# 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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