Journal URL http://www.math.washington.edu/˜ejpecp/ Paper URL http://www.math.washington.edu/˜ejpecp/EjpVol4/paper6.abs.html

We characterize those vector-valued stochastic processes (with a finite index set and defined on an arbitrary stochasic base) which can become a martingale under an equivalent change of measure.This question is important in a widely studied problem which arises in the theory of finite period securities markets with one riskless bond and a finite number of risky stocks. In this setting, our characterization gives a criterion for recognizing when a securities market model allows for no arbitrage opportunities ("free lunches"). Intuitively, this can be interpreted as saying "if one cannot win betting on a process, then it must be a martingale under an equivalent measure," and provides a converse to the classical notion that "one cannot win betting on a martingale."

We consider the wave equation in two spatial dimensions driven by space-time Gaussian noise that is white in time but has a nondegenerate spatial covariance. We give a necessary and sufficient integral condition on the covariance function of the noise for the solution to the linear form of the equation to be a real-valued stochastic process, rather than a distributionvalued random variable. When this condition is satisfied, we show that not only the linear form of the equation, but also nonlinear versions, have a real-valued process solution. We give stronger sufficient conditions on the spatial covariance for the solution of the linear equation to be continuous, and we provide an estimate of its modulus of continuity.

We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.