The purpose of this note is to give a support theorem in the Skorohod space for a one-dimensional Marcus differential equation driven by a Lévy process, without any assumption on the latter. We also give a criterion ensuring that the…

For $0<\alpha\leq 2$ and $0<\beta\leq 1$ let $X$ be the $(d,\alpha,\beta)$-superprocess, i.e. the superprocess with $\alpha$-stable spatial movement in $R^d$ and $(1+\beta)$-stable branching. Given that the initial measure $X_0$ is Lebesgue on $R^d$, Iscoe conjectured in [7] that the total…

We consider the white-noise driven stochastic heat equation on $[0,1]$ with Lipschitz-continuous drift and diffusion coefficients. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some…

In this paper, we provide the sharp asymptotics for the quantization radius (maximal radius) for a sequence of optimal quantizers for random variables $X$ in $(\mathbb{R}^d,\|\,\cdot\,\|)$ with radial exponential tails. This result sharpens and generalizes the results developed for the…

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $$ E[Yf(Y)]=\mu E…