Martingale

This is a short exploration of Martingale and its properties while looking at multiple phenomena that shows this behavior like Brownian motion and gambling.

Introduction

Martingales are a class of stochastic processes that have found wide applications in various fields, including probability theory, statistics, and mathematical finance. A martingale is a sequence of random variables X₀, X₁, X₂… such that for any time n:

1. The expected value of Xₙ is finite, i.e., E(|Xₙ|) < ∞
2. The conditional expectation of the next value , given the current and past values X₀, X₁, X₂…Xₙ is equal to the current value Xₙ, i.e E(Xₙ₊₁|X₀, X₁, X₂…Xₙ) = Xₙ.

Interpretation & Properties

The martingale property can be interpreted as a fair game condition. Given all the information up to the present, the expected value of the process at the next time step is equal to its current value. This implies that, on average, there is no tendency for the process to increase or decrease.

A classic albeit academic example is a random walk or a drunkard’s walk. The sequence of drunkard’s steps is a sequence independent and identically distributed random variables Z₀, Z₁, Z₂… where each Zᵢ takes the values +1 or -1 with equal probability. The drunkard’s current position Xₙ = Z₀ + Z₁ + Z₂ + Zₙ is a martingale.

Some interesting properties are:

1. Boundedness: If a martingale is bounded, meaning that there exists a constant M such that |Xₙ| ≤ M for all n, then the martingale converges almost surely.
2. Optional Stopping Theorem: Under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value. This theorem has important applications in finance and gambling theory.

Fun example: Gambling

Accumulated Wealth in a Fair Game: Suppose you start with an initial amount of money and you play a fair game, where you have an equal chance of winning or losing a fixed amount. If you repeatedly play such that you double the bet after every loss, your accumulated wealth after each round forms a martingale.

Say the gambler starts with $1, then loses and doubles the bet for n times and wins the round n+1. As shown in the equation, the profit is 1 which is independent of N. That means the gambler recovers all his losses and makes a profit as soon as he wins 1 spin.

The rationale is that eventually you will win, and when you do all losses will be recovered in addition to a profit. While the theory is mathematically sound, it is a highly risky strategy due to a couple of reasons. Firstly, the potential losses can quickly escalate. Secondly, the strategy relies on unlimited bankroll and no betting limits both of which are not seen in real-world gambling. With these constraints in place, it is often not as good a strategy as demonstrated in the experiments.

Serious example: Brownian motion

Imagine a tiny particle suspended in a liquid. It constantly gets bumped around by the liquid molecules, resulting in a random, jittery movement. Brownian motion is this random motion of particles suspended in a medium.

Journey of a particle suspended in a liquid

The Brownian motion can be modeled by a random walk and equivalently, a martingale. An intuitive picture is Brownian motion has no memory due to independent increments. So, past movements don’t help predict future ones. This Wikipedia image shows a representative journey of such a tiny particle.

Tailpiece — Submartingale & Doob’s Martingale Inequalities

A discrete-time submartingale is a sequence of integrable random variables satisfying X₀, X₁, X₂… such that E(Xₙ₊₁|X₀, X₁, X₂…Xₙ) ≥ Xₙ. A good model is a bacteria colony growing in a petri dish. While the number of bacteria might fluctuate due to resource availability or other factors, if the colony is generally expected to grow, the number of bacteria at time t can be modeled as a submartingale. The expected future number of bacteria, given the current count, would be greater than or equal to the current count.

An interesting result is Doob’s inequality which gives a bound on the probability that a submartingale exceeds any given value over a given interval of time. Take the model of the bacteria colony. We expect the colony to grow, but the actual number fluctuates due to various factors and has an expected maximum size. Alternatively, the colony is unlikely to get much larger than some value. And that value for those interested can be viewed by the above equation which shows how probability is extremely less with large bounds.