Stochastic Processes, Finance And Control: A Festschrift In Honor Of Robert J Elliott

A Festschrift in Honor of Robert J Elliott

Samuel N Cohen, Dilip Madan, Tak Kuen Siu, Hailiang Yang

• Publisher: WSPC
• Year: 2012
• Print ISBN: 9789814383301
• eBook ISBN: 9789814483919
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This book consists of a series of new, peer-reviewed papers in stochastic processes, analysis, filtering and control, with particular emphasis on mathematical finance, actuarial science and engineering. Paper contributors include colleagues, collaborators and former students of Robert Elliott, many of whom are world-leading experts and have made fundamental and significant contributions to these areas.

This book provides new important insights and results by eminent researchers in the considered areas, which will be of interest to researchers and practitioners. The topics considered will be diverse in applications, and will provide contemporary approaches to the problems considered. The areas considered are rapidly evolving. This volume will contribute to their development, and present the current state-of-the-art stochastic processes, analysis, filtering and control.

Contributing authors include: H Albrecher, T Bielecki, F Dufour, M Jeanblanc, I Karatzas, H-H Kuo, A Melnikov, E Platen, G Yin, Q Zhang, C Chiarella, W Fleming, D Madan, R Mamon, J Yan, V Krishnamurthy.

Contents:

• Stochastic Analysis:
  • On the Connection Between Discrete and Continuous Wick Calculus with an Application to the Fractional Black-Scholes Model (C Bender and P Parczewski)
  • Malliavin Differentiability of a Class of Feller-Diffusions with Relevance in Finance (C-O Ewald, Y Xiao, Y Zou and T K Siu)
  • A Stochastic Integral for Adapted and Instantly Independent Stochastic Processes (H-H Kuo, A Sae-Tang and B Szozda)
  • Independence of Some Multiple Poisson Stochastic Integrals with Variable-Sign Kernels (N Privault)
• Differential and Stochastic Games:
  • Strategies for Differential Games (W H Fleming and D Hernández-Hernández)
  • BSDE Approach to Non-Zero-Sum Stochastic Differential Games of Control and Stopping (I Karatzas and Q Li)
• Mathematical Finance:
  • On Optimal Dividend Strategies in Insurance with a Random Time Horizon (H Albrecher and S Thonhauser)
  • Counterparty Risk and the Impact of Collateralization in CDS Contracts (T R Bielecki, I Cialenco and I Iyigunler)
  • A Modern View on Merton's Jump-Diffusion Model (G H L Cheang and C Chiarella)
  • Hedging Portfolio Loss Derivatives with CDS's (A Cousin and M Jeanblanc)
  • New Analytic Approximations for Pricing Spread Options (J van der Hoek and M W Korolkiewicz)
  • On the Polynomial–Normal Model and Option Pricing (H Li and A Melnikov)
  • A Functional Transformation Approach to Interest Rate Modelling (S Luo, J Yan and Q Zhang)
  • S&P 500 Index Option Surface Drivers and Their Risk Neutral and Real World Quadratic Covariations (D B Madan)
  • A Dynamic Portfolio Approach to Asset Markets and Monetary Policy (E Platen and W Semmler)
  • Mean-Variance Portfolio Selection Under Regime-Switching Diffusion Asset Models: A Two-Time-Scale Limit (G Yin and Y Talafha)
• Filtering and Control:
  • Existence and Uniqueness of Solutions for a Partially Observed Stochastic Control Problem (A Bensoussan, M Çakanyildirim, M Li and S P Sethi)
  • Continuous Control of Piecewise Deterministic Markov Processes with Long Run Average Cost (O L V Costa and F Dufour)
  • Stochastic Linear-Quadratic Control Revisited (T E Duncan)
  • Optimization of Stochastic Uncertain Systems: Entropy Rate Functionals, Minimax Games and Robustness (F Rezaei, C D Charalambous and N U Ahmed)
  • Gradient Based Policy Optimization of Constrained Markov Decision Processes (V Krishnamurthy and F J Vázquez Abad)
  • Parameter Estimation of a Regime-Switching Model Using an Inverse Stieltjes Moment Approach (X Xi, M R Rodrigo and R S Mamon)
  • An Optimal Inventory-Price Coordination Policy (H Zhang and Q Zhang)

Readership: Researchers and professionals in stochastic processes, analysis, filtering and control.

Chapter 1

On the connection between discrete and continuous Wick calculus with an application to the fractional Black-Scholes model

Christian Bender and Peter Parczewski

Department of Mathematics, Saarland University,
PO Box 151150, D-66041 Saarbrücken, Germany
[email protected], [email protected]

Dedicated to Robert J. Elliott

In this chapter we discuss some questions related to the discrete Wick calculus which are motivated by a joint work of Robert Elliott with the first author [3]. One question is to what extent the connection between discrete and continuous Wick calculus can be made rigorous in terms of weak convergence results. Another question is how to sample from difference equations which involve discrete Wick products. The results are applied to financial models driven by a fractional Brownian motion.

Contents

1.1 Introduction

1.2 Continuous and discrete Wick calculus

1.2.1 Wiener integrals and Wick exponentials

1.2.2 Wick product

1.2.3 Wick powers and Wick-analytic functions

1.3 Application to the fractional Black-Scholes model

1.3.1 The fractional Black-Scholes model

1.3.2 A discrete version of the fractional Black-Scholes market

1.3.3 Weak convergence to the fractional Black-Scholes model

1.4 Simulating the discrete version of the fractional Black-Scholes model

1.4.1 Discussion of the algorithm

1.4.2 Numerical results

References

1.1. Introduction

During a research stay of the first author at Calgary in 2002, Robert Elliott suggested to study arbitrage issues in a discrete version of the fractional Black-Scholes model. The fractional Black-Scholes model can be considered as a generalization of the classical Black-Scholes model where the driving Brownian motion is replaced by a fractional Brownian motion. In the context of stochastic integration with respect to a fractional Brownian motion in terms of Wick calculus, this model was discussed by Hu and Øksendal [18] and Elliott and van der Hoek [10]. In order to avoid the technicalities of the continuous Wick calculus, in the discrete version of Bender and Elliott [3] the fractional Brownian motion is replaced by a disturbed binary random walk approximation due to Sottinen [31] and the continuous Wick product is replaced by a discrete analogue acting on binary random variables. This discrete Wick product was introduced by Holden et al. [15, 16] and motivated by formal analogies in terms of the chaos decomposition. Bender ...

Table of contents

• Cover
• Half Title
• About The Publisher
• Title
• Copyright
• Author
• Contents
• Preface
• Stochastic Analysis
• 1. On the connection between discrete and continuous Wick calculus with an application to the fractional Black-Scholes model C. Bender and P. Parczewski
• 2. Malliavin differentiability of a class of Feller-diffusions with relevance in Finance C.-O. Ewald, Y. Xiao, Y. Zou and T.K. Siu
• 3. A stochastic integral for adapted and instantly independent stochastic processes H.-H. Kuo, A. Sae-Tang and B. Szozda
• 4. Independence of some multiple Poisson stochastic integrals with variable-sign kernels N. Privault
• Differential and Stochastic Games
• 5. Strategies for differential games W.H. Fleming and D. Hernandez-Hernandez
• 6. BSDE approach to non-zero-sum stochastic differential games of control and stopping I. Karatzas and Q. Li
• Mathematical Finance
• 7. On optimal dividend strategies in insurance with a random time horizon H. Albrecher and S. Thonhauser
• 8. Counterparty risk and the impact of collateralization in CDS contracts T.R. Bielecki, I. Cialenco and I. Iyigunler
• 9. A modern view on Merton’s jump-diffusion model G.H.L. Cheang and C. Chiarella
• 10. Hedging portfolio loss derivatives with CDS’s A. Cousin and M. Jeanblanc
• 11. New analytic approximations for pricing spread options J. van der Hoek and M. W. Korolkiewicz
• 12. On the Polynomial-Normal model and option pricing H. Li and A. Melnikov
• 13. A functional transformation approach to interest rate modelling S. Luo, J. Yan and Q. Zhang
• 14. S&P 500 index option surface drivers and their risk neutral and real world quadratic covariations D.B. Madan
• 15. A dynamic portfolio approach to asset markets and monetary policy E. Platen and W. Semmler
• 16. Mean-variance portfolio selection under regime-switching diffusion asset models: A two-time-scale limit G. Yin and Y. Talafha
• Filtering and Control
• 17. Existence and uniqueness of solutions for a partially observed stochastic control problem A. Bensoussan, M. Cakanytldtrtm, M. Li and S.P. Sethi
• 18. Continuous control of piecewise deterministic Markov processes with long run average cost O.L. V. Costa and F. Dufour
• 19. Stochastic linear-quadratic control revisited T.E. Duncan
• 20. Optimization of stochastic uncertain systems: Entropy rate functionals, minimax games and robustness F. Rezaei, C.D. Charalambous and N. U. Ahmed
• 21. Gradient based policy optimization of constrained Markov decision processes V. Krishnamurthy and F.J. Vazquez Abad
• 22. Parameter estimation of a regime-switching model using an inverse Stieltjes moment approach X. Xi, M.R. Rodrigo and R.S. Mamon
• 23. An optimal inventory-price coordination policy H. Zhang and Q. Zhang
• Author Index