Random Dynamical Systems in Finance

by Anatoliy Swishchuk, Shafiqul Islam

Overview

The theory and applications of random dynamical systems (RDS) are at the cutting edge of research in mathematics and economics, particularly in modeling the long-run evolution of economic systems subject to exogenous random shocks. Despite this interest, there are no books available that solely focus on RDS in finance and economics. Exploring this emerging area, Random Dynamical Systems in Finance shows how to model RDS in financial applications.

Through numerous examples, the book explains how the theory of RDS can describe the asymptotic and qualitative behavior of systems of random and stochastic differential/difference equations in terms of stability, invariant manifolds, and attractors. The authors present many models of RDS and develop techniques for implementing RDS as approximations to financial models and option pricing formulas. For example, they approximate geometric Markov renewal processes in ergodic, merged, double-averaged, diffusion, normal deviation, and Poisson cases and apply the obtained results to option pricing formulas.

With references at the end of each chapter, this book provides a variety of RDS for approximating financial models, presents numerous option pricing formulas for these models, and studies the stability and optimal control of RDS. The book is useful for researchers, academics, and graduate students in RDS and mathematical finance as well as practitioners working in the financial industry.

Product Details

ISBN-13:	9781439867198
Publisher:	CRC Press
Publication date:	04/19/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	357
File size:	4 MB

About the Author

Swishchuk, Anatoliy ; Islam, Shafiqul

Table of Contents

Introduction

Deterministic Dynamical Systems and Stochastic Perturbations
Deterministic dynamical systems
Stochastic perturbations of deterministic dynamical systems

Random Dynamical Systems and Random Maps
Random dynamical systems
Skew products
Random maps: Special structures of random dynamical systems
Necessary and sufficient conditions for the existence of invariant measures for a general class of random maps with constant probabilities
Support of invariant densities for random maps
Smoothness of density functions for random maps
Applications in finance

Position-Dependent Random Maps
Random maps with position dependent probabilities
Markov switching position dependent random maps
Higher dimensional Markov switching position dependent random maps
Approximation of invariant measures for position dependent random maps
Applications in finance

Random Evolutions as Random Dynamical Systems
Multiplicative operator functionals (MOF)
Random evolutions
Limit theorems for random evolutions

Averaging of the Geometric Markov Renewal Processes (GMRP)
Introduction
Markov renewal processes and semi-Markov processes
The GMRP
Averaged geometric Markov renewal processes
Rates of convergence in ergodic averaging scheme
Merged geometric Markov renewal processes
Security markets and option prices using generalized binomial models induced by random maps
Applications

Diffusion Approximations of the GMRP and Option Price Formulas
Introduction
Diffusion approximation of the GMRP
Proofs
Merged diffusion geometric Markov renewal process in the case of two ergodic classes
European call option pricing formulas for diffusion GMRP
Applications

Normal Deviation of a Security Market by the GMRP
Normal deviations of the GMRP
Applications
European call option pricing formula for normal deviated GMRP
Martingale property of GMRP
Option pricing formulas for stock price modelled by GMRP
Examples of option pricing formulas modelled by GMRP

Poisson Approximation of a Security Market by the GMRP
Averaging in Poisson scheme
Option pricing formula under Poisson scheme
Application of Poisson approximation with a finite number of jump values

Stochastic Stability of Fractional RDS in Finance
Fractional Brownian motion as an integrator
Stochastic stability of a fractional (B, S)-security market in Stratonovich scheme
Stochastic stability of fractional (B, S)-security market in Hu and Oksendal scheme
Stochastic stability of fractional (B, S)-security market in Elliott and van der Hoek scheme
Appendix

Stability of RDS with Jumps in Interest Rate Theory
Introduction
Definition of the stochastic stability
The stability of the Black-Scholes model
A model of (B, S)- securities market with jumps
Vasicek model for the interest rate
The Vasicek model of the interest rate with jumps
Cox-Ingersoll-Ross interest rate model
Cox-Ingersoll-Ross model with random jumps
A generalized interest rate model
A generalized model with random jumps

Stability of Delayed RDS with Jumps and Regime-Switching in Finance
Stochastic differential delay equations with Poisson bifurcations
Stability theorems
Application in finance
Examples

Optimal Control of Delayed RDS with Applications in Economics
Introduction
Controlled stochastic differential delay equations
Hamilton-Jacobi-Bellman equation for SDDEs
Economics model and its optimization

Optimal Control of Vector-Delayed RDS with Applications in Finance and Economics
Introduction
Preliminaries and formulation of the problem
Controlled stochastic differential delay equations
Examples: optimal selection portfolio and Ramsey model

RDS in Option Pricing Theory with Delayed/Path-Dependent Information
Introduction
Stochastic delay differential equations
General formulation
A simplified problem
Appendix

Epilogue

Index

References appear at the end of each chapter.

Editorial Reviews

"… the timely publication of this book is very welcome and appreciated. There are not many books on RDS, and as far as this reviewer knows, this book is the only one that deals with the application of RDS in finance. … There are many examples from finance, and it is very interesting to see that classical problems in finance, such as pricing European call-options, are interpreted in the viewpoint of RDS. Overall, this is a very useful book to both researchers of RDS and financial engineers …"
—Mathematical Reviews, November 2014

From the Publisher