Testing the Martingale Property of an Exponential Random Walk

Summary

The document outlines how to test whether an exponential process built from a biased random walk is a martingale, submartingale, or supermartingale. It first specifies the natural filtration generated by the walk so the process is adapted, and notes that bounded increments support integrability. The central test is to compare the conditional expectation of the next process value with its current value.

Because the current value is measurable with respect to the present information, and the next increment is independent of that information, the conditional expectation reduces to the current value multiplied by the expected exponential of the next increment. The classification therefore depends on whether that multiplier equals, exceeds, or falls below one. The excerpt presents the setup and reduction but does not give the increment distribution’s full calculation or explicitly resolve the fair-walk case, so those conclusions require the missing model details.

Key ideas

• A filtration must be specified to evaluate the martingale property relative to available information.
• The process is adapted to the natural filtration generated by the random walk.
• Independence lets the conditional expectation factor into the current process value and the next increment's exponential expectation.
• Compare that expectation multiplier with one to classify the process as a martingale, submartingale, or supermartingale.
• The excerpt does not fully calculate the fair-walk case without more detail about the increment distribution.

Tags

# Martingale problem on biased random walk


# 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?

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/63496

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.