Stochastic Calculus An introduction

Preface This note contains a brief introduction to stochastic differential equations and their application in mathematical finance. It does not pretend to be complete in any respect. The topics of this note are covered in more detail in the course 04445 Statistics in Finance, and the interested reader may consult the adjacent lecture notes [8] or one of the standard texts [9, 4]. Throughout it is assumed that the reader is familiar with elementary probability theory and statistics at a level corresponding to the DTU courses 01142 Probability theory, 04041 Introductory Statistics or 04040 Introductory Statistics and Probability. No prior knowledge about economics or finance is required, and the theory is formulated in the classical sense without reference to measure theory, albeit it is now common to introduce probability measure transformations. The theory is much more conveniently formulated in terms of equivalent martingale measures and absolute continuous measure transformaations using Girsanov's theorem, but this is deemed to be outside the scope of this introductory note. See the references listed above for details. The mathematics of modern finance theory utilizes some intimate relations between stochastic differential equations and (deterministic) partial differential equations. In particular, we will focus on the Feynman-Kac representation theorems, which makes it possible solve some parabolic Cauchy problems in terms of conditional expectations. These mathematical tools are essential for the pricing of financial derivatives, where only the simplest type of European options will be considered here. Any comments about misprints or suggestions for improvement will be mostly appreciated.