## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 4 (1990), 1635-1655.

### Stochastic Volterra Equations with Anticipating Coefficients

Etienne Pardoux and Philip Protter

#### Abstract

Stochastic Volterra equations are studied where the coefficients $F(t, s, x)$ are random and adapted to $\mathscr{F}_{s\vee t}$ rather than the customary $\mathscr{F}_{s\wedge t}$. Such a hypothesis, which is natural in several applications, leads to stochastic integrals with anticipating integrands. We interpret these as Skorohod integrals, which generalize Ito's integrals to the case where the integrand anticipates the future of the Wiener integrator. We shall nevertheless construct an adapted solution, which is even a semimartingale if the coefficients are smooth enough.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 4 (1990), 1635-1655.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990638

**Digital Object Identifier**

doi:10.1214/aop/1176990638

**Mathematical Reviews number (MathSciNet)**

MR1071815

**Zentralblatt MATH identifier**

0717.60073

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H20: Stochastic integral equations

Secondary: 60H05: Stochastic integrals

**Keywords**

Stochastic Volterra equations anticipating stochastic calculus Skorohod integral

#### Citation

Pardoux, Etienne; Protter, Philip. Stochastic Volterra Equations with Anticipating Coefficients. Ann. Probab. 18 (1990), no. 4, 1635--1655. doi:10.1214/aop/1176990638. https://projecteuclid.org/euclid.aop/1176990638