Abstract
In this article, we prove that the inverse of Malliavin matrix belongs to $L^p(\Omega,\mathbb{P})$ for a class of degenerate stochastic differential equation (SDE). The conditions required are similar to Hörmander's bracket condition, but we don't need all coefficients of the SDE are smooth. Furthermore, we obtain a locally uniform estimate for the Malliavin matrix and a gradient estimate. We also prove that the semigroup generated by the SDE is strong Feller. These results are illustrated through examples.
Highlights
Introduction and NotationsIn this article, we consider the following degenerate stochastic differential equations(SDE) t xt = x +a1(xs, ys)ds, t yt y a2(xs, ys)ds + b(xs, ys)dWs. (1.1)where x ∈ Rm, y ∈ Rn, b ∈ Rn×d, Ws is a d-dimensional standard Brownian motion
The gradient estimate of the semigroup and the strongly Feller property associated to the solution should be considered, and the solution is ergodic if one knows that the solution is topological irreducible and has an invariant probability measure
The strong Feller property is very useful when we prove the uniqueness of invariant measure
Summary
We consider the following degenerate stochastic differential equations(SDE). For the special case V0 ≡ 0, in [11], Kusuoka and Stroock gave the two-sided bounds of the density for Pt(x, ) under some conditions which need some uniformity on V1, · · · , Vd. Recently, in [4], Delarue and Menozzi considered the following SDE,. Since our aim in this article is to prove the strong Feller property and give a gradient estimate of the semigroup, we don’t need the smooth conditions for all the coefficients or some uniform conditions. The local uniform estimate for Malliavin matrix is a key point to prove Theorem 3.1.
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