Abstract:

The weighted fractional Brownian motion is a limit of occupation time fluctuations of a system with independent particles moving in ℝ with respect to a symmetric stable Lévy process. It is an extension of fractional Brownian motion. We demonstrate that the law of the solution to multiplicative stochastic delay differential equations driven by weighted fractional Brownian motion is absolutely continuous with respect to Lebesgue measure on ℝ, and that the density is smooth, using Malliavin calculus. We prove the existence and uniqueness of the solution and provide an explicit expression for it. Additionally, we establish a new generalized maximum inequality in L^p for Wick integrals driven by weighted fractional Brownian motion, which bounds the moments of the solutions and their Malliavin derivatives. This is essential for achieving a smooth density. Due to the existence of Malliavin derivatives of solutions in the upper bound used for deriving the boundedness of moments, we assume some usual additional conditions on the order of derivatives of the coefficients of the equations.

2010 AMS Mathematics Subject Classification: Primary 60H07; Secondary 60H05, 60G22.

Keywords and phrases: stochastic delay differential equation, Malliavin calculus, Wick integral, weighted fractional Brownian motion, regular density.