= Stochastic differential equation
{title2=$dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dW_t$}
{wiki}
A stochastic differential equation specifies a <stochastic process> through a combination of ordinary <drift> and random increments. In the scalar <Itô integral> convention, the notation
$$
dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t
$$
means the <stochastic integral> equation $X_t=X_0+\int_0^tb(s,X_s)ds+\int_0^t\sigma(s,X_s)dW_s$, with the required measurability and local integrability. The driver $W$ is a <Brownian motion>; $b$ is the <drift coefficient> and $\sigma$ the <diffusion amplitude>.
For example, the <Ornstein-Uhlenbeck process> satisfies $dX_t=-\lambda X_tdt+\sigma dW_t$: linear restoring <drift> competes with persistent noise. A <strong solution of a stochastic differential equation> uses a prescribed driver, whereas a <weak solution of a stochastic differential equation> may construct the probability space and driver. <Pathwise uniqueness>, <uniqueness in law> and the lifetime of a <maximal local solution of a stochastic differential equation> are separate questions. If the equation uses a <Stratonovich integral>, its convention changes the drift conversion and must be stated.
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