On Carath\'eodory approximate scheme for a class of one-dimensional doubly perturbed diffusion processes

TL;DR

This paper develops a Carathéodory approximation scheme for one-dimensional doubly perturbed stochastic differential equations with parameters less than one, proving convergence, existence, and uniqueness of solutions under Lipschitz and non-Lipschitz conditions.

Contribution

It introduces a new approximation scheme for doubly perturbed SDEs and establishes convergence, existence, and uniqueness results, extending previous work by Mao et al. (2018).

Findings

Proves $L^{p}$-convergence of the scheme for all $p \\geq 2$.

Establishes existence and uniqueness of strong solutions.

Extends results to non-Lipschitz coefficients.

Abstract

In this paper, we introduce and study the convergence of new Carath\'eodory's approximate solution for one-dimensional $\alpha, \beta$-doubly perturbed stochastic differential equations (DPSDEs) with parameters $\alpha <1$ and $\beta <1$ such that $|\rho| < 1$, where $ \rho : = \frac{\alpha\beta}{(1-\alpha)(1-\beta)}$. Under Lipschitz's condition on the coefficients, we establish the $L^{p}$-convergence of the Carath\'eodory approximate solution uniformly in time, for all $p\geq 2$. As a consequence, and relying only on our scheme, we obtain the existence and uniqueness of strong solution for $\alpha, \beta$-DPSDEs. Furthermore, an extension to non-Lipschitz coefficients are also studied. Our results improve earlier work by Mao and al. (2018).