View source for Skorokhod stochastic differential equation

<!--
s1101801.png
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/S110/S.1100180 Skorokhod stochastic differential equation
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

An equation of the form

$$ \tag{a1 }
X _ {t} = X _ {0} + \int\limits _ { 0 } ^ { t }  {\sigma ( s,X _ {s} ) }  {dW _ {s} } + \int\limits _ { 0 } ^ { t }  {b ( s,X _ {s} ) }  {ds } ,
$$

where the initial condition  $  X _ {0} $
and/or the coefficients  $  \sigma $
and  $  b $
are random, the solution  $  X _ {t} $
is not adapted (cf. also [[Optional random process|Optional random process]]) to the [[Brownian motion|Brownian motion]]  $  W _ {t} $, 
and the stochastic integral is interpreted in the sense of Skorokhod (see [[Skorokhod integral|Skorokhod integral]]; [[Stochastic integral|Stochastic integral]]; [[#References|[a5]]]). One cannot use a fixed-point argument to show the existence and uniqueness of the solution, as it is done for the adapted Itô stochastic equations, because the Skorokhod integral is not continuous in the  $  L  ^ {2} $-
norm.

If  $  b \equiv 0 $, 
and  $  \sigma ( s,x ) = \sigma _ {s} x $, 
where  $  \sigma _ {s} $
is a deterministic function, (a1) has an explicit solution given by (see [[#References|[a1]]])

$$ \tag{a2 }
X _ {t} = X _ {0} \left ( \omega _  \cdot  - \int\limits _ { 0 } ^  \cdot   {\sigma _ {s} }  {ds } \right ) \times
$$

$$ 
\times 
{ \mathop{\rm exp} } \left ( \int\limits _ { 0 } ^ { t }  {\sigma _ {s} }  {dW _ {s} } - {
\frac{1}{2}
 } \int\limits _ { 0 } ^ { t }  {\sigma _ {s}  ^ {2} }  {ds } \right ) .
$$

When  $  \sigma _ {s} $
is random, a similar formula holds but the martingale exponential should be replaced by the Girsanov density associated with the anticipating shift  $  \omega _ {t} \mapsto \omega _ {t} - \int _ {0}  ^ {t} {\sigma _ {s} }  {ds } $(
see [[#References|[a3]]]).

Using the notion of [[Wick product|Wick product]], introduced in the context of [[Quantum field theory|quantum field theory]], the process (a2) can be rewritten as

$$ \tag{a3 }
X _ {t} = X _ {0} \dia e ^ {\int\limits _ { 0 } ^ { t }  {\sigma _ {s} }  {dW _ {s} } - {
\frac{1}{2}
 } \int\limits _ { 0 } ^ { t }  {\sigma  ^ {2} _ {s} }  {ds } } .
$$

Formula (a3) can be used to solve linear multi-dimensional Skorokhod equations (see [[#References|[a4]]]). One-dimensional non-linear Skorokhod stochastic differential equations are studied in [[#References|[a2]]], and a local existence and uniqueness result is obtained by means of the pathwise representation of one-dimensional diffusions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Buckdahn,   "Linear Skorohod stochastic differential equations"  ''Probab. Th. Rel. Fields'' , '''90'''  (1991)  pp. 223–240</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Buckdahn,   "Skorohod stochastic differential equations of diffusion type"  ''Probab. Th. Rel. Fields'' , '''92'''  (1993)  pp. 297–324</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Buckdahn,   "Anticipative Girsanov transformations and Skorohod stochastic differential equations" , ''Memoirs'' , '''533''' , Amer. Math. Soc.  (1994)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Buckdahn,   D. Nualart,   "Linear stochastic differential equations and Wick products"  ''Probab. Th. Rel. Fields'' , '''99'''  (1994)  pp. 501–526</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.V. Skorokhod,   "On a generalization of a stochastic integral"  ''Th. Probab. Appl.'' , '''20'''  (1975)  pp. 219–233</TD></TR></table>