Modelling with the Itô
integral or stochastic differential equations has become increasingly important in various
applied fields, including physics, biology, chemistry and finance. However, stochastic
calculus is based on a deep mathematical theory.
This book is suitable for
the reader without a deep mathematical background. It gives an elementary introduction to
that area of probability theory, without burdening the reader with a great deal of measure
theory. Applications are taken from stochastic finance. In particular, the Black-Scholes
option pricing formula is derived. The book can serve as a text for a course on stochastic
calculus for non-mathematicians or as elementary reading material for anyone who wants to
learn about Itô calculus and/or stochastic finance.
Table of Contents:
Preliminaries:
Basic Concepts from
Probability Theory
Stochastic Processes
Brownian Motion
Conditional Expectation
Martingales
The Stochastic Integral:
The Riemann and
Riemann-Stieltjes Integrals
The Itô Integral
The Itô Lemma
The Stratonovich and Other
Integrals
Stochastic Differential
Equations:
Deterministic Differential
Equations
Itô Stochastic Differential
Equations
The General Linear
Differential Equation
Numerical Solution
Applications of Stochastic
Calculus in Finance:
The Black-Scholes
Option-Pricing Formula
A Useful Technique: Change
of Measure
Appendices:
Modes of Convergence
Inequalities
Non-Differentiability and
Unbounded Variation of Brownian Sample Paths
Proof of the Existence of
the General Itô Stochastic Integral
The Radon-Nikodym Theorem
Proof of the Existence and
Uniqueness of the Conditional Expectation
Readership: Economists, financial engineers, mathematicians and physicists.
"This book under review can be determined as a very successful work ... the author's
choice of the material is done with good taste and expertise ... It can be strongly
recommended to graduate students and practitioners in the field of finance and
economics." Mathematics Abstracts, 2000
"... this is a
well-written book, which makes the difficult object of mathematical finance easy to
understand also for non-mathematicians. It might be useful for economics students and all
practitioners in the field of finance who are interested in the mathematical methodology
behind the Black-Scholes model." Statistical Papers, 2000
212 pages