Let $b:\mathbb{R}^d\rightarrow \mathbb{R}^d$ be a smooth Lipschitz function, $x \in \mathbb{R}^d$, $\sigma >0$, and consider the solution to the SDE $X_t^x$ defined by $$ dX_t^x = b(X_t^x)dt + \sigma dW_t ; \qquad X_0=x; $$ where $W_t$ is a $d$-dimensional Brownian motion defined on the stochastic base $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$.
Let $D$ be the ball about $0 \in \mathbb{R}^d$ of radius $\delta>0$ and define the exit time $\tau_D$ for $X_t^x$ from $D$ by $$ \tau_D \triangleq \left\{ t >0 :\, X_t^x \not \in D \right\} . $$ (Here, the inf of an empty set it defined to be $\infty$). Is there a large deviation principle for $$ \mathbb{P}\left( \tau_D \leq a \right) \leq \exp\left( \tau_{D}\leq F(a) \right), $$ where $a >0$ and $F:(0,\infty)\rightarrow (0,\infty)$ is a rate function.
