Viscosity Solution of System of Integro-Partial Differential Equations with Interconnected Obstacles of Non-local Type Without Monotonicity Conditions

Abstract

In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution which is unique in the class of functions with polynomial growth. In our study, the main tool is the associated of reflected backward stochastic differential equations with jumps with interconnected obstacles for which we show the existence of a unique Markovian solution.

Appendix

In the paper by Hamadène and Zhao [15], the definition of the viscosity solution of the system (18), is given as follows.

Definition 2

Let $\mathbf {u}:=(u^i)_{i\in \mathscr {I}}$ be a function of $C([0,T] \times \mathbb {R}^k ; \mathbb {R}^m)$.

(i) We say that $\mathbf {u}$ is a viscosity supersolution (resp. subsolution) of (18) if: $\forall i \in \lbrace 1,\ldots ,m\rbrace$,

$$\begin{aligned}&\text{ a) } \,u^i(T,x) \ge (\text{ resp. } \le )\,\, h_i(x) , \,\,\forall x \in \mathbb {R}^k\,\,; \\&\text{ b) } \text{ if }\,\phi \in \mathcal{C}^{1,2}([0,T] \times \mathbb {R}^k) \text{ is } \text{ such } \text{ that } (t,x) \in [0,T) \times \mathbb {R}^k \text{ a } \text{ local } \text{ minimum }\\&\quad \hbox {(resp. maximum) point of }u^i - \phi \end{aligned}$$

then

$$\begin{aligned}&\min \Big \lbrace u^i(t,x) - \displaystyle \max _{j \in {\mathscr {I}^{-i}}}(u^j(t,x)-g_{ij}(t,x)) ;\\&\quad \quad -\partial _t\phi (t,x) - \mathscr {L}\phi (t,x) - \bar{f}_i(t,x,(u^k(t,x))_{k=1,\ldots ,m},(\sigma ^\top D_x\phi )(t,x), \mathscr {B}_i\phi (t,x))\Big \rbrace \\&\quad \ge \,\,(resp. \le )\,\,0. \end{aligned}$$

(ii) We say that $\mathbf {u}:=(u^i)_{i\in \mathscr {I}}$ is a viscosity solution of (18) if it is both a supersolution and subsolution of (18).