Stochastic Equations of Hyperbolic Type and a Two-Parameter Stratonovich Calculus

Abstract

Existence, uniqueness, and a Markov property are proved for the solutions of a hyperbolic equation with a white Gaussian noise driving term. A two-parameter analog of the Stratonovich stochastic integral is introduced and is used to formulate integral versions of the hyperbolic equation. The stochastic calculus associated with the Stratonovich integral formally agrees with ordinary calculus. A class of two-parameter semimartingales is found which is closed under all the operations of a complete stochastic calculus. The class of processes which are solutions to the type of hyperbolic equation studied is closed under smooth state space transformations.