Diffusion Process

A diffusion process is a continuous-time Markov process $\{X_t\}$ with continuous sample paths, characterized locally by a drift coefficient $\mu (x,t)={\lim }_{h\to 0}\frac{1}{h}\mathbb{E}[X_{t+h}-X_t\mid X_t=x]$ and a diffusion coefficient ${\sigma }^2(x,t)={\lim }_{h\to 0}\frac{1}{h}\text{Var}(X_{t+h}-X_t\mid X_t=x)$. Equivalently, a diffusion process is any process that solves a stochastic differential equation of the form $d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t$. Brownian motion is the diffusion process with $\mu =0,\sigma =1$ (constant coefficients); geometric Brownian motion has $\mu (x,t)=\mu x$, $\sigma (x,t)=\sigma x$ (state-dependent, proportional coefficients); the Ornstein-Uhlenbeck process has $\mu (x,t)=\theta (\bar{\mu }-x)$ (mean-reverting drift). The probability density of a diffusion process evolves according to the Fokker-Planck (forward Kolmogorov) equation, the continuous-state analogue of the Kolmogorov forward equation for continuous-time Markov chains.

• 'Diffusion process' is the umbrella category, and the choice of drift and diffusion coefficient functions is what distinguishes every specific named process within it, learning the general SDE template makes each specific process (Brownian motion, GBM, OU) a special case rather than a separately memorized formula.
• The name 'diffusion' comes directly from physics, the same Fokker-Planck equation governing a diffusion process's probability density is mathematically identical in form to the heat/diffusion equation describing how a physical quantity like heat or a chemical concentration spreads out over time.
• Diffusion processes underlie the mathematics of continuous-time finance broadly (not just GBM), interest-rate models, stochastic volatility models, and credit-risk models are all built by choosing different drift and diffusion coefficient functions within this same general framework.