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.
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
Similar definitions can be given for random variables with values in other spaces, for example, Banach spaces.
A submartingale on a filtered probability space $(X,\mathcal{A},p,(\mathcal{F}_n))$ is an adapted stochastic process $(f_n)$ satisfying
for each $n\le m$.
Similarly, a supermartingale on $(X,\mathcal{A},p,(\mathcal{F}_n))$ is an adapted stochastic process $(f_n)$ satisfying
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
for each $n\le m$.
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stochastic process, random walk?, white noise
martingale convergence theorem?
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)
Last revised on July 26, 2024 at 09:34:07. See the history of this page for a list of all contributions to it.