Stochastic Delay Equations with Hereditary Drift: Estimates of the Density

Abstract

We consider a family of stochastic differential equations with a drift depending on the past history and a delayed diffusion term perturbed by a small parameter =>0. We establish the asymptotic behaviour as = a 0 for the logarithm of the corresponding family of densities at a fixed time t>0. The proof needs large deviation estimates and Malliavin calculus. Academic Press

Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

“…Ferrante et al [10], Lemma 6.1), where the symbol ∇ is the Frechét derivative in ([− , 0]; R ). Thus, the Hölder inequality and Lemma 2 enable us to get the assertions.…”

“…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 interval, while the diffusion terms depend on the state only for the edges of the finite time interval. Mohammed and Zhang in [11] studied the large deviations for the solution process under a similar situation to [10].…”

“…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 interval, while the diffusion terms depend on the state only for the edges of the finite time interval. Mohammed and Zhang in [11] studied the large deviations for the solution process under a similar situation to [10]. But, the special forms on the diffusion terms play a crucial role throughout their arguments in [10,11].…”

“…Mohammed and Zhang in [11] studied the large deviations for the solution process under a similar situation to [10]. But, the special forms on the diffusion terms play a crucial role throughout their arguments in [10,11].…”

“…Ferrante et al in [10] discussed the large deviation principle for the solution process and the asymptotic estimate of the density, in the case of ( , ) =̃( , ( − )) ( = 1, . .…”