Twisted Sheets

We give a simplified presentation of two-parameter stochastic calculus in terms of a symbolic calculus of differentials. We establish a new existence, uniqueness, and non-explosion theorem for certain hyperbolic two-parameter stochastic differential equations driven by the Brownian sheet. The theorem is applied to construct some natural path-valued processes in a Riemannian manifold, where the twisting effect of a connection makes the existing two-parameter theory inapplicable. An analysis of these path-valued processes is made using the two-parameter stochastic calculus.