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October, 1977 An Extension of Stochastic Integrals in the Plane
Eugene Wong, Moshe Zakai
Ann. Probab. 5(5): 770-778 (October, 1977). DOI: 10.1214/aop/1176995718

Abstract

For a Wiener process with a two-dimensional parameter $\{W_z, z \in R_+^2\}$, four types of stochastic integrals: $\int \phi dW, \int \psi dW dW, \int \psi dW dz, \int \psi dz dW$, have been defined under the condition $$E \int \phi^2 dz < \infty \quad \text{and} \quad E \int \psi^2 dz dz' < \infty.$$ The main purpose of this note is to extend the definition of these stochastic integrals by replacing $E(\bullet) < \infty$ with $(\bullet) < \infty$ a.s. in these conditions. Our results are in fact even more general, allowing $W$ to be replaced by a strong martingale with appropriate properties.

Citation

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Eugene Wong. Moshe Zakai. "An Extension of Stochastic Integrals in the Plane." Ann. Probab. 5 (5) 770 - 778, October, 1977. https://doi.org/10.1214/aop/1176995718

Information

Published: October, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0376.60060
MathSciNet: MR448555
Digital Object Identifier: 10.1214/aop/1176995718

Subjects:
Primary: 60H05
Secondary: 60G45

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 5 • October, 1977
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