SDEs

Stochastic differential equations (SDEs) describe the evolution of systems influenced by random effects. They extend ordinary differential equations by including terms that model noise, typically represented by Brownian motion. An SDE in n dimensions often has the form dX_t = a(X_t,t) dt + B(X_t,t) dW_t, where X_t is the state, a is the drift vector, B is the diffusion matrix, and W_t is a vector of independent Wiener processes. Solutions are stochastic processes whose paths depend on the realized noise.

Two main calculi are used to manipulate SDEs: Itô calculus and Stratonovich calculus. Itô calculus treats the stochastic integral in a non-anticipative way and leads to Itô's lemma, a stochastic chain rule essential for transforming variables. Stratonovich calculus aligns with ordinary chain rule intuition and is often preferred in certain physical models.

Mathematical analysis of SDEs addresses questions of existence and uniqueness of solutions, regularity, and long-term behavior. Sufficient conditions typically involve Lipschitz continuity and linear growth bounds on the drift and diffusion coefficients.

Numerical methods are used to simulate sample paths. The Euler–Maruyama method is the simplest explicit scheme; higher-order schemes such as Milstein improve accuracy. In practice, discretization errors and interpretation of the noise term matter for simulations.

Applications span finance, physics, biology, and engineering. In finance, asset prices are modeled by SDEs leading to models like Black–Scholes; in physics, Langevin equations describe particles under random forces; in biology and engineering, SDEs model population fluctuations or noisy control systems.