Strict Positivity of the Density for Simple Jump Processes Using the Tools of Support Theorems. Application to the KAC Equation Without Cutoff

Abstract

Consider the one-dimensional solution $X=\{X_t\}_{t \in [0,T]}$ of a possibly degenerate stochastic differential equation driven by a (non compensated) Poisson measure. We denote by $\mathcal{M}$ a set of deterministic integer-valued measures associated with the considered Poisson measure. For $m\in \mathcal{M}$, we denote by $S(m)=\{S_t(m)\}_{t\in[0,T]}$ the skeleton associated with $X$. We assume some regularity conditions, which allow to define a sort of “derivative” $D S_t(m)$ of $S_t(m)$ with respect to $m$. Then we fix $t \in\,]0,T]$, $y\in \reel$, and we prove that as soon there exists $m\in \mathcal{M}$ such that $S_t(m)=y$, $DS_t(m) \ne 0$ and $\Delta S_t(m) =0$, the law of $X_t$ is bounded below by a nonnegative measure admitting a continuous density not vanishing at $y$. In the case where the law of $X_t$ admits a continuous density $p_t$, this means that $p_t(y)>0$. We finally apply the described method in order to prove that the solution to a Kac equation without cutoff does never vanish.