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Let's consider $T=[0,+\infty]$ and let $\tau$ be markov process. I want to check if $\tau^2$ is also markov process i.e. if $\tau^2 \in \mathbb{F}_t$

My work so far

$$\{\tau^2 \le t\}=\{\tau \in [-\sqrt{t},\sqrt{t}]\}=\{\tau \le \sqrt{t}\} \cap\{\tau \ge -\sqrt{t} \}$$

First term of intersection belongs to $\mathbb{F}_{\sqrt{t}}$ which belongs to $\mathbb{F}_t$. But I'm not sure about second term. If it belongs also to $\mathbb{F}_\sqrt{t}$ then $\tau^2$ will be markov process, but I'm not able to check it. Can you give me a hand doing so ?

Lucian
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    Related: https://math.stackexchange.com/questions/27507/transformation-of-state-space-that-preserves-markov-property –  Oct 27 '20 at 21:31
  • The map $\tau\mapsto \tau^2$ is not injective, so the squared process does not possess the Markov property. It should be straightforward to find a simple counterexample - for example, a Markov chain with state space ${-1,0,1}$. – Math1000 Oct 28 '20 at 04:13
  • Hmmm okay, I get your idea. Could you please write as a answer counter example on set $T=[0,+\infty]$ ? I can't figure it out to be honest – Lucian Oct 31 '20 at 22:30

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