Martingales and Random Walk

Filtration

A filtration is a sequence of $\sigma$-algebras ${\mathcal{F}}_t$ that represents the information available up to time $t$. It is used to model the evolution of information over time. For a stochastic process $X_t$, the filtration ${\mathcal{F}}_t$ can be defined as ${\mathcal{F}}_t\equiv (X_0=x_0,X_1=x_1,...,X_t=x_t)$

Martingale

A stochastic process $X_t$ is a martingale with respect to a filtration ${\mathcal{F}}_t$ if:

1. $E[|X_t|]<\mathrm{\infty }$ for all $t$ (the expected value is finite).
2. $E[X_t|{\mathcal{F}}_s]=X_s$ with $s<t$/

INFO

Simply put, martingale property implies that the best prediction of the expectation of the future value is the current value, given all past information.

Random Walk

Random walk is a discrete stochastic process $S_t$ given by $S_t=\sum _{i=1}^t$ $X_i$, where $X_i$ are independent and identically distributed (i.i.d.) random variables with mean $E[X_i]=0$ and variance ${\sigma }^2=E[X_i^2]<\mathrm{\infty }$. The process starts at $S_0=0$. The $E[X_i]=0$ is not a requirement for random walk, but it is a commonly used condition.

Simple random walk

If $X_t$ takes values $+1$ and $-1$ with probabilities $p$ and $1-p$, respectively, then the random walk is called a simple random walk. The expected value of the position at time $t$ is given by: