Markov Processes

**symmetry7** · October 21st 2009, 03:12 AM

Hi there...

Let ${X}$ be a Markov chain and ${T}$ be a finite stopping time. Show that ${(X_{n+T})}_{n \geq 0}$ defines a Markov chain, taking care of stating the relevant filtration.

Any help would be greatly appreciated..

**Focus** · October 25th 2009, 11:53 AM

$\mathbb{P}(X_{n+T}|\sigma(X_{s+T}:s\leq m))=\sum \mathbb{P}(X_{n+i}|\sigma(X_{s+i}:s\leq m))\mathbb{P}(T=i)$
$=\sum \mathbb{P}(X_{n+i}|\sigma(X_{m+i}))\mathbb{P}(T=i) =\mathbb{P}(X_{n+T}|\sigma(X_{m+T}))$

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