Stochastic differential equations whose solutions are diffusion (or
other random) processes have been the subject of lively mathematical
research since the pioneering work of Gihman, Ito and others in the
early fifties. As it gradually became clear that a great number of real
phenomena in control theory, physics, biology, economics and other areas
could be modelled by differential equations with stochastic perturbation
terms, this research became somewhat feverish, with the results that a)
the number of theroretical papers alone now numbers several hundred and
b) workers interested in the field (especially from an applied
viewpoint) have had no opportunity to consult a systematic account. This
monograph, written by two of the world's authorities on prob- ability
theory and stochastic processes, fills this hiatus by offering the first
extensive account of the calculus of random differential equations de-
fined in terms of the Wiener process. In addition to systematically ab-
stracting most of the salient results obtained thus far in the theory,
it includes much new material on asymptotic and stability properties
along with a potentially important generalization to equations defined
with the aid of the so-called random Poisson measure whose solutions
possess jump discontinuities. Although this monograph treats one of the
most modern branches of applied mathematics, it can be read with profit
by anyone with a knowledge of elementary differential equations armed
with a solid course in stochastic processes from the measure-theoretic
point of view.
