Cross-posted from: "https://math.stackexchange.com/questions/3917971/malliavin-derivative-of-stopped-brownian-motion"

I have a small question concerning the Malliavin derivatives. It could be rather simple, but I am unsure.

Let $B_t$ stand for the standard Brownian motion in $\mathbb{R}^d$. Denote $$T = \inf\{t| \|B_t\| = 1\}.$$ That is, $T$ is the first exit time from the unit ball.

I am interested in calculating the Malliavin derivative $DB_T$. The immediate suspect is $DB_T = 1_{[0,T]}$. I could not prove it though and I suspect it is not 100% correct.

Is the functional $B_T$ even Malliavin differentiable? If so, how to obtain the derivative?


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