Martingales

A martingale is a stochastic process $\{X_n\}$ where, given all information up to time $n$, the expected future value equals the current value: $$E[X_{n+1} | X_1, X_2, \ldots, X_n] = X_n$$ Intuition: A martingale is a "fair game." Your best forecast of the future is the present. A gambler's wealth in a fair casino is a martingale — on average, each bet neither increases nor decreases your fortune. The process has "no drift." Examples of martingales: - A symmetric random walk $S_n = X_1 + X_2 + \cdots + X_n$ where $E[X_i] = 0$ - Brownian motion $W_t$ - $W_t^2 - t$ (Brownian motion squared, with the drift subtracted) - $e^{\sigma W_t - \sigma^2 t/2}$ (the exponential martingale — key in risk-neutral pricing) - A gambler's fortune in a fair game Non-examples: - A biased random walk (has drift — it's a submartingale if drift is positive) - $W_t^2$ alone (has drift $t$, so $E[W_t^2] = t \neq W_s^2$) Concrete example: You start with \$100 and repeatedly bet \$1 on a fair coin. Your wealth $W_n$ is a martingale: $E[W_{n+1}|W_n] = W_n + \frac{1}{2}(1) + \frac{1}{2}(-1) = W_n$. Key properties: - $E[X_n] = E[X_0]$ for all $n$ (constant expected value) - Under regularity conditions, $E[X_T] = E[X_0]$ for stopping times $T$ (optional stopping theorem) - Martingale convergence: bounded martingales converge almost surely When to use: Proving results about random walks and gambling (gambler's ruin, stopping times). Central to the risk-neutral pricing framework — discounted asset prices are martingales under the risk-neutral measure. If you can express a quantity as a martingale, you unlock powerful tools. This is a fundamental technique — martingale theory is one of the pillars of modern probability and mathematical finance.