Stochastic Calculus

Abstract

This chapter discusses stochastic calculus. It presents the systematic treatment of stochastic differentials for continuous semi-martingales. One of the important notions introduced is the symmetric multiplication, which corresponds to the Stratonovich integral or Fisk integral. Under this multiplication, the chain rule takes the same form as in the ordinary calculus. The stochastic integral is called the Stratonovich integral or the Fisk integral or sometimes the Fisk–Stratonovich symmetric integral. Stochastic differential equations with respect to quasimartingales are addressed in the chapter. In addition, the general theory of existence and uniqueness of the equations for semimartingales is established and explained with the help of examples. The chapter also describes moment inequalities for martingales in connection with a martingale version of the theory of HP-spaces. Fundamental inequalities for continuous local martingales are described as an application of stochastic calculus. © 1981, Kodansha Ltd.