We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itó-Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first establish the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the L 1 -contraction principle. Finally, the L 1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.2000 Mathematics Subject Classification. 45K05, 46S50, 49L20, 49L25, 91A23, 93E20.

Abstract. We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new "direct" discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Lévy measures. Furthermore, we develop a new general theory for deriving error estimates for approximate solutions of integro-PDEs, which thereafter is applied to the proposed difference-quadrature schemes. 2 I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN where Q T := (0, T ] × R N andfor smooth bounded functions φ. Equation (1.1) is convex and non-local. The coefficients a α , η α , b α , c α , f α , g are given functions taking values respectively in S N (N × N symmetric matrices), R N , R N , R, R, and R. The Lévy measure ν(dz) is a positive, possibly singular, Radon measure on R M \{0}; precise assumptions will be given later.The non-local operators J α can be pseudo-differential operators. Specifying η ≡ z and ν(dz) = K |z| N +γ dz, γ ∈ (0, 2), give rise to the fractional Laplace operator J = (−∆) γ/2 . These operators are allowed to degenerate since we allow η = 0 for z = 0. The second order differential operator L α is also allowed to degenerate since we only assume that the diffusion matrix a α is nonnegative definite. Due to these two types of degeneracies, equation (1.1) is degenerate parabolic and there is no (global) smoothing of solutions in this problem (neither "Laplacian" nor "fractional Laplacian" smoothing). Therefore equation (1.1) will have no classical solutions in general. From the type non-linearity and degeneracy present in (1.1) the natural type of weak solutions are the viscosity solutions [20,25]. For a precise definition of viscosity solution of (1.1) we refer to [27]. In this paper we will work with Hölder/Lipschitz continuous viscosity solution of (1.1)-(1.2). For other works on viscosity solutions and IPDEs of second order, we refer to [3,4,5,7,6,10,15,27,28,37,40] and references therein.Nonlocal equations such as (1.1) appear as the dynamic programming equation associated with optimal control of jump-diffusion processes over a finite time horizon (see [37,39,12]). Examples of such control problems include various portfolio optimization problems in mathematical finance where the risky assets are driven by Lévy processes. The linear pricing equations for European and Asian options in Lévy markets are also of the form (1.1) if we take A to be a singleton. For more information on pricing theory and its relation to IPDEs we refer to [18].For most nonlinear problems like (1.1)-(1.2), solutions must be computed by a numerical scheme. The construction and analysis of numerical schemes for nonlinear IPDEs is a relatively new area of research. Compared to the PDE case, there are currently only a few works available. Moreover, it is difficult to prove that such schemes converge to the correct (viscosity) s...

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng & Nualart[9] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a-priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.2000 Mathematics Subject Classification. 60H15, 35L65, 35R60, 60E15.

A zero-sum differential game with controlled jump-diffusion driven state is considered, and studied using a combination of dynamic programming and viscosity solution techniques. We prove, under certain conditions, that the value of the game exists and is the unique viscosity solution of a fully nonlinear integro-partial differential equation. In addition, we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations.where s ∈ (t, T ]; with the initial condition2000 Mathematics Subject Classification. 45K05, 46S50, 49L20, 49L25, 91A23, 93E20.