Equations for stochastic path integrals

Apart from the study of the integralwhere {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general typehave 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 typethat would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sumsay, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.