An Extension of Stochastic Integrals in the Plane

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.