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

References

1. Lions, P. L.; Sznitman, A. S. (1984). "Stochastic differential equations with reflecting boundary conditions". Communications on Pure and Applied Mathematics. 37 (4): 511. doi:10.1002/cpa.3160370408.
2. Skorokhod, A. V. (1961). "Stochastic equations for diffusion processes in a bounded region 1". Theor. Veroyatnost. i Primenen. 6: 264–274.
3. Skorokhod, A. V. (1962). "Stochastic equations for diffusion processes in a bounded region 2". Theor. Veroyatnost. i Primenen. 7: 3–23.
4. Tanaka, Hiroshi (1979). "Stochastic differential equations with reflecting boundary condition in convex regions". Hiroshima Math. J. 9 (1): 163–177.
5. Haddad, J. P.; Mazumdar, R. R.; Piera, F. J. (2010). "Pathwise comparison results for stochastic fluid networks". Queueing Systems. 66 (2): 155. doi:10.1007/s11134-010-9187-9.