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Equations for stochastic path integrals

Published online by Cambridge University Press:  24 October 2008

M. S. Bartlett
M. S. Bartlett
Affiliation:
University CollegeLondon
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Extract

Apart from the study of the integral

where {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general type

have been considered for {X(u)} either additive or Markovian (see, for example, (2), (3)), and in the Markovian case only for diffusion-type processes (Darling and Siegert (4)). More general approaches exist (e.g. (5), (6)), but seem less concerned with the investigation of specific problems. Some preliminary remarks here are therefore aimed at examining the structure of integrals of type (2), or such further extensions of the formal Riemann sum type

that would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sum

say, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 3 , July 1961 , pp. 568 - 573
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

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