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 $K<X_1<X_0$