Skorokhod stochastic differential equation

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