On a Class of Stochastic Differential Equations

Stochastic differential equations of the type dy/dt + A(t) y = x(t) (A(t) being a matrix) are studied on the assumption that the random element is introduced only through x(t). Any component xi(t) of x(t) is characterised by the fact that in any finite interval (0,t) it undergoes only a finite number of discrete transitions, xi(t) remaining a constant between any two transitions. Using this property of x(t), partial differential equations for the joint probability frequency function of xi(t) and yi(t) (i = 1,2,…, n) are derived. Methods of obtaining the moments and correlation functions of any individual yi(t) are also indicated. It is also shown how the same method can be adopted in tackling non‐linear stochastic equations of the form dyi/dt = fi(y1, y2,…,yn, x, t). Some examples are cited to illustrate the way in which such equations arise in Physics and Engineering. Copyright © 1963 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim