On some methods of solutions of stochastic differential equations

Abstract

The main theme of this thesis is to study the Langevin equations arising in different physical contexts, in the light of Ito and Stratonovich theories. The areas of investigation broadly include fluctuation, dissipation relations, stability problems, applications of path integral techniques, and smoothing approximation methods. Stochastic differential equations driven by point processes are also considered for study. the point processes are characterised in terms of certain point functions known as cumulants and product densities. They are related to new concept of combinants and the relevance of Bell-polynomials is highlighted. A short account on 'Unified calculus' stressing canonical extension method, of McShane and its generalisation of Marcus, are discussed in the thesis. A concept of Lie Series is introduced in the investigations. A new result 'Focker-Plank equation for the stochastic system' driven by random telegraph noise, and some new results on stability theory are discussed in this thesis.