Stochastic Differential Equation

A stochastic differential equation has the general form $d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t$, where $\mu (X_t,t)$ is the drift coefficient (the deterministic part of the instantaneous change), $\sigma (X_t,t)$ is the diffusion coefficient (scaling the random noise), and $W_t$ is a Wiener process. Because $W_t$ is nowhere differentiable, this expression is formal shorthand for the integral equation $X_t=X_0+{\int }_0^t\mu (X_s,s)ds+{\int }_0^t\sigma (X_s,s)d{W}_s$, where the second integral is an Ito integral, a specifically-defined stochastic integral (not an ordinary Riemann-Stieltjes integral, which doesn't apply to paths of infinite variation). A solution process $X_t$ satisfying this equation is, by definition, a diffusion process. Under standard regularity conditions on $\mu$ and $\sigma$ (Lipschitz continuity and linear growth bounds), a unique strong solution exists; Ito's lemma is the essential calculus tool for manipulating functions of the solution $X_t$.

• The notation $d{X}_t=\mu dt+\sigma d{W}_t$ is deliberately informal, because $W_t$ has no derivative, $d{W}_t$ is not literally a derivative times $dt$, the rigorous meaning is always the corresponding Ito integral equation, a distinction that matters as soon as any calculus is performed on the SDE.
• SDEs are the common language unifying every named diffusion process in this category, Brownian motion, geometric Brownian motion, and the Ornstein-Uhlenbeck process are not separate models built from scratch, they are simply particular choices of the functions $\mu (x,t)$ and $\sigma (x,t)$ plugged into the same general SDE.
• Numerically simulating an SDE (since closed-form solutions like GBM's are the exception, not the rule) typically uses discretization schemes like the Euler-Maruyama method, the direct SDE analogue of Euler's method for ordinary differential equations, adapted to handle the stochastic term.