Stochastic differential equation

for a process with respect to a Wiener process

An equation of the form

(1)

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:

(2)

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

(3)

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

References

[1]	I.I. Gikhman, A.V. Skorokhod, "Stochastic differential equations and their applications" , Springer (1972) (Translated from Russian) MR0678374 Zbl 0557.60041
[2]	R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian) MR1800858 MR1800857 MR0608221 MR0488267 MR0474486 Zbl 1008.62073 Zbl 1008.62072 Zbl 0556.60003 Zbl 0369.60001 Zbl 0364.60004

References

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[a3]	A. Freedman, "Stochastic differential equations and applications" , 1 , Acad. Press (1975)
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