A Development of Basic Stochastic Calculus and its Applications

Abstract

This paper looks to develop an understanding of stochastic calculus through basic measure theory. It begins by observing the relationships built between set theory and real-valued functions, using propositions related to ordering infinite sets through both subset placements as well as measurable value, subsequently establishing properties of measure spaces. It proceeds by looking at particular measure spaces, that is probability spaces, and properties related to set operations, such as unions, intersections, as well as complements, as it relates to probabilities. This analysis develops the concept of measurable functions in order to open the discussion into random variables on probability spaces. By building the properties of these functions, we derive various subcategories of random variables based on characteristics of their being simple, bounded, positive, arbitrary, and so on. Finally, we conclude the paper with application to financial mathematics.