Lecture 24: Stochastic Calculus

Quick Overview

Stochastic calculus extends ordinary calculus to model asset dynamics using stochastic processes, particularly Brownian motion, introducing the necessity of the Ito integral defined through nonanticipatory variables and foundational properties like the Ito isometry for convergence.

Key Points: Stochastic calculus models asset dynamics as integrals with respect to Brownian motion, accommodating randomness in increments. The increment for Brownian motion with drift (dX = mu dt + sigma dBt) follows a normal distribution with mean mu delta t and variance sigma squared delta t. The infinitesimal squared value of the Brownian motion increment has an expectation of order delta t, specifically equal to "sigma squared delta t plus little o of delta t". The Ito integral is formally defined by considering a probability process specified by a sample space (omega), a sigma field, and a probability measure, using the filtration Ft which represents information known up to time t. The Ito integral of B dBs is shown to be $\frac{1}{2} BT^2 - \frac{1}{2} T$, which differs from the real calculus result of $\frac{1}{2} x^2$ due to the term dependent on the variation of the process. Ito's formula for a function $f$ of $Bt$ with a continuous second derivative states $df(Bt) = f'(Bt)dBt + \frac{1}{2} f''(Bt)dt$, contrasting with the standard Taylor expansion by including the $\frac{1}{2} f'' dt$ term. The ruin problem for Brownian motion with drift uses martingale theory, defining a martingale $Mt = h(Xt)$ where $h$ must satisfy the PDE $\frac{\partial h}{\partial t} = -\frac{1}{2} \sigma^2 \frac{\partial^2 h}{\partial x^2}$ to find the probability of stopping at A before -B.

Context: This lecture, delivered by Peter Kempthorne, introduces stochastic calculus as an essential extension of ordinary calculus for quantitative finance, specifically focusing on modeling asset values using stochastic processes driven by Brownian motion. The discussion moves from reviewing Brownian motion with drift to formally defining the Ito integral, establishing its properties, and deriving Ito's formula for single and multiple variables, culminating in an application to the ruin problem via martingale theory.