = Vτ + ∆ + Bτ -∆ + Cτ ∈ int K, 5) If V = V π , π = (B, C) ∈ Ax, is such that V τ -∈ int K (resp. K) for some stopping time τ , then V τ -+ ∆Bτ -∆Cτ ∈ int K (resp. K), 6) x + ∆ + B 0 ∈ int K.
The three last conditions mean that portfolio manager does not deliberately get his position out of the solvency cone. It is also assumed that ∆B + 0 = ∆C 0 = ∆ + 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, 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 dt measure on R.

For every control π ∈ Ax, let us introduce the stopping time

θ π = inf t : V π t / ∈ int K , (2.4)
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. ∆ + B θ = 0 and dB t = c t = 0 for t > θ. Throughout the paper, we fix a discount coefficient β > 0.

For every control π = (B, C) ∈ Ax, x ∈ int K, we define the utility process

J π t (x) := t∧θ π 0 e -βs U (cs)ds,
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

W (x) := sup π∈Ax EJ π ∞ (x), x ∈ int K. (2.5)
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 6.2. Moreover, the continuity of W is proven in Theorem 7.4 in Appendix.

Some elementary properties of the Bellman function

We denote by the partial order defined by K, i.e. if x, y ∈ R d , x y ⇔ x -y ∈ K.

Proposition 3.1 The function W is increasing with respect to the partial order .

Proof. Suppose that x 2 x 1 . Let π = (B, C) ∈ Ax 1 and V (1) be such that

V (1) 0-= x 1 , dV (1)i
t = V (1)i t-dY i t + dB i t -dC i t , t ≥ 0, i = 1, • • • , d.

Let us define V

(2)

0-= x 2 and V (2) t = V (1)
t if t ≥ 0.
From the dynamics of V (1) we deduce that

V (2) 0-= x 2 , dV (2)
i t = V (2)i t-dY i t + d Bi t -dC i t , t ≥ 0, i = 1, • • • , d, where Bi t = B i t + (x i 1 -x i 2 )I [0,∞) (t)
is still a làdlàg and predictable process of finite variations satisfying ∆ Bt ∈ -K a.s. since

x 1 -x 2 ∈ -K. As V (2) 0-, V (1)
0-∈ int K it is straightforward that θ 2 = θ 1 where θ i , i = 1, 2, are the stopping times defined by (2.4) respectively for V (2) and V (1) . It follows that π = ( B, C) ∈ Ax 2 and we deduce that W (x 1 ) ≤ W (x 2 ). 2

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, l(x) := sup{z ∈ R + : x -ze 1 ∈ K}.

We have x -l(x)e 1 ∈ K. Then, consider the strategy ∆B 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: V t = (X t , 0) where

X t := l(x)S 1 t -S 1 t t 0 (S 1 u ) -1 cudu.
If the consumption plan is cs = κXs, then Y t := X t (S 1 t ) -1 satisfies

Y t := l(x) -κ t 0 Yudu,
i.e. Y t = l(x)e -κt . It follows that X t = S 1 t l(x)e -κt . We deduce that the process X does not hit zero, i.e. the process V stays in the interior of the solvency cone. With the strategy π := (B, C) above,

J π ∞ = ∞ 0 e -βt U (κS 1 t l(x)e -κt )dt,
and then,

W (x) ≥ sup κ>0 E ∞ 0 e -βt U (κS 1 t l(x)e -κt )dt.
In the particular case where Y 1 t = 0, i.e. S 1 t = 1, we have the following: Lemma 3.2 With S 1 ≡ 1 and the power utility function U (x) = uγ (xe 0 ), where uγ (t) = t γ /γ, γ ∈ (0, 1), we have

W (x) ≥ 1 γ κ γ-1 * l γ (x) = 1 γ β 1 -γ γ-1 l γ (x), (3.6)
where κ * := β/(1 -γ).

The HJB equation

In the following, we denote by Cp(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) ∈ Ax, 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

H(f, x) : = R d f (x + diag (x)z)I(x, z) -f (x) -f (x)diag (x)z Π(dz), (4.7)
where

I(v, z) = I v+diag (v)z∈intK . Applying a Taylor expansion to f ∈ C 1 (K), we claim that |f (x + diag (x)z)I(x, z) -f (x) -f (x)diag (x)z| ≤ Cx(|z| ∧ |z| 2 ),
where Cx is a constant depending on x. Therefore, the operator H is well defined. Let us now define

G : = (-K) ∩ ∂O 1 (0), Σ G (p) : = sup x∈G px, U * (p) : = sup x∈C (U (x) -px) , Av : = (diag v).Ξ ((diag v).Ξ) T , v ∈ R d ,
and the operators:

F 0 (X, p, H, W, x) : = 1 2 Tr A(x)X + µ T (diag x)p + H -βW, (4.8)
L 0 φ : = F 0 (φ (x), φ (x), H(φ, x), φ(x), x), (4.9)
F (X, p, H, W, x) : = max F 0 (X, p, H, W, x) + U * (p), Σ G (p) , (4.10
)

Lφ : = F (φ (x), φ (x), H(φ, x), φ(x), x). (4.11)
Let us introduce the Dirichlet problem associated to the HJB equation:

F (W (x), W (x), H(W, x), W (x), x) = 0, x ∈ int K, (4.12)
W (x) = 0, on ∂K.

(4.13)
We consider a possible solution to the HJB equation in the viscosity sense. Observe that the integro-differential operator H is not locally defined because of the operator H. Therefore, we define viscosity solutions in the global sense as follows:

Definition 4.1 A function v ∈ C(K) is called a viscosity supersolution of (4.12) on a subset K ⊆ K if for every x ∈ int K and every f ∈ C 1 (K) ∩ C 2 (x) such that v(x) = f (x) and v ≥ f on K, the inequality Lf (x) ≤ 0 holds. Definition 4.2 A function v ∈ C(K) is called a viscosity subsolution of (4.12) on a subset K ⊆ K if for every x ∈ int K and every f ∈ C 1 (K) ∩ C 2 (x) such that v(x) = f (x) and v ≤ f on K, the inequality Lf (x) ≥ 0 holds. Definition 4.3 A function v ∈ C(K)
is called a viscosity solution of (4.12) on a subset K ⊆ K if v is simultaneously a viscosity super-and subsolution on the subset K ⊆ K.

When, the subset K = 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) on a subset K ⊆ K if Lv ≤ 0 on int K. We add the adjective strict when Lv < 0 on the set int K. Lemma 4.4 A function v ∈ C(K) is a viscosity supersolution of (4.12) on a subset K ⊆ K if and only if, for every point x ∈ int K, the inequality Lf (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 of this result is the same than the proof of [START_REF] Yuri | Markets with Transaction Costs[END_REF]Lemma 4.2.4] except that we replace the notion of local minimum by the global one. 2 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 subjets J + , J -(see definition in [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]). But this is not the case in our formulation due to the nonlocal property of the integro-differential operator. Although, there is a link between the notion of viscosity solutions and super-and subjets as stated in [4, Section 7].

Theorem 4.6 Suppose that W takes finite values. Then, i) The Bellman function W is a viscosity supersolution to (4.12).

ii) The Bellman function W is a viscosity subsolution to (4.12). Theorem 4.6 implies that the Bellman function is a viscosity solution to (4.12). We shall see conditions under which W is finite.

Proof of Theorem 4.6

The proof is based on preliminary results we formulate before. The proof of the following lemma is given in [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]. Lemma 4.7 For every portfolio process V = V π , π ∈ Ax, the process

I V-∈∂K |∆Y |I [0,θ π ] is indistinguishable from zero.
Let us define for each n the compact set

Kn := {x ∈ K : |x| ≤ n} ∩ {x ∈ K : d(x, ∂K) ≥ 1 n }.
Note that (Kn) n≥1 is an increasing sequence whose union is int K. For every control π = (B, C) ∈ Ax and x ∈ int K, we define V θ n + as the stopped portfolio process.

V θ n + = V π,x,θ n + := V π,x I [0,θ n [ + V π,x θ n + I [θ n ,∞[, (4.14)
where θ n is the first instant when the portfolio exits Kn. We also define B θ n + similarly. Note that the value of V π,x θ n may be outside 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 4.7. From the dynamics of V π,x , we deduce that

V θ n + t = x + t 0 I [0,θ n ] (s)diag (V θ n + s-)(µsds + ΞdWs) + B θ n + t -C θ n t , + t 0 R d diag (V θ n + s-)zI [0,θ n ] (s) (p(dz, ds) -q(dz, ds)) .
We propose to study the quantity Xf,n t := e -βt f (V

θ n + t+ )I(V θ n + t-, ∆Y t∧θn ) + J π t , (4.15)
by means of the Ito formula. We set f (V

θ n + t+ )I(V θ n + t-, ∆Y t∧θn ) = 0 by convention when I(V θ n + t-, ∆Y t∧θn ) = 0 so that Xf,n t is well defined even if f is only defined on K. If Kn
is replaced by K we write Xf t instead of Xf,n t . We also introduce

V n t := V t+ I [0,θ n [ (t) + V θ n -I [θ n ,∞[ (t).
We have the following key result:

Lemma 4.8 (Itô expansion) Let f ∈ C 1 (K) ∩ C 2 (int K
) be an increasing function with respect to the order K . Then, we have

Xf,n t = f (x) + N t∧θ n + R t∧θ n + t∧θ n 0 e -βu L 0 f ( V n u-) -f ( V n u-)cu + U (cu) du,
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 to Xf,n t . To overcome this difficulty, instead of considering V θ n + , we study the process V θ n -defined by

V θ n -:= V I [0,θ n [ + V θ n -I [θ n ,∞[ . (4.16)
We also have a representation for V θ n -:

V θ n - t = x + t 0 I [0,θ n ] (s)diag (V θ n - s-)(µsds + ΞdWs) + B θ n - t -C θ n t + t 0 R d diag (V θ n - s-)zI [0,θ n ) (s) (p(dz, ds) -q(dz, ds)) .
For a sake of simplicity, we write

V t := V θ n - t+ = V θ n - t + ∆ + B θ n - t .
The Itô formula applied to the process e -βt f ( V t ) := e -βt f (V

θ n - t+ ) yields e -βt f ( V t ) = f (x) + t 0 e -βu f ( Vu -)d Vu -β t 0 e -βu f ( Vu -)I [0,θ n ] (s)du + 1 2 t 0 e -βs Tr A Vs -f ( Vs -)I [0,θ n ] (s)ds + s≤t e -βs f ( Vs) -f ( Vs -) -f ( Vs -)( Vs -Vs -) .
Using (4.16) and the fact that B

θ n - t + ∆ + B θ n - t = B t+ 1 [0,θ n ) + B θ n -1 [θ n ,∞) , we deduce that t 0 e -βu f ( Vu -)d Vu = t 0 e -βu f ( Vu -)I [0,θ n ) (u)diag ( V u-)(µudu + ΞdWu) + t 0 e -βu f ( Vu -)dB c u + t 0 e -βu f ( Vu -)cudu + s≤t e -βs I [0,θ n ) (s)f ( Vs -) ∆ + Bs + ∆Bs + t 0 R d e -βu f ( Vu -)diag ( Vs -)zI [0,θ n ) (s) (p(dz, ds) -q(dz, ds)) .
Note that Vs = Vs -+ diag Vs -∆Ys + ∆ + Bs + ∆Bs. We then rewrite

s≤t e -βs f ( Vs) -f ( Vs -) -f ( Vs -)( Vs -Vs -) = s≤t e -βs f ( Vs) -f ( Vs -+ diag Vs -∆Ys) I [0,θ n ) (s) + s≤t e -βs f Vs -+ diag ( Vs -)∆Ys -f ( Vs -) -f ( Vs -)diag ( Vs -)∆Ys I [0,θ n ) (s) - s≤t e -βs f ( Vs -) ∆ + Bs + ∆Bs I [0,θ n ) (s). Moreover, s≤t e -βs f Vs -+ diag ( Vs -)∆Ys -f ( Vs -) -f ( Vs -)diag ( Vs -)∆Ys I [0,θ n ) (s) = t 0 R d e -βs f ( Vs -+ diag ( Vs -)z) -f ( Vs -) -f ( Vs -)diag ( Vs -)z I [0,θ n ) (s) p(dz, ds).
Since I( Vs -, ∆Ys) = 1 for s < θ n , we may omit the indicator I within the operator H for s < θ n . We deduce that

e -βt f ( V t ) = f (x) + Ñt∧θ n + Rt∧θ n (4.17) + t∧θ n 0 e -βu L 0 f ( Vu -) -f ( Vu -)cu du, (4.18)
where

Ñt = t 0 e -βu f ( Vu -)diag ( V u-)ΞI [0,θ n ] (u)dWu (4.19) + t 0 R d e -βu f ( Vu -+ diag ( Vu -)z) -f ( Vu -) I [0,θ n ) (u) (p(dz, du) -q(dz, du)) .
The residual term is

Rt : = s≤t e -βs I [0,θ n ) (s) f Vs -+ diag ( Vs -)∆Ys + ∆B + s + ∆Bs -f Vs -+ diag ( Vs -)∆Ys + t 0 e -βu f ( Vu -)I [0,θ n ) (u)dB c u . (4.20)
The process R is decreasing due to the monotonicity of f with respect to K. Finally, observe that

Xf,n t -e -βt f ( V t ) = J π t + e -βθ n [f (V θ n + )I(V θ n -, ∆Y θn ) -f (V θ n -)] 1 t=θ n , = J π t + e -βθ n f (V θ n + ) -f (V θ n -+ diag (V θ n -)∆Y θ n ) I(V θ n -, ∆Y θn )1 t=θ n + e -βθ n f (V θ n -+ diag (V θ n -)∆Y θ n )I(V θ n -, ∆Y θn ) -f (V θ n -) 1 t=θ n . (4.21)
Plugging the r.h.s of (4.21) into (4.17), as the Lesbesgue measure dt does not charge any point, we conclude the theorem with

Ñt = t 0 e -βu f (Vu -)diag (V u-)ΞI [0,θ n ] (u)dWu (4.22) + t 0 R d e -βu [f (Vu -+ diag (Vu -)z)I(Vu -, ∆Yu) -f (Vu -)] I [0,θ n ] (u) (p(dz, du) -q(dz, du)) ,
and

Rt : = s≤t e -βs I [0,θ n ] (s) f Vs -+ diag (Vs -)∆Ys + ∆ + Bs + ∆Bs -f (Vs -+ diag (Vs -)∆Ys) I(Vs -, ∆Ys) + t 0 e -βu f (V u-)I [0,θ n ] (s)dB c s . (4.23) 2 Remark 4.9 If the function f ∈ C 2 (R d ),
we may directly apply the Ito formula to

X f t := e -βt f (V t+ ) + J π
t and we obtain the following result:

X f t = f (x) + N t + R t + t∧θ 0 e -βu L 0 f (Vu -) -f (Vu -)cu + U (cu) du,
where N is a local martingale and R is a decreasing process with R 0 = 0. Moreover,

X f t = Xf t if f vanishes outside int K.
We now formulate a strict supersolution property which is the key point to deduce that W is a global viscosity subsolution of the HJB equation by using the dynamic programming principle. We fix a closed ball such that Or(x) ⊆ O 2r (x) ⊆ int K and we define the stopping time τ π := τ π r as the exit time of V π,x from Or(x), i.e.

τ π := inf{t ≥ 0 : |V π,x t -x| ≥ r}. (4.24) Lemma 4.10 Let f ∈ C 2 (K) ∩ C 1 (K) be such that Lf ≤ -ε ≤ 0 on Or(x).
There exists a constant η := η(ε, r) and an interval (0, t 0 ] such that

sup π∈Ax E Xf,n t∧τ π ≤ f (x) -ηt, t ∈ (0, t 0 ], (4.25)
where t → Xf,n t∧τ π is defined by (4.15).

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 Kn. Therefore, τ π ≤ θ n hence we may apply Lemma

4.8 so that Xf,n t∧τ = f (x) + N t∧τ + R t∧τ + t∧τ 0 e -βu L 0 f (Vu -) + U * (V u-) du - t∧τ 0 e -βu U * (V u-) + f (Vu -)cu -U (cu) du, = f (x) + N t∧τ + R t∧τ + t∧τ 0 e -βu Lf (Vu -)du - t∧τ 0 e -βu U * (V u-) + f (Vu -)cu -U (cu) du + t∧τ 0 e -βu L 0 f (Vu -) + U * (V u-) -Lf (Vu -) du.
We deduce that

Xf,n t∧τ = f (x) + N t∧τ + R t∧τ + t∧τ 0 e -βu Lf (Vu -)du - t∧τ 0 e -βu U * (V u-) + f (Vu -)cu -U (cu) du, (4.26)
where R is a decreasing process such that R 0 = 0 and N is a local martingale. As it is shown in Lemma 8.3 [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF], the stopped process N τ is a martingale hence EN t∧τ = 0.

By assumption, Lf (y) ≤ -ε for all y ∈ Or(x) and so

Σ G (f (y)) ≤ -ε on [0, τ ]. It follows that f (y)k ≤ -ε|k| whatever k ∈ -K so that f (Or(x)) ⊆ int K * on [0, τ ]. In particular, for s ∈ [0, τ ], f (Vs -) Ḃc s ≤ -ε| Ḃc s |.
We deduce that the following term above (appearing in the expression (4.23) of R) is bounded as follows:

t∧τ 0 e -βu f (Vu -)I u≤θ Ḃc u d B c u ≤ -ε t∧τ 0 e -βu I u≤θ | Ḃc u |d B c u.
On the other hand, the other terms defining R in (4.23) can be estimated as follows:

f Vs -+ diag (Vs -)∆Ys + ∆B + s + ∆Bs -f (Vs -+ diag (Vs -)∆Ys) = (f (Vs + ) -f (Vs)) 1 ∆Bs=0 + (f (Vs + ) -f (V s-)) 1 ∆Bs =0 , = f (γs)(∆Bs + ∆ + Bs),
where γs ∈ [Vs, V s+ ] on the set {∆Bs = 0} and γs ∈ [V s-, V s+ ] on the set {∆Bs = 0}. Observe that γs ∈ Or(x) if s < τ . If s = τ we may assume without loss of generality that the controls ∆B + τ = ∆Bτ = 0 hence Vτ = V τ + as we consider the supremum given in (4.25). Therefore, we have

f Vs -+ diag (Vs -)∆Ys + ∆B + s + ∆Bs -f (Vs -+ diag (Vs -)∆Ys) I s≤τ ≤ -|∆B + s | + |∆Bs| I s≤τ .
Therefore, by Equality (4.26), we deduce that

E Xf,n t∧τ ≤ f (x) -εe -βt (t ∧ τ ) -e -βt EZ t ,
where

Z t : = t∧τ 0 r(cs, f (Vs -))ds + t∧τ 0 | Ḃc s |d B c s + s≤t∧τ |∆B + s | + |∆Bs| , r(c, p) : = U * (p) + pc -U (c). Recall that U * (p) = sup x∈C (U (x) -px) ≥ 0 since U (0) = 0. Moreover, by as- sumption, inf p∈f (Or (x)), c∈C, |c|=1 pc ≥ ε.
Since U (c)/|c| → 0 as |c| → ∞, we finally deduce that there exists a constant κ > 1

such that inf p∈f (Or (x)) r(c, p)) ≥ κ -1 |c|, ∀c ∈ C, |c| ≥ κ.
Therefore,

t∧τ 0 r(cs, f (Vs -))ds ≥ κ -1 t∧τ 0 I |cs|≥κ |cs|ds.
Moreover, the second integral defining Z dominates κ 1 B c t∧τ for some κ 1 > 0.

Indeed, recall that all norms in

R d are equivalent, in particular c -1 |.| ≤ |.| 1 ≤ c|.| for some c > 0 where |x| 1 := d i=1 |x i | and |.| is the Euclidean norm. It follows that c -1 B c ≤ Var B c ≤ c B c ,
where Var B c is the total variation of B c with respect to |.| 1 . At last, we have:

| Ḃc | 1 = d i=1 | Ḃc i | = d i=1 dB c i d B c = d i=1 dB c i dVar B c i dVar B c i d B c = dVar B c d B c .
The claimed property follows. We deduce some constant γ > 0 such that

EX f t∧τ ≤ f (x) -e -βt γ -1 E Z t ,
where

Z t := t ∧ τ + t∧τ 0 I |cs|≥κ |cs|ds + B t∧τ + .
Observe that

Z t ≥ t ∧ τ + B t∧τ + - t∧τ 0 I |cs|≤κ |cs|ds ≥ (1 -κ)t ∧ τ + B t∧τ + .
Using the stochastic formula of V , we immediately get the existence of a number t 0 > 0 and a measurable set Γ with P(Γ ) > 0 on which

|V π,x t -x| ≤ r/2 + δ B t+ , t ∈ [0, t 0 ],
whatever the control π = (B, C). Diminishing t 0 , we may assume without loss of generality that κt 0 ≤ r/(4δ). For any t ≤ t 0 , the inequality

B τ + ≥ r/(2δ) holds on the set Γ ∩ {τ ≤ t}. Therefore, if t ≤ t 0 , Z t ≥ (1 -κ)τ + 2κt 0 ≥ κt 0 ≥ t.
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. 2 Observe that, if n is large enough, τ π ≤ θ n hence X f,n t∧τ π does not depend on n.
Lemma 4.11 Suppose that W is continuous on int K. Then, for any stopping time τ ∈ T f , we have:

W (x) ≥ sup π∈Ax E J π τ (x) + e -βτ W (V x,π τ + )I τ <θ .
Proof. The proof is an adaptation of the proof of [4, Lemma 9.2]. In the latter, it suffices to replace ρ by ρ := inf{j ≥ 1 :

V x,π τ + ∈ O j }. 2
Lemma 4.12 Let T f be the set of finite stopping times. Then,

W (x) ≤ sup π∈Ax inf τ ∈T f E J π τ + e -βτ W (V x,π τ )I τ <θ . (4.27)
Proof. This lemma is proved in [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF] ). In that case, V t+ = V t for all t ≥ 0 and the Ito formula is valid as observed in Remark 4.9 but also in [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF].

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 4.10, there exists a constant η := η(ε) and an interval (0, t 0 ] such that

sup π∈Ax E J x,π t∧τ π r + e -β(t∧τ π r ) φ(V x,π t∧τ π r + )I(V x,π t∧τ π r -, ∆Y t∧τ π r ) ≤ φ(x) -ηt, t ∈ (0, t 0 ], (4.28)
where τ π := τ π r is given by (4.24). We may assume w.l.o.g. that r = r . Fix an arbitrary t ∈ (0, t 0 ]. Applying Lemma (4.12) to V + , we deduce that there exists π ∈ Ax such that

W (x) ≤ inf τ ∈T f E J x,π t∧τ + e -β(t∧τ ) W (V x,π t∧τ + )I t∧τ <θ + 1 2 ηt. (4.29)
As W ≤ φ and since

I(V x,π t∧τ π r -, ∆Y t∧τ π r ) = 0 implies that I t∧τ π r <θ = 0, we obtain from above that W (x) ≤ φ(x) -1 2 ηt, in contradiction since W (x) = φ(x). 2 5 Uniqueness theorem Definition 5.1 We say that a positive function ∈ C 1 (K) ∩ C 2 (int K) is a Lyapunov function if the following properties are satisfied: 1) (x) ∈ int K * and L 0 (x) ≤ 0 for all x ∈ int K, 2) (x) → ∞ as |x| → ∞.
In other words, is a classical supersolution of the truncated equation (excluding the term U * ), continuous up to the boundary, and increasing 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.

Condition Π 0 : ∀x ∈ int K, Π(z : x + diag xz ∈ ∂K) = 0.
Remark 5.2 This condition holds in the two dimensional case if the first component of the underlying asset is a bond B = 1 so that Π = δ 0 ⊗ π where we suppose that π does not charge the singletons. Theorem 5.3 Suppose that there exists a Lyapunov function and the Lévy measure Π is such that

Π(z : x + D xz ∈ ∂K}) = 0 ∀x ∈ int K.
Then the Dirichlet problem (4.12) has at most one viscosity solution in the class of continuous functions satisfying the growth condition

W (x)/ (x) → 0, |x| → ∞.
(5.30) Moreover, W is concave.

Proof of Theorem 5.3

Uniqueness is an immediate consequence of Theorem 5.4 below. Moreover, Theorem 5.5 ensures that the unique solution W is concave. Notice that Proposition 8.34 ensures the existence of a Lyapunov function under mild assumptions.

Theorem 5.4 (Maximum principle) Suppose that there exists a Lyapunov function and Π satisfies Condition Π 0 . Let K be a nonempty subset of K with nonempty interior. Let W be a continuous viscosity subsolution of the Dirichlet problem (4.12)

on K satisfying the growth condition

|W (x)|/ (x) → 0, |x| → ∞.
(5.31)

Let W be a continuous viscosity supersolution of the Dirichlet problem (4.12) on K such that W ≥ W on ∂ K. Suppose that either W ≥ 0 or W satisfies (5.31). Then, W ≥ W on K.

Proof. The proof is an easy adaptation of the proof of [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]Theorem 11.2 ]. Indeed, we follow the same reasoning but we consider the supremum of ∆non the set K. 2 Theorem 5.5 If the HJB equation (4.12) admits a unique global viscosity solution in C 0 (K), then the Bellman function is concave.

Proof. Consider the (modified) Bellman function W only defined by the (non empty) class of admissible strategies generating the portfolio processes evolving in (int

K)∪{0} on R + , i.e Ax := π ∈ A : V π t ∈ (int K) ∪ {0} , ∀ t ≥ 0 = ∅, 1 and
W (x) := sup π∈ Ax EJ π ∞ (x), x ∈ int K,
We also consider the corresponding stopping times

θ = θ x,π := inf{t : V x,
π : = πI [0,τ k ) + ∞ n=1 [(yn -V x,π τ k -, 0) + πn,k ]I [τ k ,∞[ I {ρ=n} I {V x,π τ k --yn∈K} I {τ k <θ} . For t < τ k , it is clear that V π t ∈ (int K) ∪ {0} as π ∈ Ax. Moreover, on {ρ = n}, V π τ k = yn ∈ int K provided that V x,π τ k --yn ∈ K and τ k < θ. Otherwise, V x,π τ k = V x,π
τ k -= 0 by Lemma 4.7. It follows that π ∈ Ax and we may conclude as in [4, 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 (the proof of it is given in Framstad et al. [START_REF] Framstad | Optimal consumption and portfolio in a jump diffusion market[END_REF]). Therefore, W is also concave. 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 (see also [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]) is a consequence of Lemma 4.8:

Lemma 6.1 Let f ∈ C 2 (R d
) be a non negative classical supersolution of (4.12) which vanishes outside int K, then f ∈ Φ.

Proof. First, we claim that a classical supersolution is increasing with respect to the partial ordering on K. Indeed, by the finite increments formula, for any x, h ∈ int K,

f (x + h) -f (x) = f (x + νh)h
for some ν ∈ [0, 1]. The right-hand side is non negative 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 all h ∈ K. As a classical supersolution of (4.12), f satisfies the inequality

L 0 f (Vu -) -f (Vu -)cu + U (cu) ≤ 0, ∀u < θ.
By Remark 4.9, as (N θ n t ) t≥0 is a local martingale bounded from below by -f (x), the local martingale N t := limn→∞ N θ n t is also bounded from below. Therefore, it is a supermartingale. In particular, N t is integrable and so is the non positive process

R + .∧θ 0 e -βu L 0 f ( Vu -) -f ( Vu -)cu + U (cu) du.
We then conclude that (X f t ) is a supermartingale. 2

The following proposition formulates finiteness and continuity up to boundary ∂K of the Bellman function in term of Φ. This corresponds to [START_REF] Vallière | Consumption-investment optimization problem in a Lévy financial model with transaction costs[END_REF]Lemma 8.1 ].

Proposition 6.2 a) If f ∈ Φ, then W ≤ f on K. Hence, if Φ = ∅, then W is finite. b) If x 0 ∈ ∂K
is such that, for every > 0, there exists f ∈ Φ with f (x 0 ) ≤ , then W is continuous at x 0 and W (x 0 ) := 0. Corollary 6.3 Let f ∈ C 2 (R d ) be a non negative classical supersolution of (4.12) which vanishes outside int K, then W is finite.

Proof. By Lemma 6.1, Φ = ∅. We conclude by Proposition 6.2. 2 7 Continuity of the value function Lemma 7.1 Let us consider x 0 ∈ int K. Then, lim sup λ→1 W (λx 0 ) ≤ W (x 0 ).

Proof. We may find πn = ( Bn , Cn ) ∈ A λx0 depending on λ such that

W (λx 0 ) ≤ E θ π n 0 e -βs U (c n s )ds + 1 n . (7.32)
Observe that, if πn = ( Bn , Cn ) ∈ A λx0 , then πn /λ := ( Bn /λ, Cn /λ) ∈ Ax 0 and θ πn /λ = θ πn . We rewrite πn as πn = λπ n where π n := (B n , C n ) ∈ Ax 0 . By (7.32), we deduce that

W (λx 0 ) ≤ E θ π n 0 e -βs U (λc n s )ds + 1 n . As λ -1 ∈]0, 1[ and x → U (x) is concave with U (0) = 0, we get that U (c n s ) = U λ -1 (λc n s ) + (1 -λ -1 ) × 0 ≥ λ -1 U (λc n s ).
We then get W (λx 0 ) ≤ λW (x 0 ) + n -1 and the conclusion follows as n → ∞. 2

Lemma 7.2 We have V t-∈ int K if t ∈ [0, θ[.
Proof. Suppose that V t-∈ ∂K for some t < θ. Then, by Lemma 4.7 and Assumption

5) of the model, we have

∆Y t = ∆B t = ∆C t = 0. Therefore, V t = V t-∈ ∂K hence a contradiction. 2
Note that we also deduce from Lemma 4.7 that the portfolio process exits K either in a continuous manner (in the case V θ-∈ ∂K) or due to a jump (in the case V θ-∈ int K).

Lemma 7.3 Let us consider a sequence xn → x 0 ∈ int K and T ∈ (0, ∞). Then, for any π ∈ Ax 0 , the sequence of portfolios

V (n) = V π,xn with initial values V (n) 0-= xn is such that T ∧ θ ≤ lim inf n θn ∧ T
where θ, θn are the stopping times defined by V := V π,x , V (n) respectively in (2.4).

Proof. Note that V (n) and V are uniquely defined by (2.3). Moreover, Therefore, there exists > 0 such that

V n := V (n) -V satisfies the dynamics d V ni t = V ni t-dY i t , V n 0-= xn -x 0 . Hence V ni t = (x i n -x i 0 )S i t or equivalently V n t = diag ((xn -x 0 )/S 0 ) S t . We deduce that sup t≤T |V (n) t -V t | ≤ |xn -x 0 | |S 0 | S *
d(Vs; ∂K) > , ∀s ∈ [0, T ∧ θ -δ].
Applying the triangular inequality d(Vs; ∂K) ≤ d(Vs; V n s )+d(V n s ; ∂K) we then deduce that

d(V n s ; ∂K) > -d(Vs; V n s ) ≥ - |xn -x 0 | |S 0 | S * T , s ∈ [0, T ∧ θ -δ].
It follows that for n large enough

d(V n s ; ∂K) > 0, s ∈ [0, T ∧ θ -δ].
We deduce that θ n ≥ T ∧ θ -δ if n is large enough. We have shown that whatever δ > 0 is small enough, there exists a.s. n 0 such that, for every

n ≥ n 0 , T ∧ θ -δ ≤ θ n . Therefore, T ∧ θ -δ ≤ inf n≥n0 θ n and finally T ∧ θ ≤ lim infn θn ∧ T . 2 Theorem 7.4 Assume that W (x 0 ) < ∞ where x 0 ∈ int K. Then, W is continuous at x 0 .
Proof. It suffices to show that W is both upper semicontinuous and lower semicontinuous.

• Let us first show that lim sup x→x0 W (x) ≤ W (x 0 ). In the contrary case, we have lim sup x→x0 W (x) > W (x 0 ). Note that lim sup x→x0 W (x) = lim k W (x k ) where x k is a subsequence converging to x 0 . As x 0 ∈ int K, we may assume that x k ∈ int K. We define

x k = (1 + k -1 )x 0 ∈ int K such that x k ∈ x 0 + int K if k is large enough. As x k → x 0 and x k -int K is an open set containing x 0 , there exists a subsequence xn k such that xn k ∈ x k -int K hence xn k x k . Since W is increasing with respect to
, we obtain that lim sup k W ( x k ) ≥ lim k W (xn k ) > W (x 0 ). On the other hand, by virtue of Lemma 7.1, W ( x k ) ≤ W (x 0 ) hence a contradiction.

• Let us show that lim infx n →x0 W (xn) ≥ W (x 0 ). For an arbitrary > 0, there exists

π ∈ Ax 0 such that W (x 0 ) ≤ + E θ π 0 e -βs U (cs)ds.
We then deduce T ∈ (0, ∞) such that

W (x 0 ) ≤ 2 + E θ π ∧T 0 e -βs U (cs)ds.
We introduce the stopping times θ n associated to the portfolios defined by the strategy π and the initial values xn. Observe the inequality

I s<lim infn θ n ∧T ≤ lim inf n I s≤θ n ∧T .
So, by virtue of Lemma 7.3 and Fatou's lemma, we then deduce that

W (x 0 ) ≤ 2 + lim inf n E θ n ∧T 0 e -βs U (cs)ds, W (x 0 ) ≤ 2 + lim inf n W (xn).
Since > 0 is arbitrarily chosen, we deduce that W (x 0 ) ≤ lim infn W (xn). 2

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 ij ) is diagonal with a ii = σ i , such that σ 0 = 0, µ 0 = 0 and σ i = 0, i = 1 . . . d, i.e. the first asset is a numéraire and the others are risky assets. We also suppose that the utility function is U (x) = uγ (xe 0 ), where uγ (t) = t γ /γ, γ ∈ (0, 1), and C = R + e 0 . In this subsection we work under the following condition

R d |z|Π(dz) < ∞.
For p ∈ int K * , we construct a Lyapunov function of the form v(x) := uη(px),where uη(x) = x η η and γ < η < 1. Note that the Bellman function is homogeneous of degree γ. Therefore, we choose η ≥ γ so that the Lyapunov function is growing faster than the Bellman function. If such a Lyapunov exists, the HJB equation admits a unique concave solution in C 1 by Theorem 5.3.

We have u η (x) = (px) η-1 p ∈ int K * as required for v to be a Lyapunov function.

Moreover,

L 0 v(x) = 1 2 A(x)p, p u η (px) + µ(x), p u η (px) -βuη(px) + R d uη(px + t pdiag (x)z)I(x, z) -uη(px) -u η (px) t pdiag (x)z Π(dz).
Let us denote the integral expression above by Hη(x). Our goal is to choose u such that L 0 v(x) ≤ 0 on intK, or equivalently

β ≥ sup x∈K p diag (x)µ u η (px) uη(px) + 1 2 A(x)p, p u η (px) uη(px) + Hη(x) uη(px) .

Let us introduce

Lη(p, x) = p diag (x)µ u η (px)

uη(px) + 1 2 A(x)p, p u η (px) uη(px) .
We have

Lη(p, x) = 1 2 η 1 -η d i=1 µ 2 i σ 2 i - 1 2 η(1 -η) d i=1 σ i p i x i px - µ i σ i (1 -η) 2 , ≤ 1 2 η 1 -η d i=1 µ 2 i σ 2 i .
We then choose β such that

β ≥ 1 2 η 1 -η d i=1 µ 2 i σ 2 i + sup x∈K Hη(x) uη(px) .
Put k(p, x) = 1 px p diag (x). We have

Hη(x) uη(px) ≤ R d (1 + k(p, x)z) η 1 {1+k(p,x)z>0} -1 -ηk(p, x)z Π(dz).
Note that k(p, tx) = 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 := K ∩ O 1 ,
where O 1 is the unit ball {x : |x| = 1}. It is easy to prove that this expression is bounded on B 1 . We then define

h(η) := inf p∈intK * sup x∈B1 R d (1 + k(p, x)z) η 1 {1+k(p,x)z>0} -1 -ηk(p, x)z Π(dz),
and we chose β such that

β > 1 2 η 1 -η d i=1 µ 2 i σ 2 i + h(η).
(8.33)

We may deduce the following: Then, the HJB equation admits a unique solution under the conditions of Theorem 5.3.

Proposition 8.1 i) If the condition (8.33) holds, then v(x) = (px) η η , 1 > η > γ is a Lyapunov function with respect to L 0 for some p ∈ K * with p 1 = 1.

Classical supersolution

The hypothesis of this section are those of Section 8 and the notations are the same. By Lemma 6.1, if we construct a non negative classical supersolution f ∈ C 2 (R d ) of (4.12) which vanishes outside int K, then f ∈ Φ. Therefore, by Proposition 6.2, W is finite.

Let us define v(x) := kuγ (px). By definition,

U * (v (x)) = sup y∈C U (y) -v (x)y = sup y∈C U (y) -ku γ (px)py , = sup y1≥0 (y 1 ) γ γ -k(px) γ-1 p 1 y 1 Then, with p 1 = 1, U * (v (x)) = k γ γ-1 (px) γ (1/γ -1). Since u ≥ 0, G ⊆ -K, and p ∈ K * \{0}, we have Σ G (v (x)) = sup x∈G u γ (p.x)px ≤ 0.
Our goal is to choose p, k so that, on intK, we have

k γ γ-1 (px) γ (1/γ -1) + k 2 A(x)p, p u γ (px) + k µ(x), p u γ (px) -kβuγ (px) + R d uγ (px + t pdiag (x)z)I(x, z) -uγ (px) -u γ (px) t pdiag (x)z Π(dz) ≤ 0.
Adapting the reasoning of the previous subsection, we choose β such that

β > 1 2 γ 1 -γ d i=1 µ 2 i σ 2 i + h(γ) + k 1 γ-1 (1 -γ).
v(x) = k γ (px) γ ,
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

κ M := 1 1 -γ β - γµ 2 2σ 2 (1 -γ) -h(γ) > 0.
Then, there exists p ∈ K * with p 1 = 1 such that the function f (x) = m(px) γ is a classical supersolution of the HJB equation and m > (1/γ)κ γ-1 M .

Combining Propositions 8.1 and 9.1, we obtain the following result Corollary 9.2 Let h * be defined as h * (γ) := max(h(γ), h(γ)).

If β > 1 2 γ 1 -γ d i=1 µ 2 i σ 2 i + h * (γ), (9.36)
there exists a classical supersolution to the HJB equation and a Lyapunov function with higher growing order than the Bellman function one.

Application to the two-asset model

We investigate the properties satisfied by the Bellman function and we construct an optimal policy in the case 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 a constant, and the risky asset follows a geometric Lévy process:

dS 1 t = 0, S 1 0 = 1, dS 2 t = S 2 t-µdt + σdW t + R x(p(dy, dt) -q(dy, dt)) , S 2 0 = 1,
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

Condition (I): ∞ -1 max (1, |t|)π(dt) < ∞.
(10.37)

The inequality (10.37) ensures that π is a finite measure such that the associated Lévy process has a finite activity. This implies that

Y 1 = 0, Y 2 t = σW t + µt + Nt i=1 χ i where N t = ∞ n=1 1
Tn≤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:

dV 1 t = dL 21 t -(1 + λ 12 )dL 12 t -c t dt, dV 2 t = V 2 t-µdt + σdW t + R
x(p(dy, dt) -q(dy, dt))

+ dL 12 t -(1 + λ 21 )dL 21 t ,
where L ij , i, j = 1, 2, are the transfer processes supposed to be làdlàg and (λ ij ) i,j=1,2 are the transaction costs coefficients. We rewrite the dynamics of a portfolio process under the vector form:

dV t = diag V t-μdt + σdW t + z(p(dz, dt) -Π(dz)dt) + dB t -dC t .
In the following, we use the notations μ = (0, µ) , σ = (0, σ) and, with z = (x, y) ,

p(dz, dt) = δ 0 (dx)dt ⊗ δ ∆Yt (dy)dN t , Π(dz) = λδ 0 (dx) ⊗ π(dy).
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

dB t = (dL 21 t -(1 + λ 12 )dL 12 t , dL 12 t -(1 + λ 21 )dL 21 t ) .
The optimization problem reads as E θ 0 e -βs u(cs)ds → max, 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: .38) In this framework, the solvency cone K is simply a sector generated by the vectors

W (νx) = ν γ W (x), ∀x ∈ K, ν ≥ 0. ( 10
g 1 = (1 + λ 12 )e 1 -e 2 , g 2 = (1 + λ 21 )e 2 -e 1 . The dual cone of K is K * = cone {p 1 , p 2 } with p 1 = (1 + λ 12 )e 2 + e 1 , p 2 = (1 + λ 21 )e 1 + e 2 .
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:

F (W (z), W (z), H(W, z), W (z), z) = 0, (10.39) W (z) = 0 on ∂K, (10.40)
where

F 0 (X, p, H, W, z) : = 1 2 σ 2 yX 22 + µyp 2 + H -βW, (10.41)
F (X, p, H, W, z) : = max F 0 (X, p, H, W, z) + U * (p), Σ G (p) . (10.42)
By simple computation, we get that where I(z, t) := 1 {(x,y+yt)∈intK} , z = (x, y) . We now formulate properties satisfied by the operator H(u, z):

U * (p) = 1 γ (p 1 ) γ γ-1 , if p = (p 1 , p 2 ) . Moreover Σ G (p) ≤ 0 ⇔ max{-g 1 p, -g 2 p} ≤ 0,
Lemma 10.1 i) If u ∈ C 1 (K) ∩ C 1 (K \ Re 1
) is a non negative concave function such that Σ G (u ) ≤ 0 on K (i.e. u is increasing with respect to the natural order on K), then the operator

H(u, z) is non positive. ii) If u ∈ C 1 (K \ Re 1 ) is homogeneous of degree γ < 1 and π(R) < ∞ then H(u, .) is continuous on K ∩ {(x, y) : y > 0}. Moreover, if uy(x, 0 + ) exists and is finite, then so is H(u, (x, 0 + )). iii) If u ∈ C 2 (K \Re 1 ) is homogeneous of degree γ < 1 and π(R) < ∞ then H(u, .) is C 1 on K ∩ {(x, y) : y > 0}.
Moreover, if uy(x, 0 + ), uyy(x, 0 + ) exist and are finite, then so is H(u, (x, 0 + )).

Proof. i) We aim to prove that P (z, t) ≤ 0 where P (z, t) := u(x, y(1 + t))I(z, t) -u(x, y) -u y (x, y)yt. (10.43) It is clear that P (z, t) = 0 in the case where y = 0. Otherwise, since u ≥ 0, we deduce that P (z, t) ≤ u(x, y(1 + t)) -u(x, y) -u y (x, y)yt. So,

P (z, t) ≤ u(x, y(1 + t)) -u(x, y) -u y (x, y)yt, ≤ u y (x, θx) -u y (x, y) yt,
where θx ∈ [y, y(1 + t)]. Since the function y → u(x, y) is concave, y → u y (x, y) is non increasing on C 1 (K \ Re 1 ) hence P (z, t) ≤ 0.

ii) Assume that (x, y) ∈ K and y > 0. Let zn := (xn, yn) ∈ K, yn > 0 be a sequence convergent to z := (x, y). Since u is locally Lipschitz on K \ Re 1 , we have for n large enough P ((xn, yn), t) -P ((x, y), t) = o(1) uniformly on t ∈ (-1, 0). We now consider t ≥ 0. Using the homogeneity of u, we obtain

u(xn, yn + ynt) -u(x, y + yt) = (1 + t) γ (u( xn 1 + t , yn) -u( x 1 + t , y)) = (1 + t) γ o(1)
,

uniformly on t ≥ 0. Finally, u(xn, yn) -u(x, y) = o(1),
and, similarly,

u y (xn, yn)ynt -u y (x, y)yt = o(t).
We deduce that P ((xn, yn), t) -P ((x, y), t) = o(1)max (1, |t|).

This implies that H(u, zn) → H(u, z) when n → ∞. The case y = 0 is proved similarly, using the boundedness of (uy(zn)) n≥1 when yn → 0.

iii) The proof is similar than ii). 2

Let us also suppose the following:

Condition (II): β > γµ 2 2σ 2 (1 -γ) + h * (γ),
where the function h * is given in Corollary 9.2. By Corollary 9.2, Condition (I) and (II) implies the existence of a Lyapunov function with higher growing order than the Bellman function and a classical supersolution to the HJB equation. Therefore, using Proposition 6.2, we deduce that the Bellman function is finite on K and continuous up to the boudary ∂K, see Theorem 7.4. Therefore, we deduce by Theorem 5.3 that the HJB equation admits a unique concave solution in the class C 1 (K). Actually, we prove that W ∈ C 2 (K) in Subsection 10.1. To do it, 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 [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF] and [START_REF] Yuri | Markets with Transaction Costs[END_REF]) may be extended to the processes with jumps, as we shall see.

Theorem 10.2 The cone K can be splitted into three non-empty open disjoint cones K i , i = 0, 1, 2, such that K = K0 ∪ K1 ∪ K2 with K0 = cone {g 1 , g2 }, K1 = cone {g 1 , g1 } and K2 = cone {g 2 , g 2 } for some vectors g1 , g2 . Moreover, K 1 contains cone (g 1 , e 1 ) and

K 2 contains cone (2µσ -2 (1 -γ) -1 (1 + λ) -1 g 2 + e 2 , g 2 ) . On K 1 ∪ K 2 , W is C ∞
and is given by

W (z) = a 1 u(p 1 z), on K 1 , (10.44) W (z) = a 2 u(p 2 z), on K 2 , (10.45)
where a 1 , a 2 are some constants such that

a 1 ≥ β 1 -γ γ-1 , a 2 ≥ β 1 -γ γ-1 1 (1 + λ) γ .
Proof. We adapt the proof of [START_REF] Yuri | Markets with Transaction Costs[END_REF]Proposition 4.8.2]. With ϕ(x) := a 1 u(p 1 x), we need to show that

L 0 ϕ(x) + u * (ϕx) + H(ϕ, x) ≤ 0,
for all x ∈ cone (g 1 , e 1 ). By Lemma 10.1 (i), we have H(ϕ, x) ≤ 0. Moreover, as in the proof of [9, Proposition 4.8.2], L 0 ϕ(x) + u * (ϕx) ≤ 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 subset K = cone (g 1 , e 1 ). Moreover, by construction, W = ϕ on K. By Theorem 5.4, we deduce that W ≤ ϕ on K. Similarly, we follow the proof of [9, Proposition 4.8.2, statement b)] using Theorem 5.4. We then define K 1 , K 2 as the largest sectors on which (10.44) and ( 10.45) hold. 2

Lemma 10.3 If W (e 1 ) = 1 γ β 1 -γ γ-1
, then the axis of the abscises is not the common boundary of K 1 and K 2 .

Proof. We follow the proof of [START_REF] Yuri | Markets with Transaction Costs[END_REF]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 ). Then, we may deduce the value of a 2 using the assumption of the lemma. Hence, we deduce with z = (x, y) that

L 0 ψ(z) + u * (ψx(z)) = a 2 (p 2 z) γ-1 y 1 2 σ 2 (γ -1) y p 2 z + µ + H(ψ, z).
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 cxy provided that y is sufficiently close to 0 and cx > 0 only depends on x, a 2 , γ, µ and λ. On the other hand, we may write

H(ψ, z) = y 2 2 M -1 ψ (z + yθ t )t 2 π(dt) + ∞ M P (z, t)π(dt), (10.46)
where |θ t | ≤ |t| for all t ≥ -1, and P is defined by (10.43). We choose M large

enough such that π([M, ∞)) is close to 0. Precisely, as ψy(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

∞ M P (z, t)π(dt) ≤ xy
where x is arbitrarily small provided that we choose M large enough. We fix M such that x ≤ cx/3. If t ∈ (-1, M ), the factor ψ (z + yθ t )|t| appearing in the first term of the r.h.s. of (10.46) 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)| ≤ (2cx/3)y provided that y is small enough. This implies that L 0 ψ(z) + u * (ψx(z)) > 0 for some z small enough. This yields a contradiction. 2 10.1 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) := W (1 -t, t), t ∈ ∆ := [-1 λ , 1 + 1 λ ]
, so that we may reconstruct W from ψ by the formula

W (x, y) = (x + y) γ ψ( y x + y ), (x, y) ∈ int K.
As in [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF] and [START_REF] Yuri | Markets with Transaction Costs[END_REF], we may show that ψ is the viscosity solution to the new HJB equation obtained by the change of variables above:

max i=0,1,2 l i (f ) = 0, (10.47)
with the two first-order operators

l 1 (f ) := -λγf + (1 + λz)f , l 2 (f ) := -λγf -(1 + λ -λz)f ,
and the second-order operator

l 0 (f ) := f 2 f + f 1 f + f 0 f + 1 -γ γ [γf -zf ] γ γ-1 + H(z, f, f (z), f (z)),
where

f 2 (z) : = 1 2 σ 2 z 2 (1 -z) 2 , f 1 (z) : = -σ 2 (1 -γ)z(1 -z)(z -θ), θ := (1 -γ) -1 µσ -2 , f 0 (z) : = 1 2 σ 2 γ(γ -1)z 2 + γµz -β, H(z, f, v, v ) : = ∞ -1 (1 + zt) γ f ( z + zt 1 + zt )1 (1-z,z(1+t))∈K -v -zt(γv + (1 -z)v ) dπ(t).
Recall that W is concave by Theorem 5.5. Then, the function ψ defined above is also concave on ∆ and its derivatives ψ , ψ exist almost everywhere. Therefore, (10.47) holds in the classical sense as stated in [4, 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 10.2, 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 ∆.

We now adapt some results from the literature about regularity of the Bellman function. We mainly focus on the extra term H(z, ψ, ψ(z), ψ (z)).

Proposition 10.4 The function ψ is continuously differentiable on the interval int ∆ except, maybe at zero. If ψ has a discontinuity at 0, then

ψ(0) = 1 γ 1 -γ β 1-γ = 1 γ κ 1-γ * . (10.48)
Proof. We adapt the proof of [9, Lemma 4.8.6]. The only difference is due to the operator H. Clearly, this does not change the proof when z / ∈ {0, 1} as H takes finite values. When z = 1, we need to show the extra property that H(zn, ψ, ψ ) → H(1, ψ, D ± ψ) as zn → 1 with zn < 1 and zn > 1 respectively. To see it, observe that 1

(1-z,z(1+t))∈K = 1, π(dt) a.s., if z ∈ [0, 1]. If the sequence zn > 1 is such that zn → 1, then 1 (1-zn,zn(1+t))∈K → 1 if t > -1. Moreover, if z bounded in a neighbourhood of 1, the quantity z(1 + t)/(1 + zt) is uniformly bounded in t > -1
and in z. As D ± ψ(1) exists, we then deduce that the integrand defining H(zn, ψ, ψ ) is uniformly bounded by c min(1, |t|) for some constant c > 0 independent of n. This implies that the dominated convergence theorem applies. Finally, when zn ↑ 1, we get that H(zn, ψ, ψ ) → H(1, ψ, D -ψ). As in [9, Lemma 4.8.6], we finally get that

f 0 (1)ψ(1) + u * (γψ(1) -D -ψ(1)) + H(1, ψ, D -ψ) ≤ 0, (10.49) f 0 (1)ψ(1) + u * (γψ(1) -D + ψ(1)) + H(1, ψ, D + ψ) ≤ 0, (10.50) where u * (x) := x γ(γ-1) -1
. Let α ∈ (0, 1) be such that p = αD -ψ + (1 -α)D + ψ. By (10.49) and ( 10.50), we deduce that

f 0 (1)ψ(1) + αu * (γψ(1) -D -ψ(1)) + (1 -α)u * (γψ(1) -D -ψ(1)) + H(1, ψ, p) ≤ 0. Since u * is strictly convexe, we get that f 0 (1)ψ(1)+u * (γψ(1)-p)+H(1, ψ, D + ψ) < 0, in contradiction with [9, Lemma 4.8.6, Inequality (4.8.20 ]
. The conclusion follows in the case where z = 1.

In the case of z = 0, we also need to verify the convergence of H(zn, ψ, ψ ) to H(0, ψ, ψ ) as zn → 0 with zn ∈ (0, 1). In this case, 1 (1-zn,zn(1+t))∈K = 1 for all n if t > -1. We verify that the quantity z(1 + t)/(1 + zt) is uniformly bounded in t > -1 and z ≥ 0. With z > 0, we have 1 + tz ≥ z + tz = z(1 + t) > 0 hence 0 ≤ z(1 + t)/(1 + zt) ≤ 1. Therefore, the dominated convergence theorem also applies and we get that f 0 (0)ψ(0) + u * (γψ(0)) + H(0, ψ, 0) ≤ 0 and finally f 0 (0)ψ(0) + u * (γψ(0)) + H(0, ψ, 0) = 0 if ψ admits a discontinuity at zero. Since H(0, ψ, 0) = 0, the conclusion follows. 2

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 10.5 The value function is C 1 on int K \ R + e 1 . If ψ is not C 1 on R + e 1 ,
then (10.48) holds. Furthermore, even if ψ is not C 1 on R + e 1 , the partial derivative

Wx is defined and continuous, and the one-sided derivatives Wy(x, 0±) are also defined and satisfy the one-sided continuity conditions

Wy(x, 0±) = lim (ξ,η)→(x,0±) Wy(ξ, η) = x γ-1 γψ(0) + D ± ψ(0) , x > 0.
Proof. The first claim is an immediate consequence of Proposition 10.4. When y = 0, W (x, y) = x γ ψ(0) so that Wx(x, y) exists and is given by Wx(x, y) = γx 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 Wx(x, y) = γ(x + y) γ-1 ψ(z(x, y)) + (x + y) γ zx(x, y)ψ (z(x, y)).

(10.51) 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 (10.51) holds everywhere. In particular, Wx is continuous.

Similarly, when y = 0, i.e. z(x, y) = 0, Wy(x, y) exists and is given by Wy(x, y) = γ(x + y) γ-1 ψ(z(x, y)) + (x + y) γ zy(x, y)ψ (z(x, y)).

(10.52)

The claim follows.

0 ≤ z 1 < z 2 < 1 + 1 λ , such that K 0 = {(x, y) ∈ int K : z 1 < y x + y < z 2 }.
Moreover, we have

ψ(z) = κ 1 (1 + λz) γ , z ∈ [-λ -1 , z 1 ], ψ(z) = κ 2 (1 + λ -λz) γ , z ∈ [z 2 , 1 + λ -1 ].
for some constants κ i , i = 1, 2.

Proposition 10. [START_REF] Guasoni | The Fundamental Theorem of Asset Pricing under Transaction Costs[END_REF] The point e 1 belongs to int K 1 .

Proof. We adapt the proof of [START_REF] Yuri | Markets with Transaction Costs[END_REF]Proposition 4.8.8]. In the case where γW (e1) > κ γ-1 * , the proof is based [9, Lemma 4.2.5.] that we need to verify for global viscosity solutions, i.e. when replacing local minimum by global minimum. As a real-valued continuous function admits a global minimum on a compact subset, we may easily adapt the proof provided that [START_REF] Yuri | Markets with Transaction Costs[END_REF]Lemma 4.2.4] also holds. To see it, we adapt the proof of the latter replacing local by global domination. At last, if γW (e1) > κ γ-1 * , we observe that H(zn, ψ, ψ ) → 0 as zn → 0. This is indeed shown in the last part of the proof of Proposition 10.4. We then conclude. 2 Corollary 10.8 The function ψ is C 1 . Proposition 10.9 We have l 1 ψ(z) < 0 and l 2 ψ(z) < 0 for all z ∈ (z 1 , z 2 ).

Proof. We follow the reasoning given in [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Section 6]. Using the same notations, as ψ is C 1 , we deduce by [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Proposition 6.2.] that the sub differential is a singleton, i.e. ∂W (x, y) = {W (x, y)}. Therefore, the functions θ + and θ -defined in [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Section 6] coincide with the function θ2 := (1 + λ) -1 g 2 W 2 . We parametrize the half-line D 2 originating at (0, g 1 ) and parallel to (0, g 2 ) as in [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF], i.e.

D 2 := (x(ρ), y(ρ)) := 1 - ρ 1 + λ , - 1 1 + λ + ρ , ρ ≥ 0 .
Since W is concave, [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Property (6.6)] holds hence the function ρ → θ 2 (x(ρ), y(ρ))

is non increasing as stated by [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Lemma 6.3,(6.7)]. Let us define the coefficient3 ρ 2 := inf{ρ : θ 2 (x(ρ), y(ρ)) = 0.

Observe that ρ 2 ∈ (0, ∞) by Theorem 10.2. By [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]Lemma 6.3 ,(6.8)], and by continuity, we get the property θ 2 (x(ρ), y(ρ) = 0 for all ρ ≥ ρ 2 since θ 2 ≥ 0 on K and ρ → θ 2 (x(ρ), y(ρ)) is non increasing. Moreover, it is trivial that θ 2 (x(ρ), y(ρ)) > 0 for all ρ < ρ 2 . This implies that the conic sector generated by the points of D 2 parametrized by ρ ≥ ρ 2 coincides with K 2 of Theorem 10.2. Similarly, We parametrize the half-line D 1 originating at the (0, g 2 ) and parallel to (0, g 1 ):

D 1 = (x(ρ), y(ρ)) := (- 1 1 + λ + ρ, 1 - ρ 1 + λ ), ρ ≥ 0 .
We also define θ 1 := (1 + λ) -1 g 1 W and

ρ 1 := inf{ρ : θ 1 (x(ρ), y(ρ)) = 0.
Similarly, we obtain that the conic sector generated by the points of D 1 parametrized by ρ ≥ ρ 1 coincides with K 1 of Theorem 10.2. Moreover, by definition of ρ 1 , we get that θ 1 (x(ρ), y(ρ)) > 0 for all ρ < ρ 1 . This means that l 1 (ψ) < 0 and l 2 (ψ) < 0 on (z 1 , z 2 ). 2

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. (10.53) where H(z, v, v ) = H(z, ψ, v, v ). Although, by doing so, ψ is no more a viscosity solution of the (new) local operator but it is only a viscosity solution in a weak sens we precise: Definition 10.10 A function v ∈ C(K) is a weak viscosity supersolution (resp. subsolution) of (4.12) on a subset K ⊆ K if and only if, for every point x ∈ int K, the inequality Lf (x) ≤ 0 (resp. Lf (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.

l0 (f ) := f 2 f + f 1 f + f 0 f + 1 -γ γ [γf -zf ] γ γ-1 + H(z, f (z), f (z)),
As usual, weak viscosity solution is both a weak viscosity supersolution and weak viscosity subsolution. Adapting the proof of [9, Lemma 4.2.5], we get the following: Proof. It suffices to repeat the proof of [START_REF] Yuri | Markets with Transaction Costs[END_REF]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. 2 Proposition 10.12 Suppose that l 1 ψ(z 0 ) < 0 and l 2 ψ(z 0 ) < 0 at some point z 0 ∈ (0, 1

+ λ -1 ]\{1}. Then, if π(R) < ∞, ψ ∈ C 2 (z 0 ).
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 equation

l0 ψ = 0 on [a, b].
First, as ψ is a viscosity supersolution to the equation max i=0,1,2 l i ψ = 0, this is also a viscosity supersolution to l 0 ψ = 0. Indeed, max i=0,1,2 l i φ ≤ 0 implies l 0 φ ≤ 0 for any test function φ.

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 consider f r := f ξ r +ψ(1-ξ r ) where, by the one dimensional version of Lemma 11.1, ξ r ∈ [0, 1] is infinitely differentiable, vanishes outside 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 = (ψ -f )ξ r . Since ξ r ≥ 0, (ψ -f r )(z) ≥ (ψ -f )(z 0 )ξ r (z). As (ψ -f )(z 0 ) < 0 and ξ r ≤ 1, (ψ -f )(z 0 )ξ r (z) ≥ (ψ -f )(z 0 ). This implies that (ψ -f r )(z) ≥ (ψ -f r )(z 0 ), i.e. ψ -f r
admits a global minimum at z 0 . Therefore, l 0 f 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 that f 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. [START_REF] Alvarez | Convex viscossity solutions and states constraints[END_REF] 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.

Precisely, if z 0 > 1, h is defined on D h = [1, ∞) with h = 0 on [1, (z 0 +1)/2]. If z 0 < 1, h is defined on D h = [0,
Therefore, the integral

∞ -1 (1 + z 0 t) γ h( z 0 + z 0 t 1 + z 0 t )1 (1-z0,z0(1+t))∈K dπ(t)
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 0 f r (z 0 ) ≤ 0 and f ≥ ψ -h on z ∈ D h , we then deduce that l0 f (z 0 ) ≤ 0 as → 0, i.e. we have replaced f 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 equation l0 ψ = 0 on [a, b] hence ψ is finally a weak viscosity solution of the equation l0

ψ = 0 on [a, b]. Let us define the local operator ζ(f, z) := l0 f (z) -f 2 (z)f (z). Since π(R) < ∞,
we deduce by Corollary 10.8 and Lemma 10.1 (ii) that z → ζ(ψ, z) is continuous on

(0, 1 + λ -1 ]. We deduce that z → G(ψ (z), ψ(z), z) := -ζ(f, z)/f 2 (z) 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 10.11,

ψ ∈ C 2 [a, b]. 2
The following result corresponds to [START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF]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 10.1 i).

Proposition 10.13 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 Finally, we deduce the following: Corollary 10.15 Suppose that π(R) < ∞. Then, the value function W is C 2 .

L 0 (W ) + U * (Wx) = 0 on K 0 \ Re 2 in

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.

Theorem 10.16 Suppose that the boundaries of K 0 are different from the x-and yaxes. Let (x, y) ∈ K0 , then the Skorokhod problem (11.62) with σ(V t ) := (-Wx(V t ) 1 γ-1 , V t ) admits a unique solution. Moreover, the portfolio process V participating in the solution of this problem is an optimal portfolio. An optimal strategy is given by the formula

B t = t 0 g(Vs)dks, (10.54)
c t = W 1 γ-1 x , (10.55)
where W is the Bellman function.

Proof. Existence of a solution to the Skorohod problem holds by Theorem 11.4.

Note that θ π = ∞ since V + t ∈ K 0 , ∀ t, and Wx is positive (hence (10.55) makes sense). We shall only consider the case where K 0 is included in the first quadrant. Otherwise, we refer to Remark 10.18. By Propositions 10.9 and 10.13, the function ψ coincides on the interval (z 1 , z 2 ) with a C 2 -function ψ defined on (-λ -1 , 1 + λ -1 ). Indeed, it suffices to replace ψ by suitable parabolic functions outside (z 1 , z 2 ). In particular, since ψ is Lipschitz on [z 1 , z 2 ] and W (z) = (x + y) γ ψ(z) where z = y/(x + y) on K 0 , we also deduce that W is also Lipschitz on K 0 . Moreover, by [9, 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 4.8, since the term R of the expansion is negative, we have

e -βt W (V + t ) + J π t ≤ W (x, y) + N t + t 0 e -βu L 0 W (Vu -) -W (Vu -)cu + U (cu) du.
Moreover, when c is defined by (10.55), we have Wx(Vu -)cu + U (cu) = U * (cu). Therefore, the last integral term in the equality above is zero by virtue of Proposition 10.14.

It remains to prove that N is a martingale and

lim n→∞ Ee -βtn W (V + tn ) → 0, (10.56)
for a sequence of real numbers tn → ∞. To prove (10.56), we observe that |W (z)| ≤ κ|z| γ and |Wx(z)| ≤ κ|z| γ-1 where κ is an upper bound of W and Wx on the intersection ∆ 0 of the set K 0 with the line x + y = 1. This is deduced from the continuity of W on ∆ 0 and the fact that ψ (0+) < ∞. We deduce the existence of a constant κ such that

E ∞ 0 e -βt W (V + t )dt ≤ κE ∞ 0 e -βt |V + t | γ dt ≤ κE ∞ 0 e -βt u(c t )dt ≤ κW (x). (10.57)
Since W is finite, this implies the existence of a sequence tn ↑ ∞ such that (10.56) holds. Details of this assertion are given in Lemma 10.17. We now prove that N t is a true martingale. Indeed, by a similar argument, we have

|zWy(z)| ≤ κ|z| γ ≤ (1 + |z|), z ∈ K 0 .
Hence, we infer that the stochastic process

• 0 e -βu Wy(V 2 u-)V 2
u-σdWu is a martingale. The second process defining N is the integral with respect to the Lévy measure. Observe that, for each fixed s, we have I(Vs -, z) = 1 (because V + s ∈ K0 , ∀ s). Moreover, using To complete the proof of the theorem, we need the following lemma Lemma 10.17 Suppose that ∞ 0 Xudu < ∞ where Xu = Ee -βu W (V u+ ) ≥ 0. Then lim t→∞ X t → 0.

Proof. Observe that e -βt W (V + t ) + J π t = W (x) + N t . With Yu = e -βu W (V u+ ), we get that for u ≥ s, Yu -Ys = Nu -Ns -(J π u -J π s ) where J π u -J π s ≥ 0. Since N is a martingale, we deduce that Xu -Xs ≤ 0, i.e. X is decreasing. Therefore, the integrablity of ∞ 0 Xudu ensures that lim t→∞ X t → 0. Indeed, if lim t→∞ X t → c > 0, then ∞ 0 Xudu ≥ ∞ 0 cdu = ∞, hence a contradiction. 2 Remark 10.18 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 point x 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 ∆B 0 := x -x to the optimal strategy given by the Skorokhod problem with the initial point x.

11 Appendix: the Skorokhod problem

Skorokhod problem for continuous diffusion processes

The construction of the optimal control for the two-dimensional optimal consumption problem we consider in Section 10 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(x) = -g i on (∂K 0 ∩∂K i )\{0} and γ(0) = 0. Let Y be the process Y t = (Y 1 0 , Y 2 0 ) + (t, W t ), t ≥ 0, where W is a standard Brownian motion. Let σ = R 2 → R 2 × R 2 be a matrix-valued function which is Lipschitz-continuous.

On the closed cone K0 of Section 10, 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 ∈ K0 and zero, such that k is non decreasing and dV t = σ(V t )dY t + γ(V t )dk t , (11.58) dk t = I Vt∈∂K0 dk t , (11.59) V t ∈ K 0 , ∀t ≥ 0. The setting of this subsection is given in Subsection 11.1. 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

N t = Ñt k=1 ∆N T k ,
where ∆N T k are i.i.d. random variables and Ñt = 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

T

  where S * T := sup t≤T |S t |. On the other hand, if δ > 0 is small enough, we may writeinf s≤T ∧θ-δ d(Vs; ∂K) = lim n d(Vs n ; ∂K), where sn ∈ [0, T ∧ θ -δ] converges to s 0 ∈ [0, T ∧ θ -δ] by compactness argument.First assume that sn ↑ s 0 . Then, inf s≤T ∧θ-δ d(Vs; ∂K) = d(Vs 0 -; ∂K) > 0, by virtue of Lemma 7.2. Otherwise, we get that inf s≤T ∧θ-δ d(Vs; ∂K) = d(Vs 0 + ; ∂K) > 0.

  that Condition (9.35) holds. Then, the function

  , y(1 + t))I(z, t) -u(x, y) -u y (x, y)yt π(dt),

Lemma 10 . 11

 1011 Let ψ ∈ C 1 (a, b) be a weak viscosity solution of the (local) equationψ (z) = G(ψ (z), ψ(z), z) on a nonempty subinterval (a , b ) ⊆ (a, b). Suppose that z → G(ψ (z), ψ(z), z) is continuous on [a , b ]. Then, ψ ∈ C 2 (a , b) and the equation holds in the classical sense.

  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 10.14 The second derivative Wyy is well defined and is continuous accross R + e 2 . Moreover, W satisfies the equation L 0 (W ) + U * (Wx) = 0 on K 0 in the classical sense.

0 Re

 0 the finite Taylor expansion, we getW ( Vs -+ diag ( Vs -)z) -W ( Vs -) ≤ |W (η)||diag ( Vs -)z|, where η ∈ [ Vs -, Vs -+ diag ( Vs -)z] satisfies |η| ≤ | Vs -|(1 + |z|). It follows that W ( Vs -+ diag ( Vs -)z) -W ( Vs -) ≤ κ|η| γ-1 |diag ( Vs -)z| ≤ κ| Vs -| γ (1 + |z|) γ-1 |z|,where the last inequality is deduced from the inequality |W (η)| ≤ κ|η| γ-1 . We then obtain thatW ( Vs -+ diag ( Vs -)z) -W ( Vs -) ≤ κ| Vs -| γ |z|.Therefore, as the Lévy process is of finite activity and (10.57) holds,E t -βu W ( Vs -+ diag ( Vs -)z)I( Vs -, z) -W ( Vs -) π(dz)ds ≤ κW (x) < ∞.By [8, Theorem I.1.33 b., p73], we deduce that the purely discontinuous local martingale N satisfies E var(N )∞ < ∞ hence is a martingale. 2

• Step 2 . 2 11. 2

 222 subset C of R 2 and for all > 0, let us define C := c∈C B(c, ),(11.61) where B(c, ) is the closed ball of radius and center c. Recall the well known result:χ 4 (x) = 1 on (C n 4 )γ n := ([ẽn, ãn] ∪ [ãn, an]) γn and χ 4 (x) = 0 on R 2 \(C n 4 ) 2γn . χ 5 (x) = 1 on (C n 5 )γ n := [cn, cn] ∪ [cn, dn] ∪ [ dn, dn]γn and χ 5 (x) = 0 on R 2 \(C n 5 ) 2γn . Let us denote by η(x) the outward normal at each point of ∂ Kn 0 . The mapping η : ∂ Kn 0 → R 2 is continuous except at the points ãn, bn, cn, dn where it admits left and righ limits we denote by η(x±). Moreover, by construction we have γn(x)η(x-) > 0 and γn(x)η(x+) > 0 for all x ∈ ∂ Kn 0 . Observe that d(x -tγn(x), ∂ Kn 0 ) ≥ min x∈∂ Kn 0 d(x -tγn(x), ∂ Kn 0 ) := m n , ∀x ∈ ∂ Kn 0 . By a compactness argument, m n = d(x∞ -tγn(x∞), ∂ Kn 0 ) for some x∞ ∈ Kn 0 . Since d(x∞ -tγn(x∞), ∂ Kn 0 ) ≥ -tγn(x∞)η(x∞±) = 2bt, where 2b := -γn(x∞)η(x∞±) > 0, we finally deduce that Condition C1 holds. By virtue of [5, Corollary 5.2], since σ is globally Lipschitz on the bounded domain Kn 0 , there exists a unique strong solution (V n , k n ), starting from x, to the reflected S.D.E. (11.58) on the domain Kn 0 . Let us introduceτ n := inf{t : V n t (1, d) = -1 n }, ρ n := inf{t : V n t (1, d) = n},and µ n := τ n ∧ ρ n . On the intervall [0, µ n ], the process (V n , k n ) is solution to (11.58) on the domain Kn+1 0 with respect to γ n+1 . Indeed, on the intervall [0, µ n ], the reflection only occurs on the boundary ∂K 0 on which γ n+1 and γ n coincides with γ.By the uniqueness property given by [5, Corollary 5.2], we deduce that (V n , k n ) = (V n+1 , k n+1 ) on [0, µ n ].It follows that µ n ≤ µ n+1 . The rest of the proof is done as in [9, page 229]. Skorokhod problem for pure-Jumps Lévy processes

2

  Lemma 10.6 The interior of K 0 is nonempty. Proof. It suffices to micmic the proof of [9, Lemma 4.8.7] by virtue of Proposition 10.4 and Lemma 10.3. 2Since K 0 = ∅ and K 1 contains cone (g 1 , e 1 ), there exists two real numbers z 1 , z 2 ∈

	∆ satisfying

Observe that Ax = ∅. Indeed, rebalance the portfolio starting from x ∈ int K so that V + = 0.

Note that the coefficient (1 + λ) in our paper corresponds to the coefficient (1 -λ) -1 = (1 -µ) -1 of[START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF].

Denoted by ρ 0 in[START_REF] Shreve | Optimal Investment and Consumption with Transaction Costs[END_REF].

Lemma 11.1 For any compact subset C of R 2 and for all > 0, there exists χ ∈ C ∞ (R 2 , [0, 1]) such that χ = 1 on C and χ vanishes outside C 2 .

The aim of this section is to show that the R.S.D.E (11.58) 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 [START_REF] Dupuis | SDEs with Oblique Reflection on Nonsmooth Domains[END_REF]):

Theorem 11.2 The Skorokhod problem (11.58) admits a solution which is trapped at zero.

Proof. Let us introduce the polygons

where n → ∞. Let D be the bisector of the cone K n 0 and d > 0 such that D := {(x, y) : -dx + y = 0}. Let x ∈ ∂K n 0 be a starting point. The case x = 0 being trivial, we assume that x = 0 hence x ∈ K n 0 if n is large enough.

• Step 1. There exists closed regions Kn

which satisfy Condition C1 for some reflection function γn such that γn(x) → γ(x) for all x ∈ ∂K 0 . Indeed, we denote by an and bn the two points of ∂K 0 ∩ {(x, y) : n = x + dy} such that ya n > y bn . Observe that bn is the symmetric of an with respect to the bisector D.

Similarly, cn and dn are the two symmetric points of ∂K 0 ∩ {(x, y) : -1 n = x + dy}. We then define Kn 0 as the polygon

and denote by ãn and bn the two points of ∂K 0 ∩ {(x, y) : ( n + n+1 )/2 = x + dy} such that y ãn > y bn . Similarly, cn and dn are the two points of ∂K 0 ∩ {(x, y) :

n+1 )/2 = x + dy} such that y cn > y dn .

Let η 1 be the outward normal to ∂K 0 ∩ K 1 and η 2 be the outward normal to

We consider a unit vector g 3 such that g 3 η 1 > 0 and g 3 (1, d) > 0. Similarly, we define g 4 as a unit vector such that g 4 η 2 > 0 and g 4 (1, d) > 0, g 5 is a unit vector such that g 5 η 1 > 0 and g 5 η 2 > 0. Let us introduce the smooth function γn

Moreover, with the notation given by (11.61) above, if γn is sufficiently closed to 0, these functions satisfy :

The goal is to show that this R.S.D.E has a solution on the set K0 . To do so, we shall use the following:

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 :

We omit the proof which is standard. It is easily observable that the direction of

Theorem 11.4 There exists a unique solution to the Skorokhod problem (11.62).

Proof. Let (T k ) k≥1 be the jump stopping times of the process . Assume that we have already constructed a solution (V, k) to (11.62) on the interval [0, T k ). Define

Let us introduce

where the projection operator P K0 is defined in Lemma 11.3. We define ∆ + k T k by the equality

Applying Theorem 11.2 and the strong markov property, there exists a solution ( Ṽ , k) to (11.62) from the starting point Ṽ0 := V T k + with respect to Ñt := N T k +t -N T k and Wt := W T k +t -W T k defined on the interval [0, T k+1 -T k ]. We then define

V t := Ṽt-T k , and k t := k T k + + kt-T k , t ∈ (T k , T k+1 ).

Uniqueness follows from uniqueness on each interval [T k , T k+1 ). 2


est destinée au dépôt

Autonomous exploration, active learning and human guidance with open-source Poppy humanoid robot platform and Explauto library

Sébastien Forestier, Yoan Mollard, Damien Caselli, Pierre-Yves Oudeyer

To cite this version:

The Demonstration

The Torso robot learns how to move its arm to control the ball, light and sounds! Interact to help him:

• On the Tablet, click an element to focus on (sound, light, ball pose...)

• Push the Demo Button and then demonstrate a motion:

¨ Move Torso's left arm to show him how to control the joystick

¨HumanoidFlowers•

  Move the left joystick to show him how to teleoperate the Ergo robot An open platform Poppy robots + Explauto library, to: • Allow non-roboticists researchers to conduct learning experiments with robots • Benchmark algorithms for active multi-tasks policy learning with robots github.com/sebastien-forestier/NIPS2016 To learn parameterized policies • To solve families of parameterized problems • Problems structured in spaces that can be • organized in a hierarchy of reusable skills github.com/sebastien-forestier/ExplorationAlgorithms Explauto provides a high-level Python API for an easy definition of: • Real and simulated robotics setups • Incremental learning of parametrized policies • Active selection of parametrized RL problems github.com/flowersteam/explauto Explauto: a library to study learning in robotics agents Poppy: an experimental platform for science "Make scientific outputs openly accessible, reproducible and cumulative" Poppy robots are accessible in terms of cost and complexity, allowing researchers to share hardware and experimental details Fast design, building and experimentation of alternative morphologies: 3D printing and rapid prototyping techniques now make it possible! poppy-project.org Robot simulator Poppy Humanoid Poppy Ergo Junior


9th International Conference on Porous Media & Annual Meeting (Rotterdam, May 8-11 2017) Advection-diffusion in porous media at low scale separation via higher-order homogenization Pascale Royer

Laboratoire de Mécanique et Génie Civil (LMGC, Univ. Montpellier, CNRS, Montpellier, France) Asymptotic multiple scale homogenization allows to determine the effective behaviour of a porous medium by starting from the pore-scale description. The thus obtained continuum descriptions are valid when the given phenomenon is considered in a medium with a large number of heterogeneities. This hypothesis of separation of scales means that the macroscopic size L, is very large in comparison with the size l of the heterogeneities. It is therefore interesting to investigate how the described theories are modified when the hypothesis of scale separation is not perfectly respected. This happens when the porous medium is macroscopically heterogeneous or when large gradients are applied to macroscopically homogeneous media. The asymptotic multiple scale homogenization method is particularly well adapted to this type of analysis. The effect of low scale separation can be obtained by exploiting higher order equations in the asymptotic homogenization procedure and then by analyzing their role in the macroscopic description. Higher-order homogenization thus allows both the determination of the conditions under which the influence of the microstructure is negligible and also how the microstructure may modify the response of the medium. While higher-order homogenizaton has been been widely applied in the field of mechanics of composite materials (e.g. [START_REF] Gambin | Higher-order terms in the homogenized stressstrain relation of periodic elastic media[END_REF], [START_REF] Boutin | Microstructural effects in elastic composites[END_REF]), porous media at low scale separation have received very little attention. However, an important work is presented in [START_REF] Goyeau | Averaged momentum equation for flow through a nonhomogeneous porous structure[END_REF], where the method of volume averaging is used to obtain the correctors of Darcy's law. The same problem is tackled via higher-order homogenization and up the third order in [START_REF] Auriault | Filtration law in porous media with poor separation of scales[END_REF]. The aim of the present study is to investigate higherorder terms of the advective-diffusive model to describe advection-diffusion in a macroscopically homogeneous porous medium at low scale separation. The advective-diffusive model is obtained by first-order homogenization when considering the convection-diffusion equations with a macroscopic Peclet number (i.e. the Peclet number measured by means of the macroscopic length L) in the order of 1 on the pore scale [START_REF] Auriault | Taylor dispersion in porous media: analysis by multiple scale expansions[END_REF]. The second and third order models are derived. The main result of the study is that the low separation of scales induces dispersion effects. In particular, for a macroscopically homogeneous medium, the second-order model is similar to the most currently used phenomenological model of dispersion: it is characacterized by a dispersion tensor which can be decomposed into a purely diffusive component and a mechanical dispersion part, whilst this property is not verifed in the homogenized dispersion model (obtained at higher Peclet numbers). The third-order description contains second and third concentration gradient terms, with a fourth order tensor of diffusion and with a third-order and a second-order tensors of dispersion.