Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary

Abstract

In this work, we study the Carathéodory approximate solution for a class of one-dimensional perturbed stochastic differential equations with reflecting boundary (PSDERB). Based on the Carathéodory approximation procedure, we prove that PSDERB have a unique solution and show that the Carathéodory approximate solution converges to the solution of PSDERB whose both drift and diffusion coefficients are non-Lipschitz. After that, we establish an explicit rate of convergence in the case of PSDERB with Lipschitz coefficients.

Keywords:

• Carathéodory approximate solution
• perturbed stochastic differential equations
• local time

AMS CLASSIFICATIONS:

• 65C30
• 60H10
• 65L20
• 60J55
• 41A25

Acknowledgements

We are thankful to the editor and the anonymous referee for very careful reading, and her/his valuable remarks and suggestions which led to improvement of the paper.