Cherchez la martingale

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

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 Dembo’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

Aalen, Odd. 1978. “Nonparametric Inference for a Family of Counting Processes.” The Annals of Statistics 6 (4): 701–26. https://doi.org/10.1214/aos/1176344247.

Adelfio, Giada, and Frederic Paik Schoenberg. 2009. “Point Process Diagnostics Based on Weighted Second-Order Statistics and Their Asymptotic Properties.” Annals of the Institute of Statistical Mathematics 61 (4): 929–48. 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 Sørensen. 1995. “Martingale Estimation Functions for Discretely Observed Diffusion Processes.” Bernoulli 1 (1/2): 17–39. https://doi.org/10.2307/3318679.

Brémaud, Pierre. 1972. “A Martingale Approach to Point Processes.” University of California, Berkeley.

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

Doob, J. L. 1949. “Application of the Theory of Martingales.” In Le Calcul Des Probabilités et Ses Applications, 23–27. 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. “High-Frequency Asymptotics for Path-Dependent Functionals of Itô Semimartingales.” Stochastic Processes and Their Applications 125 (4): 1195–1217. https://doi.org/10.1016/j.spa.2014.08.007.

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

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

Heyde, C. C., and E. Seneta. 2010. “Estimation Theory for Growth and Immigration Rates in a Multiplicative Process.” In Selected Works of C.C. Heyde, edited by Ross Maller, Ishwar Basawa, Peter Hall, and Eugene Seneta, 214–35. 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. “Complex Martingales and Asymptotic Enumeration,” April. http://arxiv.org/abs/1604.08305.

Jacod, Jean. 1997. “On Continuous Conditional Gaussian Martingales and Stable Convergence in Law.” In Séminaire de Probabilités XXXI, edited by Jacques Azéma, Marc Yor, and Michel Emery, 232–46. Lecture Notes in Mathematics 1655. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0119308.

Jacod, Jean, and Philip Protter. 1988. “Time Reversal on Levy Processes.” The Annals of Probability 16 (2): 620–41. 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. “Concentration of Measure Without Independence: A Unified Approach via the Martingale Method,” February. http://arxiv.org/abs/1602.00721.

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

Kühn, Franziska. 2018. “Existence of (Markovian) Solutions to Martingale Problems Associated with Lévy-Type Operators,” March. http://arxiv.org/abs/1803.05646.

Li, Zenghu. 2012. “Continuous-State Branching Processes,” February. http://arxiv.org/abs/1202.3223.

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

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

Raginsky, Maxim, and Igal Sason. 2012. “Concentration of Measure Inequalities in Information Theory, Communications and Coding.” Foundations and Trends in Communications and Information Theory, December. http://arxiv.org/abs/1212.4663.

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

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

Sørensen, Michael. 2000. “Prediction-Based Estimating Functions.” The Econometrics Journal 3 (2): 123–47.

Taleb, Nassim Nicholas. 2018. “Election Predictions as Martingales: An Arbitrage Approach.” Quantitative Finance 18 (1): 1–5. https://doi.org/10.1080/14697688.2017.1395230.