Asymptotic Analysis of Stochastic Variational Inequalities Modeling an Elasto-Plastic Problem with Vanishing Jumps

TL;DR

This paper analyzes the asymptotic behavior of stochastic variational inequalities used to model elasto-plastic oscillators with noise, demonstrating convergence of an approximation method involving small jumps as their size diminishes.

Contribution

It introduces a new approximation technique for stochastic variational inequalities with small jumps and proves its convergence over finite time intervals.

Findings

The approximation converges as jump size tends to zero.

Small jumps effectively separate phases in the model.

The method simplifies phase analysis in stochastic models.

Abstract

In a previous work by the first author with J. Turi (AMO, 08), a stochastic variational inequality has been introduced to model an elasto-plastic oscillator with noise. A major advantage of the stochastic variational inequality is to overcome the need to describe the trajectory by phases (elastic or plastic). This is useful, since the sequence of phases cannot be characterized easily. In particular, there are numerous small elastic phases which may appear as an artefact of the Wiener process. However, it remains important to have informations on these phases. In order to reconcile these contradictory issues, we introduce an approximation of stochastic variational inequalities by imposing artificial small jumps between phases allowing a clear separation of the phases. In this work, we prove that the approximate solution converges on any finite time interval, when the size of jumps tends to 0.