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.
Citation
Nicolas Fournier. "Strict Positivity of the Density for Simple Jump Processes Using the Tools of Support Theorems. Application to the KAC Equation Without Cutoff." Ann. Probab. 30 (1) 135 - 170, January 2002. https://doi.org/10.1214/aop/1020107763
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