A differential equation in which one or more terms are stochastic processes, typically written with a drift term and a diffusion term driven by a continuous martingale such as Brownian motion, and interpreted in a stochastic calculus sense (Itô or Stratonovich).

Principle

Model continuous-time evolution under both deterministic drift and random fluctuation by coupling an ordinary differential term with a stochastic integral; solution concepts (strong/weak) depend on adaptedness and filtration choices.

Demonstration

The Ornstein–Uhlenbeck SDE dX_t = θ(μ−X_t) dt + σ dW_t has Gaussian transition laws and an explicit stationary Gaussian distribution when θ>0.

Misapplication

Replacing stochastic integrals by ordinary Riemann integrals or ignoring the choice between Itô and Stratonovich interpretations when applying change-of-variable formulas.

Consequence

Solutions are stochastic processes (often Markov) whose laws satisfy associated forward equations for densities and whose sample paths exhibit random continuous behavior; existence and uniqueness depend on coefficient regularity.

Reversal

Setting the diffusion coefficient to zero reduces the equation to an ordinary differential equation driven by deterministic drift; conversely, removing drift yields a pure martingale-driven process.

Boundary

Requires a probability space and filtration and is distinct from random deterministic ODEs with fixed random coefficients; not all equations admit strong (pathwise) solutions and some need generalized notions of solution.

Semantic Tension

Often contrasted with deterministic forced systems with time-dependent randomness: an SDE encodes noise through stochastic integrals and filtration-adaptedness rather than fixed sample-path parameters.

Synthesis

A framework for continuous-time dynamical laws combining deterministic tendencies and stochastic forcing, equipped with stochastic calculus to define and analyze solutions and their distributions.