# Distribution after hitting

Let $X$ be a real-valued Markov process, $$\mathsf P_x\{X_1\in A\} = K(x,A)$$ is its transition kernel. Let $\tau = \inf\{n\geq 0:X_n\geq a\}$ be the first hitting time of the level $a$. I am interested in distribution $$L_k(x;A) = \mathsf P_x\{X_\tau\in A|\tau = k\}$$

In the latter event we cannot just put $\mathsf P_x\{X_\tau\in A|\tau = k\} = \mathsf P_x\{X_k\in A\}$ since it doesn't hold e.g. for $A = [a,\infty)$. Any hints are appreciated.

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I am going to assume $K$ is nicely differentiable, so that we can talk about the probability density $dK (X_i, X_{i+1})$ associated with it.

First define $B = [a, \infty)$ and $B^c = (-\infty, a)$. Then

$L_k(x, A) = P_x( X_{\tau} \in A | \tau = k ) = P_x ( X_{\tau} \in A \cap B | X_{\tau '} \in B^c \space \forall \tau ' < \tau )$ $= P_x ( X_{\tau} \in A \cap B \wedge X_{\tau '} \in B^c \space \forall \tau ' < \tau ) / P_x ( X_{\tau '} \in B^c \space \forall \tau ' < \tau )$

Each of these probabilities is expressible in terms of repeated integrals of $dK$ (over $(B^c)^{k-1}$), with the numerator having a final factor of $K(X_{k-1}, A)$ in it.

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I think this is solvable in continuous time with discrete state space (at least in principle) if you know the infinitesimal generator of the process (under some regularity conditions).

As this kind of question is closely related to barrier option in finance you might have a look at this article by Mijatovic and Pistorius