# Martingale problem on biased random walk

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?

• 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. Apr 21, 2021 at 4:07

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 $$E[ {\rm e}^{X_{n+1}}].$$

• Thanks that explains it pretty well. But how do I calculate the last expectation? Apr 21, 2021 at 9:55
• @noisyoscillator This depends on the distribution of $X_{t+1}$ and cannot be answered in general. Apr 21, 2021 at 10:09
• @Cettt So the question itself is ill-posed? Apr 21, 2021 at 12:41
• Sorry, my browser did not show the picture. Simply compute the expaction using basic probability theory Apr 21, 2021 at 12:45