Skorokhod problem

In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition.^[1]

The problem is named after Anatoliy Skorokhod who first published the solution to a stochastic differential equation for a reflecting Brownian motion.^[2][3][4]

Problem statement

The classic version of the problem states^[5] 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. ${\displaystyle \int _{0}^{t}W_{i}(s){\text{d}}Z_{i}(s)=0}$.

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