nLab martingale

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Contents

Contents

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

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

  • For every nn, the function f nf_n is n\mathcal{F}_n-measurable (i.e. the process is adapted to the filtration:);
  • For each nmn\le m, the function f nf_n is a conditional expectation of f mf_m given n\mathcal{F}_n:
    f n=𝔼(f m| 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,𝒜,p,( n))(X,\mathcal{A},p,(\mathcal{F}_n)) is an adapted stochastic process (f n)(f_n) satisfying

f n𝔼(f m| n) f_n \;\le\; \mathbb{E}(f_m|\mathcal{F}_n)

for each nmn\le m.

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

f n𝔼(f m| n) f_n \;\ge\; \mathbb{E}(f_m|\mathcal{F}_n)

for each nmn\le m.

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

f m=𝔼(f n| m) f_m \;=\; \mathbb{E}(f_n|\mathcal{F}_m)

for each nmn\le m.

Properties

(…)

See also

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

Last revised on July 26, 2024 at 09:34:07. See the history of this page for a list of all contributions to it.