ABSTRACT
This chapter introduces stochastic differential equations and considers analytical solution methods. As for ordinary differential equations, Lipschitz and bounded growth conditions must be imposed on the drift and diffusion terms in order to obtain existence and uniqueness of solutions. Generally, it is difficult to obtain closed form solutions to stochastic differential equations. However, the Ito formula, that in all other aspects complicates analytical calculations considerably, may be valuable as an intermediary step in obtaining closed form solutions. The chapter describes a close relationship between stochastic differential equations and parabolic partial differential equations. It introduces the concepts of (probability) measures, the Radon–Nikodym derivative and the Girsanov theorem, which enables to change (probability) measures in continuous-time models. The chapter also discusses the Girsanov measure transformation as the theoretical foundation of a modern application of the well-known Maximum Likelihood method.
