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

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 $\mathrm{\mid }\rho \mathrm{\mid }<1$, 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.