Stochastic differential equation
for a process with respect to a Wiener process
An equation of the form
where and are non-anticipative functionals, and the random variable plays the part of the initial value. There are two separate concepts for a solution of a stochastic differential equation — strong and weak.
Let be a probability space with an increasing family of -algebras , and let be a Wiener process. One says that a continuous stochastic process is a strong solution of the stochastic differential equation (1) with drift coefficient , diffusion coefficient and initial value , if for every with probability one:
where it is supposed that the integrals in (2) are defined.
The first general result on the existence and uniqueness of a strong solution of a stochastic differential equation of the form
was obtained by K. Itô. He demonstrated that if for every the functions and satisfy a Lipschitz condition with respect to and increase not faster than linearly, then a continuous solution of the equation (3) exists, and this solution is unique in the sense that if is another continuous solution, then
If , the measurability and boundedness of the drift coefficient (vector) guarantees the existence and uniqueness of a strong solution of (3). The equation , generally speaking, does not have a strong solution for any bounded non-anticipative functional .
When studying the concept of a weak solution of the stochastic differential equation (1), the probability space with the family of -algebras , the Wiener process and the random variable are not fixed in advance, but the non-anticipative functionals , , defined for continuous functions , and the distribution function (so to speak, the initial value) are fixed. Then by a weak solution of the equation (1) with given , and one understands a set of objects
where is a Wiener process relative to , and and are related by
and . The term "weak solution" sometimes applies only to the process that appears in the set . A weak solution of equation (3) exists under weaker hypotheses. It is sufficient, for example, that , and that be continuous in , that be measurable in , and that .
The development of the theory of stochastic integration (see Stochastic integral) using semi-martingales (cf. Semi-martingale) and random measures has led to the study of more general stochastic differential equations, where semi-martingales and random measures are used as generators (along with a Wiener process). The following result is typical. Let be a probability space, let be an increasing family of -algebras, let be an -dimensional semi-martingale, and let be a matrix consisting of non-anticipative functionals such that
where the do not increase too rapidly (in ). Then the stochastic differential equation , , has a unique strong solution.
If the functions and , , , satisfy a Lipschitz condition (in ) and do not increase faster than linearly, then the solution of equation (3) (unique up to stochastic equivalence) will be a Markov process. If, moreover, and are continuous in all variables, then this will be a diffusion process. Using stochastic differential equations, starting only from a Wiener process, it is thus possible to construct Markov and diffusion processes.
Given certain extra conditions of smoothness on the functions and , the solution of equation (3) with initial condition is such that the function , given a sufficiently smooth function , satisfies the backward Kolmogorov equation
in the domain , , with the boundary condition
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Stochastic differential equation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stochastic_differential_equation&oldid=26949