On Carathéodory approximate scheme for a class of one-dimensional doubly perturbed diffusion processes

Abstract

In this paper, we introduce and study the convergence of new Carathéodory'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 Carathéodory approximate solution uniformly in time, for all $p\ge 2$. 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).

21 pages