--- title: Cherchez la martingale subtitle: Stuff about probability and orthogonality date: '2019-11-25T18:09:40+01:00' date-modified: '2019-11-30T18:09:40+01:00' link-citations: true bibupdated: '2023-12-16T16:55:12' slug: martingales nocite: - '@*' bibliography: martingales.yaml bibliography-alt: martingales.bib csl: minimalist.csl reference-section-title: References categories: - dynamical systems - Hilbert space - linear algebra - probability - SDEs - signal processing - time series polish: 0 novelty: 0 certainty: 0 usefulness: 0 images: - /images/faun_fairy_trail.png image: /images/faun_fairy_trail.png --- Cherchez la martingale — The Dan MacKinlay stable of variably-well-consider’d enterprises

Cherchez la martingale

Stuff about probability and orthogonality

November 26, 2019 — December 1, 2019

dynamical systems
Hilbert space
linear algebra
probability
SDEs
signal processing
time series
Figure 1

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

Aalen. 1978. Nonparametric Inference for a Family of Counting Processes.” The Annals of Statistics.
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.
Duembgen, and Podolskij. 2015. High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications.
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.
Isaev, and McKay. 2016. Complex Martingales and Asymptotic Enumeration.” arXiv:1604.08305 [Math].
Jacod. 1997. On Continuous Conditional Gaussian Martingales and Stable Convergence in Law.” In Séminaire de Probabilités XXXI. Lecture Notes in Mathematics 1655.
Jacod, and Protter. 1988. Time Reversal on Levy Processes.” The Annals of Probability.
Komorowski, Landim, and Olla. 2012. Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Grundlehren Der Mathematischen Wissenschaften : A Series of Comprehensive Studies in Mathematics 345.
Kontorovich, and Raginsky. 2016. Concentration of Measure Without Independence: A Unified Approach via the Martingale Method.” arXiv:1602.00721 [Cs, Math].
Kühn. 2018. Existence of (Markovian) Solutions to Martingale Problems Associated with Lévy-Type Operators.” arXiv:1803.05646 [Math].
Kurtz. 1980. Representations of Markov Processes as Multiparameter Time Changes.” The Annals of Probability.
Li. 2012. Continuous-State Branching Processes.” arXiv:1202.3223 [Math].
McCauley, Bassler, and Gunaratne. 2008. Martingales, Nonstationary Increments, and the Efficient Market Hypothesis.” Physica A: Statistical and Theoretical Physics.
Podolskij, and Vetter. 2010. Understanding Limit Theorems for Semimartingales: A Short Survey: Limit Theorems for Semimartingales.” Statistica Neerlandica.
Raginsky, and Sason. 2012. Concentration of Measure Inequalities in Information Theory, Communications and Coding.” Foundations and Trends in Communications and Information Theory.
Rakhlin, Sridharan, and Tewari. 2014. Sequential Complexities and Uniform Martingale Laws of Large Numbers.” Probability Theory and Related Fields.
Robbins, and Siegmund. 1971. A Convergence Theorem for Non Negative Almost Supermartingales and Some Applications.” In Optimizing Methods in Statistics.
Sørensen. 2000. Prediction-Based Estimating Functions.” The Econometrics Journal.
Taleb. 2018. Election Predictions as Martingales: An Arbitrage Approach.” Quantitative Finance.
van de Geer. 1995. Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.” The Annals of Statistics.