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

TL;DR

The paper addresses convergence and well-posedness of a Carathéodory-type approximation for one-dimensional $\alpha$,$\beta$-doubly perturbed SDEs with past maximum and minimum terms. It introduces a three-component Carathéodory scheme that uses the full history of the maximum and minimum processes to compute perturbations, and proves $L^{p}$ convergence for $p\ge 2$ under $|\rho|<1$ with $\rho=\dfrac{\alpha\beta}{(1-\alpha)(1-\beta)}$, including a non-Lipschitz extension via a concave modulus $\rho$ with $\int_{0^{+}}\frac{du}{\rho(u)}=\infty$. The key contributions are the existence and uniqueness of a strong solution under the new scheme without the restrictive condition $|\alpha|+|\beta|<1$, explicit convergence rates in the Lipschitz case, and an extension to non-Lipschitz coefficients leveraging a Bihari-type argument. This work improves upon Mao et al. (2018) by relaxing parameter restrictions and broadening applicability to DPSDEs with path-dependent perturbations, offering a robust numerical approximation framework for such models.

Abstract

In this paper, we introduce and study the convergence of new Carathéodory's approximate solution for one-dimensional $α, β$-doubly perturbed stochastic differential equations (DPSDEs) with parameters $α<1$ and $β<1$ such that $|ρ| < 1$, where $ ρ: = \frac{αβ}{(1-α)(1-β)}$. Under Lipschitz's condition on the coefficients, we establish the $L^{p}$-convergence of the Carathéodory 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 $α, β$-DPSDEs. Furthermore, an extension to non-Lipschitz coefficients are also studied. Our results improve earlier work by Mao and al. (2018).

Theorems & Definitions (14)

• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• Remark 2.5
• Lemma 2.6
• Remark 2.7
• Remark 2.8
• Proposition 2.9
• Corollary 2.10