On solutions for stochastic differential equations with Hölder coefficients

Rongrong Tian, Liang Ding, Jinlong Wei

Published 2017-12-08Version 1

Consider the following stochastic differential equation (SDE for short) $$X_t=x+\int\limits_0^tb(s,X_s)ds+\int\limits_0^t\sigma(s,X_s)dW_s,\quad t>0, \, x\in\mathbb{R}^d,$$ where $\{W_s\}_{s\geq 0}$ is a $d$-dimensional standard Wiener process, $b\in L^q_{loc}(\mathbb{R}_+;\mathcal{C}_b^\alpha(\mathbb{R}^d))$, $\sigma\in \mathcal{C}(\mathbb{R}_+;\mathcal{C}_b^\alpha(\mathbb{R}^d))$ with $\alpha\in (0,1), q\in [1,2]$. Suppose that $1+\alpha-2/q>0$, and $\sigma\sigma^\top$ meets uniformly elliptic condition, then there exits a weak solution to the above equation. Furthermore, if $q=2$, the weak solution is unique, the Markov semi-group has the strong Feller property and there is a density associated with the above SDE. Moreover, if $|\nabla\sigma|\in L^2_{loc}(\mathbb{R}_+;L^\infty(\mathbb{R}^d))$ in addition, the path-wise uniqueness holds and the unique solution $X$ lies in $L^p(\Omega;\mathcal{C}([0,T];\mathcal{C}_{b}^\beta(B_R)))$ for every $p\geq 1$, every $R>0$, every $T>0$ and every $\beta\in (0,1)$.