On a Class of Stochastic Differential Equations

Stochastic differential equations of the type d y /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 x i (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, x i (t) remaining a constant between any two transitions. Using this property of x (t), partial differential equations for the joint probability frequency function of x i (t) and y i (t) (i = 1,2,…, n) are derived. Methods of obtaining the moments and correlation functions of any individual y i (t) are also indicated. It is also shown how the same method can be adopted in tackling non‐linear stochastic equations of the form dy i /dt = f i (y 1 , y 2 ,…,y n , x, t). Some examples are cited to illustrate the way in which such equations arise in Physics and Engineering.