Cherchez la martingale
One weird trick in stochastic processes
2019-11-25 — 2019-11-30
Wherein the Ubiquity of Martingales in Stochastic Processes Is Expounded, and the Classical Double‑down Gambling Strategy Is Identified as a Local Martingale, While Possible Applications to Limit Theorems and Estimators Are Suggested
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.
1 Fair games
A martingale formalizes a “fair” game: \(\mathbb{E}[X_{n+1} \mid X_1, \ldots, X_n] = X_n\) — my expected holdings after the next round are my holdings now.
Consider a multiplicative coin flip: my wealth \(W_n\) doubles on heads, but halves on tails. In log coordinates the game is fair in the martingale sense: \(\log_2 W_n\) steps \(\pm 1\) with equal probability. This is a simple symmetric random walk, hence a martingale. In level (i.e. primal) coordinates the same game looks favourable: \(\mathbb{E}[W_{n+1} \mid W_n] = \tfrac12\left(2 + \tfrac12\right) W_n = \tfrac54 W_n\), which makes \(W_n\) a submartingale.1 (“Submartingale” is a fancy way of saying “favourable game”: my expected wealth grows over time. The naming was chosen to confuse the hell out of people.)
Multiply wealth by \(1.5\) on heads and \(0.6\) on tails, and the level process is a submartingale (\(\mathbb{E}[W_n]\) grows 5% per round) while the log process is a supermartingale (\(\mathbb{E}[\log W_n]\) shrinks by \(\tfrac12\log\tfrac{10}{9}\) per round, so it is an unfavourable game). Ergodicity economics dines out on this observation.
2 Local martingales
The classical gambling strategy (double-down until you win) is in fact a local martingale. 🚧TODO🚧