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,
W ( t ) = X ( t ) + R Z ( t ) ≥ 0 {\displaystyle W(t)=X(t)+RZ(t)\geq 0}
Z ( 0 ) = 0 {\displaystyle Z(0)=0} and d Z ( t ) ≥ 0 {\displaystyle dZ(t)\geq 0}
∫ 0 t W i ( s ) d Z i ( s ) = 0 {\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
References