Brownian motion
A continuous random path with independent Gaussian increments.
Library note. Check the assumptions and further reading before applying a formula.
What Is This?
Standard Brownian motion starts at zero. Over any time interval, its change is normally distributed with mean zero and variance equal to the interval length. Changes on disjoint intervals are independent. Its paths are continuous but almost surely nowhere differentiable. This makes it useful for modelling accumulated noise, while also requiring different calculus from smooth motion.
Try an example
Over four time units, a standard Brownian increment has standard deviation 2, not 4. The scale grows with the square root of elapsed time.
Where it needs care
Brownian motion is an idealised model. Financial returns can jump, have changing volatility and show heavier tails. Standard Brownian motion itself is not a realistic price model.
Historical Context
Robert Brown described irregular particle motion in 1827. Einstein and Smoluchowski explained its physical basis; Norbert Wiener later gave a mathematical construction.
Real-World Applications
- Model diffusion and accumulated small disturbances.
- Build stochastic differential equations.
