Difference between revisions of "Skorokhod stochastic differential equation"
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An equation of the form | 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 | + | 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 | + | 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 | + | 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 | 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. | 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. | ||
Latest revision as of 08:14, 6 June 2020
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) to the Brownian motion $ W _ {t} $, and the stochastic integral is interpreted in the sense of Skorokhod (see Skorokhod integral; Stochastic integral; [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 [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 [a3]).
Using the notion of Wick product, introduced in the context of 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 [a4]). One-dimensional non-linear Skorokhod stochastic differential equations are studied in [a2], and a local existence and uniqueness result is obtained by means of the pathwise representation of one-dimensional diffusions.
References
| [a1] | R. Buckdahn, "Linear Skorohod stochastic differential equations" Probab. Th. Rel. Fields , 90 (1991) pp. 223–240 |
| [a2] | R. Buckdahn, "Skorohod stochastic differential equations of diffusion type" Probab. Th. Rel. Fields , 92 (1993) pp. 297–324 |
| [a3] | R. Buckdahn, "Anticipative Girsanov transformations and Skorohod stochastic differential equations" , Memoirs , 533 , Amer. Math. Soc. (1994) |
| [a4] | R. Buckdahn, D. Nualart, "Linear stochastic differential equations and Wick products" Probab. Th. Rel. Fields , 99 (1994) pp. 501–526 |
| [a5] | A.V. Skorokhod, "On a generalization of a stochastic integral" Th. Probab. Appl. , 20 (1975) pp. 219–233 |
Skorokhod stochastic differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skorokhod_stochastic_differential_equation&oldid=17111