Skorokhod equation

Skorohod equation

A stochastic equation describing a reflecting Brownian motion. Given a one-dimensional Brownian motion on , the reflecting Brownian motion is defined by

which is a Markov process on with continuous sample paths.

A.V. Skorokhod discovered that the reflecting Brownian motion , , is identical in law with the solution , , of the equation

where the triple is a system of real continuous stochastic processes (cf. also Stochastic process) required to have the following properties:

is a one-dimensional Brownian motion starting at and independent of ;

for all ;

is increasing in with and .

In fact, the solution of this Skorokhod equation can be described uniquely and deterministically by the given Brownian motion as

a formula due to P. Lévy in case that . Further, is twice the Lévy local time of at the origin.

The Skorokhod equation has been extended to the higher-dimensional case , , to describe a normally reflecting Brownian motion on the closure of a domain . In this case, the equation takes the form

where is a -dimensional Brownian motion starting at the origin, , , is the inward normal vector field on the boundary and is a real increasing process such that , . The third term at the right-hand side of the equation expresses a singular drift, keeping the process inside against the isotropic nature of the Brownian motion . For a bounded convex domain in the Skorokhod equation has a unique solution. For other domains, the Skorokhod equations are studied not only from the point of view of stochastic differential equations, but also in relation to other principles, e.g. submartingale problems or Dirichlet forms. Obliquely reflecting Brownian motions, where the vector fields in the Skorokhod equations are different from the normal vector field, also arise naturally in the diffusion approximation in stochastic network theory.

References

[a1]	N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusions" , North-Holland (1989) (Edition: Second)