nLab martingale

Idea

In probability theory, a martingale is a stochastic process where, intuitively, one takes averages over the data which are not yet known. As time progresses, one's knowledge increases, and so the “amount of averaging” decreases.

It is one of the most widely used concepts in probability theory.

Definition

Let $(X,\mathcal{A},p,(\mathcal{F}_n))$ be a filtered probability space.

A martingale on $(X,\mathcal{A},p,(\mathcal{F}_n))$ is a stochastic process (collection of random variables) $f_n:X\to\mathbb{R}$ such that

• For every $n$, the function $f_n$ is $\mathcal{F}_n$-measurable (i.e. the process is adapted to the filtration:);
• For each $n\le m$, the function $f_n$ is a conditional expectation of $f_m$ given $\mathcal{F}_n$:
  $$f_n \;=\; \mathbb{E}(f_m|\mathcal{F}_n) .$$

Similar definitions can be given for random variables with values in other spaces, for example, Banach spaces.

Variations

A submartingale on a filtered probability space $(X,\mathcal{A},p,(\mathcal{F}_n))$ is an adapted stochastic process $(f_n)$ satisfying

$$f_n \;\le\; \mathbb{E}(f_m|\mathcal{F}_n)$$

for each $n\le m$.

Similarly, a supermartingale on $(X,\mathcal{A},p,(\mathcal{F}_n))$ is an adapted stochastic process $(f_n)$ satisfying

$$f_n \;\ge\; \mathbb{E}(f_m|\mathcal{F}_n)$$

for each $n\le m$.

A backward martingale or inverse martingale can be seen as a martingale in reverse time. Explicitly, given a probability space $(X,\mathcal{A},p)$ with a decreasing filtration $(\mathcal{F}_n)$ of sub-sigma-algebras (i.e. $\mathcal{F}_n\supseteq\mathcal{F}_m$ for $n\le m$), a backward martingale is an adapted process $(f_n)$ satisfying

$$f_m \;=\; \mathbb{E}(f_n|\mathcal{F}_m)$$

for each $n\le m$.

Properties

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See also

• probability theory, categorical probability

• random variable, expectation value, conditional expectation

• stochastic process, random walk?, white noise

• martingale convergence theorem?

• probability monad, mixture, empirical mean

References

• Ruben Van Belle, A categorical treatment of the Radon-Nikodym theorem and martingales, 2023. (arXiv)

• Paolo Perrone and Ruben Van Belle, Convergence of martingales via enriched dagger categories, 2024. (arXiv)

Introductory material

• Paolo Perrone, Convergence of martingales via enriched dagger categories, video talk, 2024. (YouTube)

category: probability