Publication Date

7-9-1979

Abstract

In this paper we study the double stochastic integral or stochastic quadratic form

Q =integreal(integral(phi(x,y)Z(dx)Z(dy) ))

with respect to a Gaussian random spectral measure Z. Letting F be the corresponding spectral distribution function, the kernel function is required to satisfy a) the function phi(x,x) is integrable respect to F and b) phi(x,y) is square integrable with respect to FxF. A definition of Q is given in terms of a carefully constructed sequence of step functions converging to phi. Iterated integral formulae, a discussion of the stochastic contribution to Q due to the diagonal values of phi, and an improvement of a bound on the tail probability of Q due to Koopmans and Qualls [(1972) Annals of Math. Statist., Vol. 43, No. 6] are given. The eigenfunction expansion of the kernel, the corresponding expansion for Q, and a related tail probability inequality are discussed. The differences among the double stochastic integral of phi, the eigenfunction expansion for Q, and Ito's stochastic multiple integral of phi are explained. Also, double. stochastic integrals with respect to Poisson random measures are briefly explored.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Clifford Ray Qualls

Second Committee Member

Richard Jerome Griego

Third Committee Member

Lambert Herman Koopmans

Language

English

Document Type

Dissertation

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