# Cherchez la martingale

### Stuff about probability and orthogonality

Usefulness: đź”§
Novelty: đź’ˇ
Uncertainty: đź¤Ş đź¤Ş đź¤Ş
Incompleteness: đźš§ đźš§ đźš§

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 soemthing therefby; This idea connects and solves a bunch of tricky problems at once.

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

I am indebted to Saif for setting my head straight about the utility of martingales, and Kevin Ross who, in part of Amir Demboâ€™s course materials, was the one whose explanation of the orthogonality interpretation of martingales finally communicated the neatness of this idea to even my meagre intelligence.

TBC.

# Refs

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Adelfio, Giada, and Frederic Paik Schoenberg. 2009. â€śPoint Process Diagnostics Based on Weighted Second-Order Statistics and Their Asymptotic Properties.â€ť Annals of the Institute of Statistical Mathematics 61 (4): 929â€“48. https://doi.org/10.1007/s10463-008-0177-1.

Athreya, Krishna B, and S. N Lahiri. 2006. Measure Theory and Probability Theory. New York: Springer. http://link.springer.com/chapter/10.1007/978-0-387-35434-7_19.

Bibby, Bo Martin, and Michael SĂ¸rensen. 1995. â€śMartingale Estimation Functions for Discretely Observed Diffusion Processes.â€ť Bernoulli 1 (1/2): 17â€“39. https://doi.org/10.2307/3318679.

BrĂ©maud, Pierre. 1972. â€śA Martingale Approach to Point Processes.â€ť University of California, Berkeley.

Burgess, Nicholas. 2014. â€śMartingale Measures & Change of Measure Explained.â€ť SSRN Scholarly Paper ID 2961006. Rochester, NY: Social Science Research Network. https://papers.ssrn.com/abstract=2961006.

Doob, J. L. 1949. â€śApplication of the Theory of Martingales.â€ť In Le Calcul Des ProbabilitĂ©s et Ses Applications, 23â€“27. Colloques Internationaux Du Centre National de La Recherche Scientifique, No. 13. Centre National de la Recherche Scientifique, Paris. http://www.ams.org/mathscinet-getitem?mr=0033460.

Duembgen, Moritz, and Mark Podolskij. 2015. â€śHigh-Frequency Asymptotics for Path-Dependent Functionals of ItĂ´ Semimartingales.â€ť Stochastic Processes and Their Applications 125 (4): 1195â€“1217. https://doi.org/10.1016/j.spa.2014.08.007.

Geer, Sara van de. 1995. â€śExponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes.â€ť The Annals of Statistics 23 (5): 1779â€“1801. https://doi.org/10.1214/aos/1176324323.

Heyde, C. C. 1974. â€śOn Martingale Limit Theory and Strong Convergence Results for Stochastic Approximation Procedures.â€ť Stochastic Processes and Their Applications 2 (4): 359â€“70. https://doi.org/10.1016/0304-4149(74)90004-0.

Heyde, C. C., and E. Seneta. 2010. â€śEstimation Theory for Growth and Immigration Rates in a Multiplicative Process.â€ť In Selected Works of C.C. Heyde, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214â€“35. Selected Works in Probability and Statistics. Springer New York. http://link.springer.com/chapter/10.1007/978-1-4419-5823-5_31.

Isaev, Mikhail, and Brendan D. McKay. 2016. â€śComplex Martingales and Asymptotic Enumeration,â€ť April. http://arxiv.org/abs/1604.08305.

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Jacod, Jean, and Philip Protter. 1988. â€śTime Reversal on Levy Processes.â€ť The Annals of Probability 16 (2): 620â€“41. https://doi.org/10.1214/aop/1176991776.

Komorowski, Tomasz, Claudio Landim, and Stefano Olla. 2012. Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Grundlehren Der Mathematischen Wissenschaften : A Series of Comprehensive Studies in Mathematics 345. Heidelberg [Germany] ; New York: Springer.

Kontorovich, Aryeh, and Maxim Raginsky. 2016. â€śConcentration of Measure Without Independence: A Unified Approach via the Martingale Method,â€ť February. http://arxiv.org/abs/1602.00721.

Kurtz, Thomas G. 1980. â€śRepresentations of Markov Processes as Multiparameter Time Changes.â€ť The Annals of Probability 8 (4): 682â€“715. https://doi.org/10.1214/aop/1176994660.

KĂĽhn, Franziska. 2018. â€śExistence of (Markovian) Solutions to Martingale Problems Associated with LĂ©vy-Type Operators,â€ť March. http://arxiv.org/abs/1803.05646.

Li, Zenghu. 2012. â€śContinuous-State Branching Processes,â€ť February. http://arxiv.org/abs/1202.3223.

McCauley, Joseph L, Kevin E Bassler, and Gemunu H Gunaratne. 2008. â€śMartingales, Nonstationary Increments, and the Efficient Market Hypothesis.â€ť Physica A: Statistical and Theoretical Physics 387 (15): 3916â€“20. https://doi.org/10.1016/j.physa.2008.01.049.

Podolskij, Mark, and Mathias Vetter. 2010. â€śUnderstanding Limit Theorems for Semimartingales: A Short Survey: Limit Theorems for Semimartingales.â€ť Statistica Neerlandica 64 (3): 329â€“51. https://doi.org/10.1111/j.1467-9574.2010.00460.x.

Raginsky, Maxim, and Igal Sason. 2012. â€śConcentration of Measure Inequalities in Information Theory, Communications and Coding.â€ť Foundations and Trends in Communications and Information Theory, December. http://arxiv.org/abs/1212.4663.

Rakhlin, Alexander, Karthik Sridharan, and Ambuj Tewari. 2014. â€śSequential Complexities and Uniform Martingale Laws of Large Numbers.â€ť Probability Theory and Related Fields 161 (1-2): 111â€“53. https://doi.org/10.1007/s00440-013-0545-5.

Robbins, H., and D. Siegmund. 1971. â€śA Convergence Theorem for Non Negative Almost Supermartingales and Some Applications.â€ť In Optimizing Methods in Statistics, edited by Jagdish S. Rustagi, 233â€“57. Academic Press. https://doi.org/10.1016/B978-0-12-604550-5.50015-8.

SĂ¸rensen, Michael. 2000. â€śPrediction-Based Estimating Functions.â€ť The Econometrics Journal 3 (2): 123â€“47.

Taleb, Nassim Nicholas. 2018. â€śElection Predictions as Martingales: An Arbitrage Approach.â€ť Quantitative Finance 18 (1): 1â€“5. https://doi.org/10.1080/14697688.2017.1395230.