Brownian Motion and Stochastic Calculus
Brownian motion is the continuous random process whose increments are independent and Gaussian; the stochastic calculus built upon it provides the rules for integrating and differentiating along its erratic paths.
Definition
Brownian motion is a continuous-time process with independent stationary Gaussian increments and continuous nowhere-differentiable paths, and stochastic calculus is the theory of integration and differentiation with respect to such processes, centred on the Ito integral and Ito's change-of-variables formula.
Scope
This area covers the Wiener process and its path properties, the Ito stochastic integral and Ito's formula, stochastic differential equations and diffusion processes, the link to partial differential equations through Feynman-Kac and the Fokker-Planck equation, the Girsanov change of measure, and the extension to Levy processes with jumps.
Sub-topics
Core questions
- What properties characterise Brownian motion and make its paths so irregular?
- How is integration defined against Brownian motion despite its infinite variation?
- What is Ito's formula and how does it replace the ordinary chain rule?
- How do stochastic differential equations and Levy processes extend the framework?
Key theories
- Ito integral and Ito's formula
- The Ito integral defines integration against Brownian motion by exploiting the martingale property and the quadratic variation that equals elapsed time, and Ito's formula gives a change-of-variables rule with an extra second-derivative term reflecting that variation.
- Diffusions and the link to partial differential equations
- Solutions of stochastic differential equations are Markov diffusions whose transition densities solve the Fokker-Planck and backward Kolmogorov equations, and the Feynman-Kac formula represents solutions of parabolic equations as expectations over diffusion paths.
Clinical relevance
Brownian motion and stochastic calculus model the diffusion of particles and heat, the random fluctuation of asset prices in the Black-Scholes theory of option pricing, noise in physical and engineering systems, and filtering of noisy signals, making them indispensable across physics, finance, and control.
History
Brown observed the erratic motion of pollen grains in 1827, Einstein and Smoluchowski gave its physical theory around 1905, Bachelier had already used it for finance in 1900, Wiener constructed it rigorously in 1923, and Ito created the stochastic calculus in the 1940s that turned it into a computational tool.
Key figures
- Robert Brown
- Albert Einstein
- Norbert Wiener
- Kiyosi Ito
Related topics
Seminal works
- oksendal2003
- karatzasShreve1991
Frequently asked questions
- Why can't ordinary calculus be used for Brownian motion?
- Brownian paths have infinite total variation and are nowhere differentiable, so ordinary integrals and the classical chain rule fail; Ito's stochastic calculus supplies replacements that account for the quadratic variation.
- What is Ito's formula?
- It is the stochastic analogue of the chain rule for functions of Brownian motion or diffusions, including an additional term involving the second derivative that arises from the nonzero quadratic variation of the paths.