Skorokhod problem

In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition.

The problem is named after Anatoliy Skorokhod who first published the solution to a stochastic differential equation for a reflecting Brownian motion.

Problem statement

The classic version of the problem states that given a càdlàg process {X(t), t ≥ 0} and an M-matrix R, then stochastic processes {W(t), t ≥ 0} and {Z(t), t ≥ 0} are said to solve the Skorokhod problem if for all non-negative t values,

1. $W(t)=X(t)+RZ(t)\geq 0$
2. $Z(0)=0$ and $dZ(t)\geq 0$
3. $\int _{0}^{t}W_{i}(s){\text{d}}Z_{i}(s)=0$.

The matrix R is often known as the reflection matrix, W(t) as the reflected process and Z(t) as the regulator process.

See also

• List of things named after Anatoliy Skorokhod