CHAPTER 11
Stochastic Calculus
Extend ordinary calculus to continuous-time systems whose evolution contains both systematic change and random fluctuations.
STOCHASTIC CALCULUS
Calculus with random change
Why Ordinary Calculus Is Not Enough
Understand why continuous random fluctuations require an extension of ordinary differential calculus.
Learn → 02Brownian Motion and Brownian Increments
Study Brownian motion, normally distributed increments and the \(\sqrt{\Delta t}\) scaling of random change.
Learn → 03Stochastic Differential Equations
Combine deterministic drift and Brownian diffusion in a continuous-time stochastic model.
Learn → 04The Itô Integral
Define integration with respect to Brownian motion and give meaning to stochastic accumulation.
Learn → 05Itô's Lemma
Understand the stochastic chain rule and the extra term caused by Brownian quadratic variation.
Learn → 06Euler–Maruyama Method
Approximate stochastic differential equations numerically and generate sample paths.
Learn → 07Moments of Stochastic Processes
Study expectation, second moments, variance, covariance and moment equations.
Learn → 08The Fokker–Planck Equation
Describe how the probability density of a stochastic diffusion evolves through time.
Learn → 09From Jump Processes to Diffusion Approximations
See how discrete random event models can lead systematically to drift and diffusion terms.
Learn → 10Stochastic Modelling and Interpretation
Interpret stochastic trajectories, uncertainty, threshold probabilities and modelling assumptions.
Learn → 11Deterministic and Stochastic Models Compared
Compare what deterministic and stochastic models describe and when uncertainty changes conclusions.
Learn →The central idea
The drift term describes systematic change. The diffusion term describes continuous random fluctuation. Stochastic calculus provides the rules needed to analyse both together.