I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips!
(I edited the entire thing to make it clearer)
The conjecture is the following;
Let $\bar{x}\in\mathbb{R}^n$ and $\bar{t}\in\mathbb{R}$. Let B be an $n$-dimensional Brownian motion with $B_{t_0}=x_0$, i.e. the motion starts at time $t_0$ at location $x_0$. Let $\eta>0$, and suppose that $(x_0,t_0)$ is in the interior of the $\eta$-ball with center $(\bar{x},\bar{t})$. Define the following stopping time.
$$\tau(x_0,t_0):=\inf \lbrace t : \sqrt{|B_t-\bar{x}|^2+|t-\bar{t}|^2 }\geq \eta \rbrace$$
Let a continuous function $f$ be defined on the boundary of the $\eta$-ball with center $(\bar{x},\bar{t})$. Let $U$ be the interior of the $\eta$-ball with center $(\bar{x},\bar{t})$. Then define the function $F:U\rightarrow\mathbb{R}$ by
$$F(x_0,t_0):=\int_{\omega} f(B_{\tau(x_0,t_0)}(\omega))dW(\omega)$$
The conjecture is that $F$ is differentiable at the point $(\bar{x},\bar{t})$.
I know from the literature on harmonic functions that the conjecture is true if the stopping time is instead given by $\tau(x_0,t_0):=\inf[t : \sqrt{|B_t-\bar{x}|^2}\geq \eta ]$ and $f$ does not depend on $t$, because in this case $F$ is a harmonic function. I have not found how to use the techniques from that literature in the present case, however.
It would be awesome if someone could give me a hint as to how to go about this!
Or, if someone knows of another stochastic process that has this property that would also be great!
