Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

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. 1. Introduction Stochastic functional differential equations, or stochastic delay differential equations, determine non-Markovian processes, because the current states of the process in the equation depend on the past histories of the process. Such kind of equations was initiated by It？ and Nisio [1] in their pioneering work about 50 years ago. As stated in [2], there are some difficulties to study such equations, because we cannot use any methods in analysis, partial differential equations, and potential theory at all. On the other hand, it seems to be more natural to consider the models determined by the solutions to the stochastic functional differential equations in finance, physics, biology, and so forth, because such processes include their past histories and can be recognized to reflect real phenomena in various fields much more exactly. The Malliavin calculus is well known as a powerful tool to study some properties on the density function by a probabilistic approach. There are a lot of works on the densities for diffusion processes by many authors, from the viewpoint of the Malliavin calculus (cf. [3]). Moreover it is also applicable to the case of solutions to stochastic functional differential equations, regarding as one of the examples of the Wiener functionals. Kusuoka and Stroock in [4] studied the application of the Malliavin calculus to the solutions to stochastic functional differential equations and obtained the result on the existence of the smooth density for the solution with respect to the Lebesgue measure. On the other hand, it is well known that the Malliavin calculus is very fruitful to study the asymptotic behavior of the density function related to the large deviations theory (cf. Léandre [5–8] and Nualart [9]). In fact, the Varadhan-type estimate of the density function for the diffusion processes can be also obtained from this viewpoint. Ferrante et al. in [10] discussed such problem in the case of stochastic delay differential equations, where the drift term depends on the whole past histories on the finite time