A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process.
Use in physics
For example a general, coupled set of first-order SDEs (note that there are standard techniques for transforming a higher-order equation into several coupled first-order equations by introducing new unknowns) is often written in the form:
where 
is the set of unknowns, the fi and gi are arbitrary functions and the ηm are random functions of time, often referred to as "noise terms". If the gi are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise.
The main method of solution is by use of the Fokker-Planck equation, which provides a deterministic equation satisfied by the time dependant probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integration have also been used, drawing on the analogy between statistical physics and quantum mechanics (for example the Fokker-Planck equation can be transformed into the Schrodinger equation by rescaling a few variables).
Use in probability and financial mathematics
The notation used in the context of financial mathematics as well as in probability theory is slightly different. A typical equation is of the form
This equation should be interpreted as a slightly colloquial way of expressing the corresponding integral equation
The equation above characterises the behavior of the continuous time stochastic process Xt as the sum of an ordinary Lebesgue integral and an Itô integral.
The formal interpretation of an SDE is given in terms of what constitutes a solution to the SDE. There are two main definitions of a solution to an SDE, a strong solution and a weak solution. Both require the existence of
a process Xt that solves the integral equation version of the SDE. The difference between the two lies in the underlying probability space 
. A weak solution consists of a
probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space.
An important example is the equation for geometric Brownian motion
which is the equation for the dynamics of the price of a stock in the Black Scholes options pricing model of financial mathematics.