Introductory Stochastic Analysis for Finance and Insurance

CHAPTER 1

INTRODUCTION

The theory of stochastic processes studies sequences of time-related random events, termed as stochastic processes, governed by probability laws. It provides mathematical tools to model and analyze these events in both qualitative and quantitative ways. Stochastic calculus concerns a specific class of stochastic processes that are stochastically integrable and are often expressed as solutions to stochastic differential equations. Applications of stochastic processes and stochastic calculus may be found in many disciplines such as physics, engineering, finance and insurance.

Early financial applications of stochastic processes date back to at least year 1900 when the French mathematician Louis Bachelier applied a special stochastic process called Brownian motion or a Wiener process to describe stock prices in his Ph.D. dissertation¹. Significant financial applications of stochastic processes and stochastic calculus in particular came in the late 1960s and early 1970s when a series of seminal research papers in option pricing were published by Fisher Black, Robert Merton and Myron Scholes, among others. In recent years, stochastic processes and stochastic calculus have been applied to a wide range of financial problems. Besides the valuation of options, they have been used to model interest rates and evaluate fixed income securities including interest rate futures, interest rate options, interest rate caps and floors, swaps and swaptions, and mortgage-backed ...