Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

Abstract

The goal of this paper is to show that under some assumptions, for a d-dimensional fractional Brownian motion with Hurst parameter H > 1/2, the density of the solution of the stochastic differential equationUnder the framework of this present work, the Laplace method can be obtained in general hypoelliptic case and without imposing the structure equations on vector fields in Theorem 1.1. These two assumptions are imposed to obtain the correct Riemannian distance in the kernel expansion.Remark 1.3. When H > 1/2, to obtain a short-time asymptotic formula for the density of solution to equation (1.1) but with drift, one need to work on a version of Laplace method with fractional powers of ε, which will be very heavy and tedious in computation.Remark 1.4. When the present work was almost completed, we noticed that a proof for the Laplace method for stochastic differential equation driven by fractional Brownian motion with Hurst parameter 1/3 < H <

Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory

“…In this final section we give a sufficient condition for our main result (Theorem 2.2) on the off-diagonal asymptotics and compare it with a preceding result by Baudoin and Ouyang (Theorem 1.2, [2]), which is probably the only paper on this kind of problem. Proposition 9.1 Assume (A1) at the starting point a ∈ R n .…”

“…Second, cancellation of "odd terms" as in p. 20 and p. 34, [21] does not happen in general in our case. (If the drift term in Young ODE (2.1) is zero, then this kind of cancellation takes place as in [1,2]). …”

“…This kind of asymptotics was already studied in [1,2], but without Malliavin calculus. In [1], they showed on-diagonal short time asymptotics when V 0 ≡ 0.…”

“…In Section 9, we prove that, under the ellipticity assumption at the stating point, our main result (the off-diagonal asymptotics) holds when the end point is close enough to the starting point. We also make sure that Baudoin and Ouyang's result in [2] is basically included in ours.…”

“…In [1], they showed on-diagonal short time asymptotics when V 0 ≡ 0. In [2], by using Laplace's method, they showed off-diagonal short time asymptotics when V 0 ≡ 0 and the vector fields V i 's satisfy a rather special condition. Our results is a generalization of these preceding ones.…”