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

In this paper, we introduce and study the convergence of new Carathéodory’s approximate solution for one-dimensional [Formula: see text]-doubly perturbed stochastic differential equations (DPSDEs) with parameters [Formula: see text] and [Formula: see text] such that [Formula: see text], where [Formula: see text]. Under Lipschitz’s condition on the coefficients, we establish the [Formula: see text]-convergence of the Carathéodory approximate solution uniformly in time, for all [Formula: see text]. As a consequence, and relying only on our scheme, we obtain the existence and uniqueness of strong solution for [Formula: see text]-DPSDEs. Furthermore, an extension to a class of non-Lipschitz coefficients also studied. Our results improve earlier work by Mao et al. [Approximate solutions for a class of doubly perturbed stochastic differential equations, Adv. Differential Equations 2018(1) (2018) 1–17].