Stochastic integral equations with respect to semimartingales
Stochastic integral equations were first developed by mathematicians as a tool for the explicit construction of the paths of diffusion processes for given coefficients of drift and diffusion. Since many physical, engineering, biological as well as social phenomena can be modelled by stochastic integral equations, the theory of stochastic integral equations has become one of the most active fields of mathematical research. This thesis considers stochastic integral equations with respect to semimaningales, which in some sense forms the most general case. This thesis consists of five chapters. In Chapter I we first develop the theory of existence and uniqueness of solutions to stochastic integral equations with respect to semimartingales (SIES) and delay SIES. Chapter II presents the explicit representation of the solutions to linear SIES. Chapter ITIcontains the theory of stochastic stability and boundedness. Chapter IV is for comparison theorems. Chapter V is devoted to the transformation formula which transforms stochastic integrals with respect to continuous local martingales into classical Ito's integrals with respect to Brownian motion. This formula is then applied to study properties of stochastic integrals, SIES and stability.