Suppose I have the following two stochastic differential equations ($t\geq 0$) $$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$ where $X = (X_t)$, $Z = (Z_t).$
Note that
- $W=(W_t)$ is a scalar Brownian motion
- $X$ is in $\mathbb{R}$.
- Fix $(\Omega, \mathcal{F},\mathbb{P})$ and $\mathbb{F} = (\mathcal{F}_t)$ be a filtration for $W$.
Define $$\Gamma(x,z;t)dt :=\mathbb{P}(Z_{\tau}\in dt \mid X_0 = x, Z_0=z), \ \ \ \tau:=\inf\{t\geq0:X_t=m\}.$$ $\tau$ here is the first hitting time of $X$ (to the specific boundary $m$).
My question is how to derive a PDE and boundary condition for the transition density $\Gamma(x,z;t)$?
Thanks in advance.