Stochastic Integral Equations in Life Sciences and Engineering

Abstract

Our aim in this presentation is to introduce the theoretical and applied scientists to the area of Stochastic Integral Equations. We hope to convey the manner in which such equations arise, the mathematical difficulties encountered, their usefulness to life sciences and engineering, and a survey of the concerned efforts in recent years to develop the theory of stochastic integral equations using as the principal tools the methods of probability theory, functional analysis and topology. Due to the nondeterministic nature of phenomena in the general areas of the biological, engineering, oceanographic and physical sciences, the mathematical descriptions of such phenomena frequently result in random or stochastic equations. These equations arise in various ways, and in order to understand better the importance of developing the theory of such equations and its application, it is of interest to consider how they arise. Usually the mathematical models or equations used to describe physical phenomena contain certain parameters or coefficients which have specific physical interpretations, but whose values are unknown. As examples, we have the diffusion coefficient in the theory of diffusion, the volume-scattering coefficient in underwater acoustics, the coefficient of viscosity in fluid mechanics, the propagation coefficient in the theory of wave propagation, and the modulus of elasticity in the theory of elasticity, among others. The mathematical equations are solved using as the value of the parameter or coefficient the mean value of a set of observations experimentally obtained. However, if the experiment is performed repeatedly, then the mean values found will vary, and if the variation is large, the mean value actually used may be quite unsatisfactory. Thus in practice the physical constant is not really a constant, but a random variable whose behaviour is governed by some probability distribution. It is thus advantageous to view these equations as being random rather than deterministic, to search for a random solution, and to study its statistical properties. There are many other ways in which random or stochastic equations arise. Stochastic differential equations appear in the study of diffusion processes and Brownian motion (I. I. Gikhmann and A. V. Skorokhod [24]). The classical Ito random integral equation (K. Ito [27]) may be found in many texts, for example in Doob [19], which is a Stieltjes integral with respect to the Brownian motion process. Integral equations with random kernels arise in random eigenvalue problems (A. T. Bharucha-Reid [11]). Stochastic integral equations