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A scalar diffusion $X=(X_t)_{0\leq t \leq T}$ is the unique strong solution of the SDE $$\text{d}X_t=\text{d}W_t$$ where $W=(W_t)_{0\leq t \leq T}$ is a scalar Brownian motion.

Assume that the state space of $X$ is $R$. Let $Y=(Y_t)_{t\ge 0}$ be given by $\text{d}Y_t=\text{d}t$.

Define $p(x,y;t)\text{d}t:=P(Y_{\tau}\in \text{d}t|X_0=x,Y_0=y), \tau:=\text{inf}\{t\ge 0:X_t=m\}$

Find $p(x,y;t)$.

I'm only vaguely familiar with stochastic processes, should I use Kolmogorov forward equation or backward equation to solve it?

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  • $\begingroup$ Essentially you need to compute the density of the distribution of $\tau$. Ever heard of the reflection principle (for Brownian motion)? $\endgroup$
    – saz
    Commented Mar 14, 2020 at 7:09
  • $\begingroup$ @saz Yes. I know the density of the first hitting time of Brownian motion. Is it $p(x,y;t)$? $\endgroup$
    – KKKcat
    Commented Mar 14, 2020 at 20:07

1 Answer 1

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Indeed as mentioned in the comments, this is simply the density for Brownian motion

$$p(x,y;t)\text{d}t:=P(Y_{\tau}\in \text{d}t|X_0=x,Y_0=y)=P[y+\tau\in dt| X_{0}=x].$$

This has been computed here Density of first hitting time of Brownian motion from its CDF.

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