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.

Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem without monotonicity condition

“…. Finally, using a result by Hamadène-Zhao [14], we deduce that (u i ) i∈I is a solution in viscosity sense of the following system of IPDE with interconnected obstacle:…”

“…Theorem 3.1 (see [14]). Assume that the deterministic functions (f i ) i∈I , (g ij ) i,j∈I , (h i ) i∈I and (γ) i∈I verify Assumptions (H1)-(H3) and (H4).…”

“…Before doing so, we precise our meaning of the definition of the viscosity solution of this system. It is not exactly the same as in [14] (see also Definition (4.4) in the Appendix). − ∂ t φ(t, x) − Lφ(t, x) −f i (t, x, (u k (t, x)) k=1,...,m , (σ ⊤ D x φ)(t, x), B i u i (t, x)) ≥ (resp.…”

“…We mention that the system of IPDEs (1.1) and related to optimal switching problems of jump-diffusion process have been studied in [5], [16], [13], [14] or [17].…”

“…We show in Section 4 that the function (u i ) i=1,m is the unique viscosity solution of (1.1) in the class of continuous functions with polynomial growth. In the Appendix, we give another definition of the viscosity solution of system (1.1) which is inspired by the work by Hamadène-Zhao in [14]. ✷…”