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I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not.

Showing the square of a Markov process is or isn't Markov

I guess I can repeat the method since to be Markov it must satisfy the discrete.

Thanks

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For every $t\geqslant0$, $(B_{t+s}^2)_{s\geqslant0}$ is distributed as $(X_s)_{s\geqslant0}$, where, for every $s\geqslant0$, $$X_s=B_t^2+2\sqrt{B_t^2}\cdot W_s+W_s^2,$$ where $(W_s)_{s\geqslant0}$ is a Brownian motion independent of $(B_u)_{0\leqslant u\leqslant t}$. Thus, indeed, $(B^2_t)_{t\geqslant0}$ is a Markov process.

The discrete analogue of this result is that, if $(X_n)_{n\geqslant0}$ is a random walk with $\pm1$ steps of equal probabilities $\frac12$, then $(|X_n|)_{n\geqslant0}$ is also Markov.

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  • $\begingroup$ Thank you. Did you use that $2B_{t}W_{s}\stackrel{d} {\sim}2\sqrt{B^{2}_{t}}W_{s}$? $\endgroup$ – TKM Nov 3 '14 at 23:13
  • $\begingroup$ Yes, provided one adds a missing factor 2 on the RHS and one passes to the level of identity in distribution of processes, not only of random variables. $\endgroup$ – Did Nov 3 '14 at 23:14
  • $\begingroup$ what do you mean by "level of identity in distribution of processes"? $\endgroup$ – TKM Nov 3 '14 at 23:25
  • $\begingroup$ how can $B_{t}\stackrel{d}{\sim} |B_{t}|$? The latter is not even Gaussian. $\endgroup$ – TKM Nov 3 '14 at 23:26
  • $\begingroup$ You do not need only the identity in distribution mentioned in your first comment to hold for each fixed $s$, but the identity $$(B_tW_s)_{s\geqslant0}\stackrel{d}{=}(|B_t|\cdot W_s)_{s\geqslant0}.$$ $\endgroup$ – Did Nov 3 '14 at 23:27
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Hint:

Use $B_{t_n}=sign(B_{t_n})|B_{t_n}|$ and $\sigma(B_s ^2)= \sigma(|B_{s}| )$ and independent increment property of B.M to show

$$P(B_t^2 \leq x | B_{t_1}^2 ,..., B_{t_n}^2)=P(B_t^2 \leq x | B_{t_n}^2) $$ where $0 <t_1<t_2 <...<t_n<t$ and $x \in \mathbb R$ .

In the process, you need to show $sign(B_{t_n})$ is independent with $\sigma(|B_{t_1}|,|B_{t_2}|,..., |B_{t_n}| )$ by calculating marginal distributions.

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