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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

  1. $W=(W_t)$ is a scalar Brownian motion
  2. $X$ is in $\mathbb{R}$.
  3. 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.

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  • $\begingroup$ When "defining" $\Gamma(x,z;t)dt= \mathbb P(Z_\tau \in dt | X_0=x,Z_0=z) $, do you mean $\mathbb P(Z_\tau \in [t,t+dt) | X_0=x, Z_0=z)$? $\endgroup$
    – Mick
    Mar 18, 2020 at 15:46
  • $\begingroup$ @Mick Yes this is what it means. $\endgroup$ Mar 18, 2020 at 19:11
  • $\begingroup$ did you mean to write $dZ_{t}=dt$? This is just deterministic. $\endgroup$ Apr 10, 2020 at 1:39

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