Cherchez la martingale

Stuff about probability and orthogonality

November 26, 2019 — December 1, 2019

Like Markov processes, a weirdly useful class of stochastic processes. Often you can find a martingale within some stochastic process, or construct a martingale from a stochastic process and prove something nifty thereby; This idea connects and solves a bunch of tricky problems at once.

TODO: examples, maybe a CLT and something else wacky like the life table estimators of (Aalen 1978).

I am indebted to Saif Syed for setting my head straight about the utility of martingales, and Kevin Ross who, in part of Amir Dembo’s course materials, was the one whose explanation of the orthogonality interpretation of martingales finally communicated the neatness of this idea to me.

TBC.

1 Local martingales

The classical gambling strategy (double-down until you win) is in fact a local martingale.

2 References

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Adelfio, and Schoenberg. 2009. Point Process Diagnostics Based on Weighted Second-Order Statistics and Their Asymptotic Properties.” Annals of the Institute of Statistical Mathematics.
Athreya, and Lahiri. 2006. Measure theory and probability theory.
Bibby, and Sørensen. 1995. Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli.
Bielecki, Jakubowski, and Niewęgłowski. 2020. Fundamentals of the Theory of Structured Dependence Between Stochastic Processes. Encyclopedia of Mathematics and Its Applications.
Brémaud. 1972. “A Martingale Approach to Point Processes.”
Burgess. 2014. Martingale Measures & Change of Measure Explained.” SSRN Scholarly Paper ID 2961006.
Doob. 1949. Application of the Theory of Martingales.” In Le Calcul Des Probabilités Et Ses Applications. Colloques Internationaux Du Centre National de La Recherche Scientifique, No. 13.
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Heyde. 1974. On Martingale Limit Theory and Strong Convergence Results for Stochastic Approximation Procedures.” Stochastic Processes and Their Applications.
Heyde, and Seneta. 2010. Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde. Selected Works in Probability and Statistics.
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Jacod, and Protter. 1988. Time Reversal on Levy Processes.” The Annals of Probability.
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van de Geer. 1995. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.” The Annals of Statistics.