Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump

2007/04/30 by Fournier, Nicolas
#60H10 #60J75 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0704.3922

Abstract

We consider a one-dimensional jumping Markov process \X^x_t\_{t ≥ 0}, solving a Poisson-driven stochastic differential equation. We prove that the law of X^x_t admits a smooth density for t>0, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the map x ↦ X^x_t is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments.

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