Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles of Non-Local Type

Abstract

In this paper, we study systems of nonlinear second-order variational inequalities with interconnected bilateral obstacles with non-local terms. They are of min–max and max–min types and related to a multiple modes zero-sum switching game in the jump-diffusion model. Using systems of penalized reflected backward SDEs with jumps and unilateral interconnected obstacles, and their associated deterministic functions, we construct for each system a continuous viscosity solution which is unique in the class of functions with polynomial growth.

Appendix: Alternative Definition of the Viscosity Solution of System (2.1)

The following result inspired by the work by Barles–Imbert [3] is another definition of the viscosity solution of system (2.1). We do not give its proof since it is an adaptation of the one given in ([19], Proposition 5.2, pp. 1656) as the function $q\longmapsto g^{ij}(t,x,\vec {y},z,q)$ is non-decreasing, $\beta$ is a bounded function, $\gamma ^{ij}$ is non-negative which then imply $I_{ij}(t,x,\phi )\le I_{ij}(t,x,\psi )$, $I_{ij}^{1,\delta }(t,x,\phi )\le I_{ij}^{1,\delta }(t,x,\psi )$ and $I_{ij}^{2,\delta }(t,x,\phi )\le I_{ij}^{2,\delta }(t,x,\psi )$ for any $\phi \le \psi$ such that $\phi (t,x)=\psi (t,x)=u^{ij}(t,x)$ ($\delta >0$ and $(i,j)\in \Gamma$ are fixed).

Proposition 4.6

A function $\vec {u}=(u^{ij}(t,x))_{(i,j)\in \Gamma }\,\,:[0,T]\times \mathbf{R}^k \rightarrow \mathbf{R}^{m_1\times m_2}$ such that for any $(i,j)\in \Gamma$, $u^{ij}\in \Pi _g$ is lsc (resp. usc) is a viscosity supersolution (resp. subsolution) of (2.1) if:

• (i) $v^{ij}(T,x_0)\ge \,\, (resp. \le ) \,\, h^{ij}(x_0)$, $\forall x_0\in \mathbf{R}^k$ ;

• (ii) For any $(t_0,x_0)\in (0,T)\times \mathbf{R}^k$, $\delta \in (0,1)$ and a function $\phi \in \mathcal{C}^{1,2}([0,T]\times \mathbf{R}^k)$ such that $u^{ij}(t_0,x_0)=\phi (t_0,x_0)$ and $u^{ij}-\phi$ has a global minimum (resp. maximum) at $(t_0,x_0)$ on $(0,T)\times B(x_0,\delta K_\beta )$ where $K_\beta$ is the bound of $\beta$ (see the first inequality of (A0)-(ii)), we have:

  $$\begin{aligned} \left\{ \begin{array}{ll} \min \{(u^{ij}-L^{ij}[\vec {u}])(t_0,x_0); \max \{(u^{ij}-U^{ij}[\vec {u}])(t_0,x_0);\\ -\,\partial _t\phi (t_0,x_0)-b(t_0,x_0)^{\top } D_x\phi (t_0,x_0)-\frac{1}{2}\text{ Tr }(\sigma \sigma ^\top (t_0,x_0)D^2_{xx}\phi (t_0,x_0))\\ \quad \,-\,I^1_\delta (t_0,x_0,\phi )-I^2_\delta (t_0,x_0,D_x\phi ,u^{ij})\\ -\,g^{ij}(t_0,x_0,(u^{kl}(t_0,x_0))_{(k,l)\in A^1\times A^2},\sigma (t_0,x_0)^\top D_x\phi (t_0,x_0),I^{1,\delta }_{ij}(t_0,x_0,\phi )\\ \quad \, +\,I^{2,\delta }_{ij}(t_0,x_0,u^{ij}))\}\}\ge (resp. \le )\,0. \end{array} \right. \end{aligned}$$

Remark 4.2

In taking $\bar{g}_{jl}\equiv +\infty$ (resp. $\underline{g}_{ik}\equiv +\infty$) for any $j,l\in A_2$ (resp. $i,k\in A_1$) we obtain an alternative definition of the viscosity solution of the system of variational inequalities with interconnected lower (resp. upper) obstacles. $\square$