Stochastic Analysis, Stochastic Systems, and Applications to Finance 8197.9789814355704-tp.indd 1 5/19/11 12:05 PM NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI World Scientic Allanus Tsoi University of Missouri, Columbia, USA David Nualart University of Kansas, USA George Yin Wayne State University, Michigan, USA Edited by Stochastic Analysis, Stochastic Systems, and Applications to Finance 8197.9789814355704-tp.indd 2 5/19/11 12:05 PM This page is intentionally left blank British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4355-70-4 ISBN-10 981-4355-70-4 All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Printed in Singapore. STOCHASTIC ANALYSIS, STOCHASTIC SYSTEMS, AND APPLICATIONS TO FINANCE He Yue - Stochastic Analysis.pmd 5/11/2011, 3:59 PM1 May 13, 2011 11:8 WSPC - Proceedings Trim Size: 9in x 6in cnts v Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Contributors and Addresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part I. Stochastic Analysis and Systems 1. Multidimensional Wick-Itˆo Formula for Gaussian Processes . . . . . . . . 3 D. Nualart and S. Ortiz-Latorre 2. Fractional White Noise Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 27 A. H. Tsoi 3. Invariance Principle o f Regime-Switching Diffusions . . . . . . . . . . . 43 C. Zhu and G. Yin Part II. Finance and Stochastics 4. Real Options and Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 A. Bensoussan, J. D. Diltz, and S. R. Hoe 5. Finding Exp e ctations of Monotone Functions of Binary Random Var iables by Simulation, with Applications to Reliability, Finance, and Round Robin Tournaments . . . . . . . . . . . . . . . . . . . . . 101 M. Brown, E. A. Pek¨oz, and S. M. Ross 6. Filtering with Counting Process Observations and Other Facto rs: Applications to Bond Price Tick Data . . . . . . . . . . . . . . . . . 115 X. Hu, D. R. Kuipers, and Y. Zeng May 13, 2011 11:8 WSPC - Proceedings Trim Size: 9in x 6in cnts vi Contents 7. Jump Bond Markets Some Steps towards General Models in Applications to Hedging and Utility Problems . . . . . . . . . . . . . . . 145 M. Kohlmann and D. Xiong 8. Recombining Tree for Regime-Switching Model: Algorithm and Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .193 R. H. Liu 9. Optimal Re insurance under a Jump Diffusion Model . . . . . . . . . . . 215 S. Luo 10. Applications of Counting Processes and Martingales in Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 J. Sun 11. Stochastic Algorithms and Numerics for Mean-Reverting Asset Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Q. Zhang, C. Zhuang, and G. Yin April 27, 2011 9:46 WSPC - Proceedings Trim Size: 9in x 6in 01-preface vii Preface This volume contains 11 chapters. It is an expanded version of the papers presented at the first K ansas–Missouri Winter School of Applied Probabil- ity, which was organized by Allanus Tsoi and was held at the University of Missouri, February 14 and 15, 2008. It brought together resea rchers from different parts of the country to review and to update the recent advances, and to identify future directions in the areas of applied proba bility, stochas- tic pro cesses, and their applications. After the successful conference was over, there was a strong support of publishing the paper s delivered in the conference as an archival volume. Based on the support, we began the preparation on this project. In addition to paper s reported at the conference, we have invited a number of collea gues to contribute additional papers. As an archive, this volume presents some of the highlights of the con- ference, as well as some of most recent developments in stochastic sys tems and applications . This book is naturally divided into two parts. The first part contains some recent results in stochastic analysis, stochastic processes and related fields. It explores the Itˆo formula for multidimensional Gaussian processes using the Wick integral, introduces the notion of fractional white noise multiplication, and discusses the LaSalle type of invariance principles for hybrid sw itching diffusions. The second part of the book is devoted to fi- nancial mathematics, insurance models, and applications. Included here are optimal investment policies for irreversible capital investment projects un- der uncertainty in monopoly and Stackelberg le ader-follower environments, April 27, 2011 9:46 WSPC - Proceedings Trim Size: 9in x 6in 01-preface viii Preface finding expectations of monotone functions of binary random variables by simulation, with applications to reliability, finance, and round robin tour- naments, jump bond markets with general models in applications to hedg- ing and utility proble ms , algorithm and weak convergence for reco mbining tree in a regime-switching model, applications of counting processes and martingales in survival analysis, extended filtering micro-movement model with counting process observations a nd applicatio ns to bond price tick data, optimal reinsurance for a jump diffusion mo del, recursive algorithms and numerical studies for mean-reverting asset trading. Without the encouragement and assistance of many colleag ue s, this vol- ume would have never come into being. We thank all the authors of this volume, and all of the speakers of the conference for their contributions. The financial support provided by the University of Mis souri for this conference is also greatly acknowledged. Allanus Tsoi Columbia, Missouri David Nualart Lawrence, Kansas George Yin Detroit, Michigan April 21, 2011 16:30 WSPC - Proceedings Trim Size: 9in x 6in names ix Contributors and Addresses • Alain Bensoussan, School of Management, University of Texas at Dallas, Richardson, TX 75083-068 8, USA. & The Ho ng Kong Poly- technic University, Hong Kong. Email: alain.bensoussan@utdallas. edu • Mark Brown, Department of Mathematics, City Colle ge, CUNY, New York, NY, USA. Email: cy bergarf@aol.c om • J. David Diltz, Department of Finance and Real Estate, The Uni- versity of Texas at Arlington, Arlington, TX 76019, USA. Email: diltz@uta.edu • SingRu Hoe, School of Management, University of Texas at Dallas, Richardson, TX 75083-0688, USA. Email: celinehoe@utdallas.edu • Xing Hu, Department of Economics, Princeton University, Prince- ton, 08544, USA. Email: xinghu@princeton.edu • Michael Kohlmann, Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany. Email: michael.kohlmann@uni-konstanz.de • David R. Kuipers, Department of Finance, Henry W. B loch School of Business and Public Administration, University of Mis- souri at Kansas City, Kansas City, MO 64 110, USA. Email: kuip e rsd@umkc.edu • Ruihua Liu, Department of Mathematics, University of Dayton, 300 College Park, Dayton, OH 45469-23 16, USA. Email: rui- hua.liu@notes.udayton.edu April 21, 2011 16:30 WSPC - Proceedings Trim Size: 9in x 6in names x Contributors and Addresses • Shangzhen Luo, Department of Mathematics, University of North- ern Iowa, Cedar Falls, Iowa, 5 0614-0 506, USA. Email: luos @uni.edu • David Nualart, Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA. Email: nualart@math.ku.edu • Salvador Ortiz-Latorre, Departament de Probabilitat, L`ogica i Es- tad´ıstica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. Email: sortiz@ub.edu • Erol A. Pekoz, School of Management, Boston University, 595 Commonwealth Avenue, Boston, MA 02215, USA. Email: pekoz@bu.edu • Sheldon M. Ross, Department of Industrial and Systems Engineer- ing, University of Souther n California, Los Angeles, CA 90089, USA. Email: smross@usc.e du • Jianguo Sun, Department of Statistics, University of Mis souri, USA. Email: sunj@missouri.edu • Allanus Hak-Man Tsoi, Department of Mathematics, University o f Missouri, Columbia, MO 65211, USA. Email: tsoia@missouri.edu • Dewen Xiong, Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China. Email: xiongdewen@sjtu.edu.cn • George Yin, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email: gyin@math.wayne.edu • Yong Zeng, Department of Mathematics and Statistics, University of Missouri at Kansas City, Kansas City, MO 64110, USA. Email: zengy@umkc.edu • Qing Zhang, Department of Mathematics, University of Georgia, Athens, GA 30602, USA. Email: qingz@math.uga.edu • Chao Zhu, Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA. Email: zhu@uwm.edu • Chao Zhuang, Marshall School of Business, University of Southern California, Los Angeles, CA 90089, USA. Email: czhuang@usc.edu [...]... fBm and other related processes are suitable input noises in practical problems arising in a variety of fields including finance, telecommunications and hydrology (see, for instance, Mandelbrot and Van Ness7 and Sottinen13 ) A possible definition of the stochastic integral with respect to the fBm is based on the divergence operator appearing in the stochastic calculus of variations This approach to define... This approach to define stochastic integrals started from the ¨ u work by Decreusefond and Ust¨nel3 and was further developed by Carmona ∗ Supported by the NSF Grant DMS-0604207 May 11, 2011 10:49 4 WSPC - Proceedings Trim Size: 9in x 6in 01-nu D Nualart and S Ortiz-Latorre and Coutin2 and Duncan, Hu and Pasik-Duncan4 (see also Hu5 and Nualart9 for a general survey papers on the stochastic calculus for... O Mazet and D Nualart, Stochastic calculus with respect to Gauso sian processes, Ann Probab 29, 766–801 (2001) 2 P Carmona and L.Coutin, Stochastic integration with respect to fractional Brownian motion, Ann Inst H Poincar´ 39, 27–68 (2003) e ¨ u 3 L Decreusefond and A S Ust¨nel, Stochastic analysis of the fractional Brownian motion, Potential Analysis 10, 177–214 (1998) 4 T E Duncan, Y Hu and B Pasik-Duncan,... 01-nu D Nualart and S Ortiz-Latorre 8 W Margrabe, The value of an option to exchange one asset for another, The Journal of Finance 33, 177-186 (1978) 9 D Nualart, Stochastic integration with respect to fractional Brownian motion and applications, Contemporary Mathematics 336, 3–39 (2003) 10 D Nualart, The Malliavin Calculus and Related Topics, (Springer-Verlag, Berlin, 2006) 11 D Nualart and M.S Taqqu,... 2006) 11 D Nualart and M.S Taqqu, Wick-Itˆ formula for regular processes and applio cations to the Black and Scholes formula, Stochastics and Stochastics Reports, to appear 12 D Nualart and M.S Taqqu, Wick-Itˆ formula for Gaussian processes, J o Stoch Anal Appl 24, 599–614 (2006) 13 T, Sottinen, Fractional Brownian motion in finance and queueing, Ph.D Thesis, University of Helsinki (2003) May 11, 2011... the solution to equation (11) is C (t, z) = zN (d1 ) − N (d2 ), where d1 := ln z + 1 2 T t T t θ (s) ds T , d2 := d1 − θ (s) ds, t θ (s) ds and N (x) is the N (0, 1) cumulative distribution function Finally, taking into account the values of Vt1,1 , Vt2,2 and Vt1,2 , we get 1 2 1 2 C t, St , St = St N (d1 ) − St N (d2 ) , 1 2 where d1 and d2 are obtained from d1 and d2 making z = St /St and T t 2 2... fractional Brownian motion We also use this Itˆ formula to compute the price of o an exchange option in a Wick-fractional Black-Scholes model Keywords: Wick-Itˆ formula; Gaussian processes; Malliavin calculus o 1 Introduction The classical stochastic calculus and Itˆ’s formula can be extended to semio martingales There has been a recent interest in developing a stochastic calculus for Gaussian processes which... Pasik-Duncan, Stochastic calculus for fractional Brownian motion I Theory, SIAM J Control Optim 38, 582–612 (2000) 5 Y Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Memoirs of the AMS 175 (2005) 6 S Janson, Gaussian Hilbert Spaces (Cambridge University Press, Cambridge, 1997) 7 B B Mandelbrot and J W.Van Ness, Fractional Brownian motions, fractional noises and applications, ... 1/2) The lower bound for H is a natural one, see Al`s, Mazet and Nualart.1 o The aim of this paper is to generalize the results of Nualart and Taqqu12 to the multidimensional case We introduce the multidimensional Wick-Itˆ o integral as a limit of forward Riemann sums and prove a Wick-Itˆ formula o under conditions similar to those in Nualart and Taqqu,12 allowing infinite quadratic variation processes... space Dm,2 is the completion of S with respect to the norm F m,2 defined by m F 2 m,2 = E[F 2 ] + E[ Di F i=1 2 ] H ⊗i May 11, 2011 10:49 6 WSPC - Proceedings Trim Size: 9in x 6in 01-nu D Nualart and S Ortiz-Latorre The Wick product F X (h) between a random variable F ∈ D1,2 and the Gaussian random variable X (h) is defined as follows Definition 2.2 Let F ∈ D1,2 and h ∈ H Then the Wick product F X (h) is . London WC2H 9HE Printed in Singapore. STOCHASTIC ANALYSIS, STOCHASTIC SYSTEMS, AND APPLICATIONS TO FINANCE He Yue - Stochastic Analysis.pmd 5/11/2011, 3:59. State University, Michigan, USA Edited by Stochastic Analysis, Stochastic Systems, and Applications to Finance 8197.9789814355704-tp.indd 2 5/19/11