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

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).