Skorokhod problem explained
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 and an M-matrix R, then stochastic processes and are said to solve the Skorokhod problem if for all non-negative t values,
- W(t) = X(t) + R Z(t) ≥ 0
- Z(0) = 0 and dZ(t) ≥ 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
Notes and References
- Lions . P. L. . Sznitman . A. S. . Alain-Sol Sznitman. 10.1002/cpa.3160370408 . Stochastic differential equations with reflecting boundary conditions . Communications on Pure and Applied Mathematics . 37 . 4 . 511 . 1984 .
- A. V. . Skorokhod . Anatoliy Skorokhod . Stochastic equations for diffusion processes in a bounded region 1 . Theor. Veroyatnost. I Primenen. . 6 . 1961 . 264–274.
- A. V. . Skorokhod . Anatoliy Skorokhod . Stochastic equations for diffusion processes in a bounded region 2 . Theor. Veroyatnost. I Primenen. . 7 . 1962 . 3–23.
- Stochastic differential equations with reflecting boundary condition in convex regions . Hiroshi . Tanaka . Hiroshima Math. J. . 9 . 1 . 1979 . 163–177 . 10.32917/hmj/1206135203 . free .
- 10.1007/s11134-010-9187-9. Pathwise comparison results for stochastic fluid networks. Queueing Systems. 66. 2. 155. 2010. Haddad . J. P. . Mazumdar . R. R. . Piera . F. J. .
