In this paper, we study the existence and uniqueness of viscosity solutions for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case of this system is the deterministic version of the Verification Theorem of the Markovian optimal m-states optimal switching problem in finite horizon. The switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central valuation in cases where such organizations or state give grants or financial assistance to power plants that promotes green energy in their production activity or that uses less polluting modes in their production. Our main tools is an approximation scheme and the notion of systems of reflected backward stochastic differential equations.

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of L p (p ≥ 2)-solutions in the case of a fixed terminal time under suitable p-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension 1.

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domainWe prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of D, via a penalization method. 1 case of fixed convex domains were for the first time studied in [3]. Actually, the authors study multidimensional RBSDEs in the case of fixed convex domain C = {C t , t ∈ [0, T ]}, of the form:where K t is continuous, increasing and of bounded total variation |K| satisfying K 0 = 0. The last condition insures that K is minimal in the sense that it increases only when Y is at the boundary of C. In fact, the process K is inward normal to C at Y , precisely K t = t 0 η s d|K| s such that η s ∈ N (Y s ) and where N (Y s ) is the inward normal unit vector to C at Y s . Actually, when Y is at the boundary it is pushed into the domain along η ∈ N (Y ). The authors provide existence and uniqueness for such RBSDEs via a penalization method. Later, [12] extended the result of [3] to the case of jumps (i.e. whose noise includes a Poisson random measure part). The author studied RBSDE of Wiener-Poisson type in fixed convex domain C, for which he established existence and uniqueness using a penalization method. They considered RBSDEs of the following form:• M: the space of IR m×d -valued, F-progressively measurable processes (Z t ) 0≤t≤T such that T 0 Z s 2 ds < ∞ P-a.s, and equipped with the metric θ(Z, Z ′ ) = E T 0 Z s − Z ′ s 2 ds ∧ 1 . • L: the set of mappings V : Ω × [0, T ] × U → IR m which are P ⊗ U -measurable, such that T 0 U |V s (e)| 2 λ(de)ds < ∞ P-a.s, and equipped with the metric ̟(V, V ′ ) = E T 0 U |V s (e) − V ′ s (e)| 2 λ(de)ds ∧ 1 .

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