stochastic differential equation

Consider the ordinary differential equationMathworldPlanetmath, for example, the population growth model


where a(t) is the relative rate of growth at time t, and X(t) is the solution-trajectory of the system.

But we may want to take into account, in our model, the randomness or the uncertainty of our knowledge of the data. In this case we may introduce the data a(t) as:


where N(t) is a noise term, represented by a random variableMathworldPlanetmath with some postulated probability distribution.

In general, stochastic differential equations can be posed in the case that the infinitesimal increment dX(t) is a Gaussian random variable. (Other types of random variables are also possible, but require extensionsPlanetmathPlanetmath of the basic theory.) A stochastic differential equation (SDE) is an equation of the form:


where ω lives in some probability spaceMathworldPlanetmath, and W(t) is a Wiener processMathworldPlanetmath on that probability space. The real-valued functions μ and σ are to satisfy certain measurability requirements, and are usually assumed to be known, with the process X(t) being sought.

The argument ω is usually suppressed in the notation:

dX(t)=μ(t)dt+σ(t)dW(t), (1)

with the understanding that X(t), W(t), μ(t) and σ(t) denote random variables for each time t.

The interpretationMathworldPlanetmathPlanetmath of the stochastic differential equation (1) is that a process X(t) satisfies it if and only if it satisfies this relationMathworldPlanetmath amongst integrals:

X(t1)-X(t0)=t0t1μ(t)𝑑t+t0t1σ(t)𝑑W(t) (2)

for all times t0 and t1. The last integral is an Itô integral.

In many cases, the coefficients μ and σ depend on X(t) itself:


In this case, equation (2) does not give an explicit solution for the stochastic differential equation. Nevertheless, there are theorems analogous to those of ordinary differential equations, that guarantee existence of solutions given certain bounds on the growth of the coefficients μ(t,x) and σ(t,x).

In simpler cases, stochastic differential equations that involve unknowns on the right-hand side may still be solved explicitly using changes of variables (often called Itô’s formulaMathworldPlanetmathPlanetmath in this context). For example,


(for any initial conditionMathworldPlanetmath X0) provides a solution to:



  • 1 Bernt Øksendal. , An Introduction with Applications. 5th ed. Springer 1998.
  • 2 Lawrence Evans. . Department of Mathematics, U.C. Berkeley.
Title stochastic differential equation
Canonical name StochasticDifferentialEquation
Date of creation 2013-03-22 16:10:07
Last modified on 2013-03-22 16:10:07
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 13
Author stevecheng (10074)
Entry type Definition
Classification msc 60H10
Classification msc 34-00
Synonym SDE
Related topic ItoIntegral
Related topic WienerProcess
Related topic BrownianMotion