Abstract
By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density.Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.
Highlights
Consider the following stochastic differential equation in Rd: dXt = b(Xt)dt + A1dWt + A2dLt, X0 = x, (1.1)where b : Rd → Rd is a smooth vector field, A1 and A2 are two constant d × d-matrices, Wt is a d-dimensional standard Brownian motion and Lt is a purely jump d-dimensionalLévy process with Lévy measure ν(dz)
In the continuous diffusion case (i.e. A2 = 0 and A1 = A1(x)), under Hörmander’s conditions, Malliavin [13] proved that SDE (1.1) has a smooth density by using the stochastic calculus of variations
About the smoothness of distributional density for degenerate SDEs driven by purely jump noises, Takeuchi [20], Cass [6] and Kunita [10] have already studied this problem under different Hörmander’s conditions
Summary
In the continuous diffusion case (i.e. A2 = 0 and A1 = A1(x)), under Hörmander’s conditions, Malliavin [13] proved that SDE (1.1) has a smooth density by using the stochastic calculus of variations (nowadays, it is called the Malliavin calculus, and a systematic introduction about the Malliavin calculus is referred to the book [14]). About the smoothness of distributional density for degenerate SDEs driven by purely jump noises, Takeuchi [20], Cass [6] and Kunita [10] have already studied this problem under different Hörmander’s conditions. Their results do not cover the present general case (see [23, 24] for some related works). Convention: The letter C or c with or without subscripts will denote an unimportant constant, whose value may be different in different places
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