## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 3 (1991), 907-1397

### Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations

Thomas G. Kurtz and Philip Protter

#### Abstract

Assuming that $\{(X_n,Y_n)\}$ is a sequence of cadlag processes converging in distribution to $(X,Y)$ in the Skorohod topology, conditions are given under which the sequence $\{\int X_n dY_n\}$ converges in distribution to $\int X dY$. Examples of applications are given drawn from statistics and filtering theory. In particular, assuming that $(U_n,Y_n) \Rightarrow (U,Y)$ and that $F_n \rightarrow F$ in an appropriate sense, conditions are given under which solutions of a sequence of stochastic differential equations $dX_n = dU_n + F_n(X_n)dY_n$ converge to a solution of $dX = dU + F(X)dY$, where $F_n$ and $F$ may depend on the past of the solution. As is well known from work of Wong and Zakai, this last conclusion fails if $Y$ is Brownian motion and the $Y_n$ are obtained by linear interpolation; however, the present theorem may be used to derive a generalization of the results of Wong and Zakai and their successors.

#### Article information

**Source**

Ann. Probab. Volume 19, Number 3 (1991), 1035-1070.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176990334

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176990334

**Mathematical Reviews number (MathSciNet)**

MR1112406

**Zentralblatt MATH identifier**

0742.60053

**Subjects**

Primary: 60H05: Stochastic integrals

Secondary: 60F17: Functional limit theorems; invariance principles 60G44: Martingales with continuous parameter

**Keywords**

Stochastic integrals stochastic differential equations weak convergence Skorohod topology filtering symmetric statistics Wong-Zakai correction

#### Citation

Kurtz, Thomas G.; Protter, Philip. Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations. The Annals of Probability 19 (1991), no. 3, 1035--1070. doi:10.1214/aop/1176990334. http://projecteuclid.org/euclid.aop/1176990334.