## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 4 (1976), 570-586.

### Weak Martingales and Stochastic Integrals in the Plane

#### Abstract

This paper continues the development of a stochastic calculus for two-parameter martingales, and particularly for the two-parameter Wiener process. Whereas in an earlier paper we showed that two types of stochastic integrals were necessary for representing functionals and martingales of a Wiener process, introduction of two mixed area integrals is necessary to complete the stochastic calculus. These mixed integrals are weak martingales in the sense of Cairoli and Walsh, and are necessary in a general representation for weak martingales and transformations of weak martingales. Stopping times are introduced for two-parameter processes, and a characterization of strong martingales in terms of stopping times is given.

#### Article information

**Source**

Ann. Probab. Volume 4, Number 4 (1976), 570-586.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176996028

**Digital Object Identifier**

doi:10.1214/aop/1176996028

**Mathematical Reviews number (MathSciNet)**

MR517927

**Zentralblatt MATH identifier**

0359.60053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G45

Secondary: 60H05: Stochastic integrals

**Keywords**

Martingales 2-parameter processes stochastic integral random field Wiener process

#### Citation

Wong, Eugene; Zakai, Moshe. Weak Martingales and Stochastic Integrals in the Plane. Ann. Probab. 4 (1976), no. 4, 570--586. doi:10.1214/aop/1176996028. http://projecteuclid.org/euclid.aop/1176996028.