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Random Dynamical Systems in Finance

By Anatoliy Swishchuk, Shafiqul Islam

Chapman and Hall/CRC – 2013 – 357 pages

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    978-1-43-986718-1
    April 23rd 2013

Description

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.

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.

Name: Random Dynamical Systems in Finance (Hardback)Chapman and Hall/CRC 
Description: By Anatoliy Swishchuk, Shafiqul Islam. 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,...
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