Vol. 37, No. 2, 1971

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Gaussian Markov expectations and related integral equations

John A. Beekman and Ralph A. Kallman

Vol. 37 (1971), No. 2, 303–317

Let {X(w),s w t} be a Gaussian Markov stochastic process with continuous sample functions. Examples of such processes are the Wiener, Orns+ˇein-Uhlenbeck, and Doob-Kac processes. An operator valued function space integral is defined for each process. This was done for the Wier.er process by R. H. Cameron and D. A. Storvick. For functionals of the form F(x) = exp{ st𝜃(t w,x(w))dw} where 𝜃(t,u) is bounded and almost everywhere continuous, the special inlegrals satisfy integral equations related to the generalized Schroedinger equations studied by the first author. For the Wiener process, a “backwards time” equation is coupled with the Cameron-Storvick equation to give a pair of integral equations.

Mathematical Subject Classification 2000
Primary: 28A40
Secondary: 60H20
Received: 22 April 1970
Published: 1 May 1971
John A. Beekman
Ralph A. Kallman