I am struggling to understand the martingale property of exponential of a biased random walk. For example, in the following problem how do I verify whether the following is a martingale, submartingale or supermartingale? What happens when p=q=1/2?
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1$\begingroup$ Hi: You need to calculate $E(S_{n+1} | S_{n})$. If that is $S_n$, then $S_{n}$ is a martingale, If it's greater than$S_{n}$, then it's a super- martingale and so on and so forth. It's good practice to do the calculation yourself using conditional probability. $\endgroup$– mark leedsApr 21, 2021 at 4:07
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1 Answer
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A filtration needs to be specified. In this case the natural one is:
$$ {\cal F}_n = \sigma(S_i | 1\leq i\leq n) $$
This makes $S$ adapted. Integrability comes from the boundness of $X$ and $S$.
As per comments, we need then to compute
$$ E[S_{n+1}|{\cal F}_n] $$ and see how it compares with $S_n$.
Note that $$ E[S_{n+1}|{\cal F}_n] = E[S_{n} {\rm e}^{X_{n+1}}|{\cal F}_n] $$ $$ = S_n E[{\rm e}^{X_{n+1}}|{\cal F}_n] = S_n E[ {\rm e}^{X_{n+1}}], $$
where we have used the fact that $S_n$ is ${\cal F}_n$-measurable and that $X_{n+1}$ is independent of ${\cal F}_n$.
So the key is the computation of expectation $$E[ {\rm e}^{X_{n+1}}].$$
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$\begingroup$ Thanks that explains it pretty well. But how do I calculate the last expectation? $\endgroup$ Apr 21, 2021 at 9:55
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$\begingroup$ @noisyoscillator This depends on the distribution of $X_{t+1}$ and cannot be answered in general. $\endgroup$– CetttApr 21, 2021 at 10:09
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$\begingroup$ @Cettt So the question itself is ill-posed? $\endgroup$ Apr 21, 2021 at 12:41
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1$\begingroup$ Sorry, my browser did not show the picture. Simply compute the expaction using basic probability theory $\endgroup$– CetttApr 21, 2021 at 12:45