Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness

Abstract

In this paper we study viscosity solutions to the system

$$\begin{array}{ll} \min \{ -\mathcal{H}u_i(x,t)-\psi _i(x,t),u_i(x,t) - \max_{j \neq i} (-c_{i ,j} (x,t) + u_j (x,t)) \} = 0,\\ u_i(x,T)=g_i (x), \, i \in \{1,\ldots , d \},\end{array}$$

where ${(x,t)\in{\mathbb{R}}^{N} \times [0,T]}$. Concerning ${{\mathcal{H}}}$, we assume that ${{\mathcal{H}}={\mathcal{L}}+{\mathcal{I}}}$ where ${{\mathcal{L}}}$ is a linear, possibly degenerate, parabolic operator of second order and ${{\mathcal{I}}}$ is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator ${{\mathcal{H}}}$ and on the continuous functions ${\psi_i}$, c _i,j , and g _i . Using the comparison principle we prove the existence of a unique viscosity solution (u ₁, . . . , u _d ) to the system by Perron’s method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c _{i, j} } and allowing c _{i, j} to depend on x as well as t.