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) + R Z(t) ≥ 0
2. Z(0) = 0 and dZ(t) ≥ 0
3. $\underset{0}{\overset{t}{\int }}{W}_i(s)\text{d}{Z}_i(s)=0$.

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

References

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