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On a Class of Stochastic Differential Equations
Author(s) -
Srinivasan S. K.
Publication year - 1963
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19630430603
Subject(s) - stochastic partial differential equation , mathematics , interval (graph theory) , differential equation , constant (computer programming) , stochastic differential equation , class (philosophy) , function (biology) , stochastic matrix , mathematical physics , type (biology) , mathematical analysis , matrix (chemical analysis) , physics , pure mathematics , combinatorics , statistics , ecology , materials science , artificial intelligence , evolutionary biology , computer science , markov chain , composite material , biology , programming language
Abstract 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.