First-order weak balanced schemes for bilinear stochastic differential equations

TL;DR

This paper introduces first-order weak balanced schemes for bilinear stochastic differential equations that are stable and do not require complex stochastic integrals, demonstrated through theoretical development and numerical experiments.

Contribution

The paper develops new stable, first-order weak schemes for multidimensional bilinear SDEs that avoid multiple stochastic integrals, using optimization and heuristic methods.

Findings

Schemes achieve almost sure stability.

Numerical experiments confirm promising performance.

Methods simplify implementation of weak schemes.

Abstract

We use the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak balanced schemes based on the addition of stabilizing functions to the drift terms. Then, we design balanced schemes for multidimensional bilinear SDEs achieving the first order of weak convergence, which do not involve multiple stochastic integrals. To this end, we follow two methodologies to find appropriate stabilizing weights; through an optimization procedure or based on a closed heuristic formula. Numerical experiments show a promising performance of the new numerical schemes.