Appl Math Optim 36:263–289 (1997)
1997 Springer-Verlag New York Inc.
Anticipating Stochastic Differential Equations of Stratonovich Type
and J. A. Le´on
Departamento de Matem´aticas, CINVESTAV-IPN,
Apartado Postal 14-740, 07000 M´exico D.F., Mexico
Department d’Economia, Universitat Pompeu Fabra,
Ram´on Trias Fargas 25-27, 08005 Barcelona, Spain
Abstract. We prove the existence and uniqueness of Stratonovich stochastic dif-
ferential equations where the coefﬁcients and the initial condition may depend on
the whole path of the driving Wiener process. Our main hypothesis is that the dif-
fusion coefﬁcient satisﬁes the Frobenius condition. The solution is given in terms
of solutions of ordinary differential equations and the Wiener process. We use this
representation to study properties of the solution.
Key Words. Anticipating calculus, Stratonovich integral, Stochastic differential
AMS Classiﬁcation. 60H05, 60H10.
In recent years,several authors have layedout the basic theory for an anticipating stochas-
tic calculus. Many papers have appeared dealing with different aspects of this theory.
In particular, this theory allows for the study of stochastic differential equations (SDEs)
where the coefﬁcients and the initial condition may depend on the future (see, e.g., 
and the references therein).
In this article we consider the Stratonovich SDE
) ds +
, 0≤ t ≤ T. (1)
This research was partially supported by CONACyT Grant No. 2059-E9302. The ﬁrst author was
partially supported by CONACyT Beca Patrimonial No. 920295-R93.