Stochastic differential variational inequalities with applications

TL;DR

This paper introduces stochastic differential variational inequalities (SDVI), proves solution existence and uniqueness, demonstrates Euler scheme convergence, and applies these results to electrical circuits and bridge collapse problems in stochastic settings.

Contribution

It presents the first comprehensive study of SDVI, including solution theory, numerical scheme convergence, and practical applications in engineering problems.

Findings

Existence and uniqueness of SDVI solutions established.

Euler scheme convergence proved under mild conditions.

Applications to electrical circuits and bridge collapse problems demonstrated.

Abstract

In this paper, we introduce and study a stochastic differential variational inequality (SDVI) which consists of a stochastic differential equation and a stochastic variational inequality. We obtain the existence and uniqueness of the solutions for SDVI by using the iteration method and Gronwall's inequality. Moreover, we show the convergence of Euler scheme for solving SDVI under some mild conditions. Finally, we apply the obtained results to solve the electrical circuits with diodes and the collapse of the bridge problems in stochastic environment.