Real Analysis By Royden [Ch 3 – 4]
Stochastic Calculus For Finance Vol II By Steven E Shreve [Ch 1 – 4]
Principles Of Real Analysis By Aliprantis And Burkinshaw [Ch 3 – 4]
Basic Stochastic Processes By Brzezniak And Zastawniak
Stochastic Calculus By Richard Durrett
Introduction To Stochastic Integration By Hue Kuo
After studying this course the students should be able to: • understand sigma-algebra, Borel sigma-algebra, measurable sets and measureable functions • understand the concept of Lebesgue integration • calculate expectation of a given function (pdf) • calculate conditional expectation of a given function (pdf) • prove properties of conditional expectation (in discrete and continuous time) • understand concepts of stochastic process and Brownian motion • prove certain properties of Brownian motion • define Ito’s integral ad use it to understand Stochastic Differential Equations • prove basic results from Ito’s Calculus • understand Fundamental Theorem of Stochastic Calculus • calculate integrals of functions of Brownian motion.
In Stochastic Calculus we study functions of random variables and their integrals. These functions are called stochastic processes and are used to model random phenomena. The course only assumes knowledge of real analysis as pre-requisite. We will discuss in detail the stochastic process called Brownian motion and its properties, and then the construction (derivation) and applications of Ito’s integral. Fundamental theorem of stochastic calculus will also be discussed.