Problems of Control and Information Theory 7. (Budapest, 1978)
1978 / 2. szám - Smith, T. L. - Tsokos, C. P.: On extending a method of Puri for the study of certain integral functionals of stochastic processes
I Problems of Control ami Information Theory, Vol. 7 (2), pp. 57—71 (1978) ON EXTENDING A METHOD OF PURI FOR THE STUDY OF CERTAIN INTEGRAL FUNCTIONALS OF STOCHASTIC PROCESSES* T. L. SMITH, C. P. TSOKOS (Tampa) (Received May 10, 1977) 1. Introduction One often encounters, in the stochastic formulation of a problem, the necessity of dealing with processes which represent a cumulative effect of another stochastic process. Interest might be focused, for instance, on determining the variation in the total amount of toxin excreted by a bacterial population whose size can be treated as a stochastic process. Another similar problem is that of estimating the total cost of an epidemic whose spread is characterized as a stochastic process. In both the above cases (and in many others) we are interested in studying a process, say Y(t, oo), defined by Y(t,o>) = j f\X(t,co), r]dr, (1.1) Ö where X(t, w) is a process with known behavior. For our purposes we shall assume that for a finite value of t, the trajectory of X and the integral defining Y are almost surely bounded. Puri [3] has proposed a method for studying the distribution of integral functionals of the form (5.1). Section 2 will be devoted to a brief description of his derivation of the results when X has a discrete state space along with minor generalizations and Section 3 will consider the extension to a continuous state space. It is also often of interest to study integral functionals of the form Y(t,co)= $f[X(r, d),r,t]dr (1.2) о * This research was sponsored by the United States Air Force Office of Scientific, Grant Number AFOSR-74-2711.