## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 4 (1992), 1902-1931.

### Large Deviations for a Class of Anticipating Stochastic Differential Equations

A. Millet, D. Nualart, and M. Sanz

#### Abstract

Consider the family of perturbed stochastic differential equations on $\mathbb{R}^d$, $X^\varepsilon_t = X^\varepsilon_0 + \sqrt{\varepsilon} \int^t_0\sigma(X^\varepsilon_s)\circ dW_s + \int^t_0 b(X^\varepsilon_s) ds,$ $\varepsilon > 0$, defined on the canonical space associated with the standard $k$-dimensional Wiener process $W$. We assume that $\{X^\varepsilon_0, \varepsilon > 0\}$ is a family of random vectors not necessarily adapted and that the stochastic integral is a generalized Stratonovich integral. In this paper we prove large deviations estimates for the laws of $\{X^\varepsilon_., \varepsilon > 0\}$, under some hypotheses on the family of initial conditions $\{X^\varepsilon_0, \varepsilon > 0\}$.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 4 (1992), 1902-1931.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989535

**Digital Object Identifier**

doi:10.1214/aop/1176989535

**Mathematical Reviews number (MathSciNet)**

MR1188048

**Zentralblatt MATH identifier**

0769.60053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Secondary: 60F10: Large deviations

**Keywords**

Large deviations anticipating stochastic differential equations stochastic flows

#### Citation

Millet, A.; Nualart, D.; Sanz, M. Large Deviations for a Class of Anticipating Stochastic Differential Equations. Ann. Probab. 20 (1992), no. 4, 1902--1931. doi:10.1214/aop/1176989535. https://projecteuclid.org/euclid.aop/1176989535