On Malliavin differentiability and absolute continuity of one-dimensional doubly perturbed diffusion processes
Rachid Belfadli, Lahcen Boulanba, Youssef Ouknine
TL;DR
The paper analyzes a one-dimensional SDE with past maximum and minimum perturbations, under the regime $\alpha<1$, $\beta<1$, and $|\rho|<1$, establishing Malliavin differentiability $X_t\in \mathbb{D}^{1,2}$ for all $t>0$, and proving absolute continuity of the law of $X_t$ with respect to Lebesgue measure. It then shows, under regularity on the coefficients, that $X_t$ has a smooth density for small times $t\in(0,t_0)$, with an explicit time bound in the additive-noise case and a multiplicative-noise counterpart via a transformation. The proofs rely on Picard iteration, Skorokhod mapping arguments for the max/min terms, and standard Malliavin calculus techniques to transfer differentiability into density properties. These results generalize prior singly perturbed cases and strengthen the available regularity theory for doubly perturbed diffusions in one dimension.
Abstract
In this paper, we establish Malliavin differentiability and absolute continuity for $α, β$-doubly perturbed diffusion process with parameters $α<1$ and $β<1$ such that $|ρ| < 1$, where $ ρ: = \frac{αβ}{(1-α)(1-β)}$. 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.