On Malliavin differentiability and absolute continuity of one-dimensional doubly perturbed diffusion processes

TL;DR

This paper proves Malliavin differentiability and absolute continuity for a class of doubly perturbed diffusion processes, establishing conditions under which the solution has a smooth density, extending previous results using Malliavin calculus techniques.

Contribution

It introduces new conditions for Malliavin differentiability and smooth density of doubly perturbed diffusions, generalizing earlier findings.

Findings

Established Malliavin differentiability for the process.

Proved the existence of a smooth density under certain conditions.

Extended previous results by Yue and Zhang (2015) and Xue, Yue, and Zhang (2016).

Abstract

In this paper, we establish Malliavin differentiability and absolute continuity for $\alpha, \beta$-doubly perturbed diffusion process with parameters $\alpha <1$ and $\beta <1$ such that $|\rho| < 1$, where $ \rho : = \frac{\alpha\beta}{(1-\alpha)(1-\beta)}$. Furthermore, under some regularity conditions on the coefficients, we prove that the solution $X_t$ has a smooth density for all $t\in(0, t_0)$ for some finite number $t_0>0$. Our results recover earlier works by Yue and Zhang (2015) and Xue, Yue and Zhang (2016), and the proofs are based on the techniques of the Malliavin calculus.