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Let $W$ be a $d$-dimensional Brownian motion and $X$ the strong solution to $$\mathrm{d} X = \mu(X)\mathrm{d} t + \sigma(X)\mathrm{d} W,$$ starting from some $x$, where $\mu$ and $\sigma$ are Lipschitz functions on $\mathbb{R}^d$. Let $B$ be an open ball in $\mathbb{R}^d$. Suppose that the smallest eigenvalue of $\sigma \sigma^T$ is bounded below by a positive constant on $B$. Let $$V(x) = \mathbb{E}_{X_0 = x}[f(X_{\inf\{t\ge 0: X_t \notin B\}})],$$ where $f$ is a bounded, measurable function on $\partial B$.

Then is $V$ continuous on $B$? I know that $V$ is measurable and it is continuous when $X$ is a Brownian motion or when $d=1$.

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