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…