Lecture 14: Stochastic Processes II

Quick Overview

The lecture defines Brownian motion mathematically, detailing its properties such as normally distributed, independent, and continuous increments, and establishes its formal definition as the limiting process of a normalized random walk, independent of the underlying step distribution.

Key Points: Albert Einstein formalized the Brownian motion model mathematically, and Norbert Wiener developed its mathematical foundation. One-dimensional Brownian motion $Bt$ starts at 0, has increments that are normally distributed with mean 0 and variance proportional to $\Delta t$, and possesses independent, disjoint increments. Brownian motion is a Markov process, and the variance of $Bt$ increases with time, resulting in a standard deviation proportional to $\sigma\sqrt{t}$. The covariance between $Bs$ and $Bt$ (with $s < t$) equals $\sigma^2 s$, depending on the overlap of the time intervals. Brownian motion can be formally defined as the limiting stochastic process corresponding to a random walk where the normalized partial sums converge to a Gaussian distribution, regardless of the initial step distribution. The probability of the maximum of Brownian motion up to time $t$ exceeding $x$ is twice the probability that the ending value $Bt$ exceeds $x$, leading to the formula $P(M(t) > x) = 2 P(Bt > x)$. The quadratic variation of Brownian motion over an interval $[0, T]$ equals $T$ with probability 1, highlighting a key regularity in the process.

Context: This lecture, titled "Lecture 14: Stochastic Processes II," introduces and mathematically analyzes Brownian motion, a stochastic process named after botanist Robert Brown, who observed the dynamics in nature. Peter Kempthorne outlines the historical context, crediting Einstein for the mathematical formalization and Wiener for developing the foundation, before transitioning into the rigorous definition and properties of the process in one dimension.

Detailed Analysis

The lecture meticulously defines one-dimensional Brownian motion ($Bt$) as a continuous-time stochastic process starting at 0, characterized by normally distributed, independent, and disjoint increments whose variance is proportional to the time interval length $\Delta t$. Key properties discussed include its Markov nature, zero expected value (if starting at 0), and variance growing linearly with time, $\text{Var}(Bt) = \sigma^2 t$. The covariance function is derived as $\text{Cov}(Bs, Bt) = \sigma^2 \min(s, t)$. The speaker then connects the probability model to the diffusion equation, noting its relation to the heat equation and Fokker-Planck equation. Crucially, Brownian motion is formally defined as the limit of a normalized random walk as the number of steps increases, a limit which is invariant to the underlying step distribution. Advanced properties include the reflection principle, which yields the result that the probability of the maximum exceeding a level $x$ is twice the probability of the endpoint exceeding $x$. Extensions discussed include reflected Brownian motion, absorbed Brownian motion, and the Brownian Bridge, which arises from the normalized empirical distribution function of uniform random variables. Finally, the lecture covers Brownian motion with drift ($\mu$) and its application to solving the gambler's ruin problem via a second-order differential equation for the hitting probability function $u(x)$, and emphasizes the non-differentiability of Brownian motion contrasted with its constant quadratic variation equal to the time elapsed, $T$.