Probabilistic Representation and Approximation for Coupled Systems of Variational Inequalities

TL;DR

This paper develops a probabilistic framework and numerical methods for solving coupled systems of variational inequalities, which are driven by different operators and include non-linear dependencies and structural constraints.

Contribution

It introduces a novel probabilistic representation of solutions via constrained BSDEs with jumps, enabling new numerical approximation schemes for these complex systems.

Findings

Probabilistic representation of coupled variational inequalities as constrained BSDEs.

Development of a natural probabilistic numerical scheme for these systems.

Application to optimal switching problems.

Abstract

Our study is dedicated to the probabilistic representation and numerical approximation of solutions to coupled systems of variational inequalities. The dynamics of each component of the solution is driven by a different linear parabolic operator and suffers a non-linear dependence in all the components of the solution. This dynamics is combined with a global structural constraint between all the components of the solution including the practical example of optimal switching problems. In this paper, we interpret the unique viscosity solution to this type of coupled systems of variational inequalities as the solution to one-dimensional constrained BSDEs with jumps introduced recently in [6]. In the spirit of [3], this new representation allows for the introduction of a natural entirely probabilistic numerical scheme for the resolution of these systems.