Skorokhod equation

Skorohod equation

A stochastic equation describing a reflecting Brownian motion. Given a one-dimensional Brownian motion $X _ { t }$ on $\mathbf{R} ^ { 1 } = ( - \infty , \infty )$, the reflecting Brownian motion $X _ { t } ^ { + }$ is defined by

\begin{equation*} X _ { t } ^ { + } = | X _ { t } | , t \geq 0, \end{equation*}

which is a Markov process on $[ 0 , \infty )$ with continuous sample paths.

A.V. Skorokhod discovered that the reflecting Brownian motion $X _ { t } ^ { + }$, $t \geq 0$, is identical in law with the solution $Y _ { t }$, $t \geq 0$, of the equation

\begin{equation*} Y _ { t } = Y _ { 0 } + B _ { t } + \text{l} _ { t } , t \geq 0, \end{equation*}

where the triple $\{ Y _ { t } , B _ { t } , \text{l} _ { t } \}$ is a system of real continuous stochastic processes (cf. also Stochastic process) required to have the following properties:

$B _ { t }$ is a one-dimensional Brownian motion starting at $0$ and independent of $Y _ { 0 }$;

$Y _ { t } \geq 0$ for all $t \geq 0$;

$ \operatorname{ l}_t$ is increasing in $t \geq 0$ with $\mathbf{l} _ { 0 } = 0$ and $\int _ { 0 } ^ { t } I _ { ( 0 ) } ( Y _ { s } ) d \text{l} _ { s } = \text{l} _ { t }$.

In fact, the solution $Y _ { t }$ of this Skorokhod equation can be described uniquely and deterministically by the given Brownian motion $B _ { t }$ as

\begin{equation*} Y _ { t } = B _ { t } - \operatorname { min } _ { 0 \leq s \leq t } B _ { s } \bigwedge 0, \end{equation*}

a formula due to P. Lévy in case that $Y _ { 0 } = 0$. Further, $ \operatorname{ l}_t$ is twice the Lévy local time of $B _ { t }$ at the origin.

The Skorokhod equation has been extended to the higher-dimensional case $\mathbf{R} ^ { d }$, $d \geq 2$, to describe a normally reflecting Brownian motion $Y _ { t }$ on the closure $\overline{ D }$ of a domain $D \subset \mathbf R ^ { d }$. In this case, the equation takes the form

\begin{equation*} Y _ { t } = Y _ { 0 } + B _ { t } + \int _ { 0 } ^ { t } \mathbf{n} ( Y _ { s } ) d \text{l} _ { s } ,\; t \geq 0, \end{equation*}

where $B _ { t }$ is a $d$-dimensional Brownian motion starting at the origin, $\mathbf{n} ( x )$, $x \in \partial D$, is the inward normal vector field on the boundary $\partial D$ and $\operatorname{l}$ is a real increasing process such that $\int _ { 0 } ^ { t } I _ { \partial D } ( Y _ { s } ) d \text{l} _ { s } = \text{l} _ { t }$, $t \geq 0$. The third term at the right-hand side of the equation expresses a singular drift, keeping the process $Y _ { t }$ inside $\overline{ D }$ against the isotropic nature of the Brownian motion $B _ { t }$. For a bounded convex domain in $\mathbf{R} ^ { d }$ the Skorokhod equation has a unique solution. For other domains, the Skorokhod equations are studied not only from the point of view of stochastic differential equations, but also in relation to other principles, e.g. submartingale problems or Dirichlet forms. Obliquely reflecting Brownian motions, where the vector fields $\bf n$ in the Skorokhod equations are different from the normal vector field, also arise naturally in the diffusion approximation in stochastic network theory.

References

[a1]	N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusions" , North-Holland (1989) (Edition: Second)