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May, 1982 Stochastic Equations of Hyperbolic Type and a Two-Parameter Stratonovich Calculus
Bruce Hajek
Ann. Probab. 10(2): 451-463 (May, 1982). DOI: 10.1214/aop/1176993869


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.


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Bruce Hajek. "Stochastic Equations of Hyperbolic Type and a Two-Parameter Stratonovich Calculus." Ann. Probab. 10 (2) 451 - 463, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0478.60069
MathSciNet: MR647516
Digital Object Identifier: 10.1214/aop/1176993869

Primary: 60H75
Secondary: 60H60

Keywords: hyperbolic differential equation , multiple parameter random processes , Stochastic differential equation , stochastic integral , Stratonovich integral

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • May, 1982
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