Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

Abstract

Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument.

Joint distributions for stochastic functional differential equations

“…Proposition 1 (cf. [10,12]) For −r ≤ t ≤ T , the random variable X (t) is smooth in the Malliavin sense. Moreover, the Malliavin derivative DX (t) = D u X (t) ; 0 ≤ u ≤ T of X (t) and the Malliavin covariance matrix V (t) := DX (t), DX (t) H m 0 for X (t) can be computed as follows:…”

“…They obtained in [2,3] the existence of the smooth density under the degeneracy condition on ∑ m i=1 Âi  * i by using the delay structure of the equation and conditioning the past history of the process, which is weaker than the uniformly elliptic condition on ∑ m i=1 A i A * i . Furthermore, Kitagawa and Takeuchi [10] studied the asymptotic behavior of the density such as the Varadhan-type estimate for diffusion processes, by the large deviation theory and the Malliavin calculus, in which the constant r, called the delay parameter, plays a crucial role.…”