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I am looking for the first hitting time probability of the following Brownian motion:

$dX=\mu X dt+ \sigma (X) X dW$

assuming $X(0)=X_0$ and $\sigma(X)= \sigma_1$ if $X>X_1$ and $\sigma(X)=\sigma_2$ if $X<X_1$ and we are looking for

$F(t,K)$ which is the probability that the first time $X$ touches $K$ happens before time $t$.

also assume $K<X_1<X_0$

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