Random Processes pp 182-199 | Cite as

# Martingales

Chapter

## Abstract

We consider a family of processes called martingales. Though a number of other illustrative examples will be given, the most immediate interpretation is that of a fair game. The discrete parameter case is considered for simplicity. The parameter almost surely, the sequence {

*n*is assumed to run over all the integers, the positive integers or the negative integers. Let*X*_{ n }be a sequence of random variables with*B*_{ n }a corresponding nondecreasing sequence of Borei subfields of*F*. The random variable*X*_{ n }is assumed to be measurable with respect to*B*_{n}. In fact,*B*_{ n }is often taken to be the Borei field generated by*X*_{ k },*k*≤*n*, though this is not necessarily the case in our discussion. Also*let**E*[|*X*_{n}|] < ∞*for each n. The sequence*{*X*_{n}}*is called a martingale with respect to the Borel fields*{*B*_{n}}*if*$$ {X_m} = E\left[ {{X_n}|{\mathcal{B}_m}} \right],\quad m < n, $$

(1)

*almost surely*. Think of a sequence of gambles at the times*n*.*B*_{n}can be thought of as corresponding to the information available to the player at time*n*.*X*_{ n }is the cumulative gain (or loss) of the player up to and including time*n*. Condition (1) then states that the sequence of plays is “fair” in the sense that the conditional net gain from time*m*to*n*given information up to and including time*m*is zero. If the condition (1) is replaced by$$ {X_m} \geqslant E\left[ {{X_n}|{\mathcal{B}_n}} \right],\quad m < n, $$

(2)

*X*_{ n }} is called a*supermartingale relative to*{*B*_{ n }}.## Preview

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## References

- 2.P. Lévy (see his
*Theorié de l’Addition des Variables Aléatoires*, 1937) obtained the earliest results on martingales. Much of the development of interest in martingales and their application is due to the research and influence of J. L. Doob [12].Google Scholar

## Copyright information

© Springer-Verlag New York Inc. 1974