From Stochastic Calculus to Mathematical...

Unformatted text preview: From Stochastic Calculus to Mathematical Finance Yu. Kabanov R. Lipster J. Stoyanov From Stochastic Calculus to Mathematical Finance The Shiryaev Festschrift With 15 Figures ABC Yuri Kabanov Robert Liptser Université de Franche-Comté 16, route de Gray 25030 Besançon Cedex France e-mail: [email protected] Department of Electrical Engineering-Systems Tel Aviv University P.O.B. 39040, Ramat Aviv Tel Aviv 69978 Israel e-mail: [email protected] Jordan Stoyanov School of Mathematics & Statistics University of Newcastle Newcastle upon Tyne NE1 7RU United Kingdom e-mail: [email protected] Library of Congress Control Number: 2005938923 Mathematics Subject Classification (2000): 60-XX, 93-XX ISBN-10 3-540-30782-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30782-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006
Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the Authors and SPI Publisher Services using a Springer LATEX macro package Cover art: Margarita Kabanova Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11591221 41/3100/SPI 543210 To Albert Shiryaev with love, admiration and respect Preface This volume contains a collection of articles dedicated to Albert Shiryaev on his 70th birthday. The majority of contributions are written by his former students, co-authors, colleagues and admirers strongly influenced by Albert’s scientific tastes as well as by his charisma. We believe that the papers of this Festschrift reflect modern trends in stochastic calculus and mathematical finance and open new perspectives of further development in these fascinating fields which attract new and new researchers. Almost all papers of the volume were presented by the authors at The Second Bachelier Colloquium on Stochastic Calculus and Probability, Metabief, France, January 9-15, 2005. Ten contributions deal with stochastic control and its applications to economics, finance, and information theory. The paper by V. Arkin and A. Slastnikov considers a model of optimal choice of an instant to launch an investment in the setting that permits the inclusion of various taxation schemes; a closed form solution is obtained. M.H.A. Davis addresses the problem of hedging in a “slightly” incomplete financial market using a utility maximization approach. In the case of the exponential utility, the optimal hedging strategy is computed in a rather explicit form and used further for a perturbation analysis in the case where the option underlying and traded assets are highly correlated. The paper by G. Di Masi and L. Stettner is devoted to a comparison of infinite horizon portfolio optimization problems with different criteria, namely, with the risk-neutral cost functional and the risk-sensitive cost functional dependent on a sensitivity parameter γ < 0. The authors consider a model where the price processes are conditional geometric Brownian motions, and the conditioning is due to economic factors. They investigate the asymptotics of the optimal solutions when γ tends to zero. An optimization problem for a onedimensional diffusion with long-term average criterion is considered by A. Jack and M. Zervos; the specific feature is a combination of absolute continuous control of the drift and an impulsive way of repositioning the system state. VIII Preface Yu. Kabanov and M. Kijima investigate a model of corporation which combines investments in the development of its own production potential with investments in financial markets. In this paper the authors assume that the investments to expand production have a (bounded) intensity. In contrast to this approach, H. Pham considers a model with stochastic production capacity where accumulated investments form an increasing process which may have jumps. Using techniques of viscosity solutions for HJB equations, he provides an explicit expression for the value function. P. Katyshev proves an existence result for the optimal coding and decoding of a Gaussian message transmitted through a Gaussian information channel with feedback; the scheme considered is more general than those available in the literature. I. Sonin and E. Presman describe an optimal behavior of a female decisionmaker performing trials along randomly evolving graphs. Her goal is to select the best order of trials and the exit strategy. It happens that there is a kind of Gittins index to be maximized at each step to obtain the optimal solution. M. R´ asonyi and L. Stettner consider a classical discrete-time model of arbitrage-free financial market where an investor maximizes the expected utility of the terminal value of a portfolio starting from some initial wealth. The main theorem says that if the value function is finite, then the optimal strategy always exists. The paper by I. Sonin deals with an elimination algorithm suggested earlier by the author to solve recursively optimal stopping problems for Markov chains in a denumerable phase space. He shows that this algorithm and the idea behind it can be applied to solve discrete versions of the Poisson and Bellman equations. In the contribution by five authors — O. Barndorff-Nielsen, S. Graversen, J. Jacod, M. Podolski, and N. Sheppard — a concept of bipower variation process is introduced as a limit of a suitably chosen discrete-time version. The main result is that the difference between the approximation and the limit, appropriately normalizing, satisfies a functional central limit theorem. J. Carcovs and J. Stoyanov consider a two-scale system described by ordinary differential equations with randomly modulated coefficients and address questions on its asymptotic stability properties. They develop an approach based on a linear approximation of the original system via the averaging principle. A note of A. Cherny summarizes relationships with various properties of martingale convergence frequently discussed at the A.N. Shiryaev seminar. In another paper, co-authored with M. Urusov, A. Cherny, using a concept of separating times makes a revision of the theory of absolute continuity and singularity of measures on filtered space (constructed, to a large extent by A.N. Shiryaev, J. Jacod and their collaborators). The main contribution consists in a detailed analysis of the case of one-dimensional distributions. B. Delyon, A. Juditsky, and R. Liptser establish a moderate deviation principle for a process which is a transformation of a homogeneous ergodic Markov Preface IX chain by a Lipshitz continuous function. The main tools in their approach are the Poisson equation and stochastic exponential. A. Guschin and D. Zhdanov prove a minimax theorem in a statistical game of statistician versus nature with the f -divergence as the loss functional. The result generalizes a result of Haussler who considered as the loss functional the Kullback–Leibler divergence. Yu. Kabanov, Yu. Mishura, and L. Sakhno look for an analog of Harrison– Pliska and Dalang–Morton–Willinger no-arbitrage criteria for random fields in the model of Cairolli–Walsh. They investigate the problem for various extensions of martingale property for the case of two-parameter processes. Several studies are devoted to processes with jumps, which theory seems to be interested from the point of view of financial applications. To this class belong the contributions by J. Fajardo and E. Mordecki (pricing of contingent claims depending on a two-dimensional L´evy process) and by D. Gasbarra, E. Valkeila, and L. Vostrikova where an enlargement of filtration (important, for instance, to model an insider trading) is considered in a general framework including the enlargement of filtration spanned by a L´evy process. The paper by H.-J. Engelbert, V. Kurenok, and A. Zalinescu treats the existence and uniqueness for the solution of the Skorohod reflection problem for a one-dimensional stochastic equation with zero drift and a measurable coefficient in the noise term. The problem looks exactly like the one considered previously by W. Schmidt. The essential difference is that instead of the Brownian motion, the driving noise is now any symmetric stable process of index α ∈]0, 2]. C. Kl¨ uppelberg, A. Lindner, and R. Maller address the problem of modelling of stochastic volatility using an approach which is a natural continuoustime extension of the GARCH process. They compare the properties of their model with the model (suggested earlier by Barndorff-Nielsen and Sheppard) where the squared volatility is a L´evy driven Ornstein–Uhlenbeck process. A survey on a variety of affine stochastic volatility models is given in a didactic note by I. Kallsen. The note by R. Liptser and A. Novikov specifies the tail behavior of distribution of quadratic characteristics (and also other functionals) of local martingales with bounded jumps extending results known previously only for continuous uniformly integrable martingales. In their extensive study, S. Lototsky and B. Rozovskii present a newly developed approach to stochastic differential equations. Their method is based on the Cameron–Martin version of the Wiener chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise. Existence, uniqueness, regularity, and probabilistic representation of generalized solutions are established for a large class of equations. Applications to non-linear filtering of diffusion processes and to the stochastic Navier–Stokes equation are also discussed. X Preface The short contribution by M. Mania and R. Tevzadze is motivated by financial applications, namely, by the problem of how to characterize varianceoptimal martingale measures. To this aim the authors introduce an exponential backward stochastic equation and prove the existence and uniqueness of its solution in the class of BMO-martingales. The paper by J. Obl´ oj and M. Yor gives, among other results, a complete characterization of the “harmonic” functions H(x, x ¯) for two-dimensional ¯ ) where N is a continuous local martingale and N ¯ is its runprocesses (N, N ¯ ning maximum, i.e. Nt := sups≤t Nt . Resulting (local) martingales are used to find the solution to the Skorohod embedding problem. Moreover, the paper contains a new interesting proof of the classical Doob inequalities. G. Peskir studies the Kolmogorov forward PDE corresponding to the solution of non-homogeneous linear stochastic equation (called by the author the Shiryaev process) and derives an integral representation for its fundamental solution. Note that this equation appeared first in 1961 in a paper by Shiryaev in connection with the quickest detection problem. In statistical literature one can meet also the “Shiryaev–Roberts procedure” (though Roberts worked only with a discrete-time scheme). The note by A. Veretennikov contains inequalities for mixing coefficients for a class of one-dimensional diffusions implying, as a corollary, that processes of such type may have long-term dependence and heavy-tail distributions. N. Bingham and R. Schmidt give a survey of modern copula-based methods to analyze distributional and temporal dependence of multivariate time series and apply them to an empirical studies of financial data. Yuri Kabanov Robert Liptser Jordan Stoyanov Contents Albert SHIRYAEV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV Publications of A.N. Shiryaev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XXI On Numerical Approximation of Stochastic Burgers’ Equation ¨ Aureli ALABERT, Istv´ an GYONGY ............................... 1 Optimal Time to Invest under Tax Exemptions Vadim I. ARKIN, Alexander D. SLASTNIKOV . . . . . . . . . . . . . . . . . . . . . . 17 A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales Ole E. BARNDORFF–NIELSEN, Svend Erik GRAVERSEN, Jean JACOD, Mark PODOLSKIJ, Neil SHEPHARD . . . . . . . . . . . . . . . . . . . . . 33 Interplay between Distributional and Temporal Dependence. An Empirical Study with High-frequency Asset Returns Nick H. BINGHAM, Rafael SCHMIDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Asymptotic Methods for Stability Analysis of Markov Dynamical Systems with Fast Variables Jevgenijs CARKOVS, Jordan STOYANOV . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Some Particular Problems of Martingale Theory Alexander CHERNY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 On the Absolute Continuity and Singularity of Measures on Filtered Spaces: Separating Times Alexander CHERNY, Mikhail URUSOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Optimal Hedging with Basis Risk Mark H.A. DAVIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 XII Contents Moderate Deviation Principle for Ergodic Markov Chain. Lipschitz Summands Bernard DELYON, Anatoly JUDITSKY, Robert LIPTSER . . . . . . . . . . . . 189 Remarks on Risk Neutral and Risk Sensitive Portfolio Optimization Giovanni B. DI MASI, L 3 ukasz STETTNER . . . . . . . . . . . . . . . . . . . . . . . . 211 On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes Hans-J¨ urgen ENGELBERT, Vladimir P. KURENOK, Adrian ZALINESCU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 A Note on Pricing, Duality and Symmetry for Two-Dimensional L´ evy Markets Jos´e FAJARDO, Ernesto MORDECKI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Enlargement of Filtration and Additional Information in Pricing Models: Bayesian Approach Dario GASBARRA, Esko VALKEILA, Lioudmila VOSTRIKOVA . . . . . 257 A Minimax Result for f -Divergences Alexander A. GUSHCHIN, Denis A. ZHDANOV . . . . . . . . . . . . . . . . . . . . 287 Impulse and Absolutely Continuous Ergodic Control of One-Dimensional Itˆ o Diffusions Andrew JACK, Mihail ZERVOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 A Consumption–Investment Problem with Production Possibilities Yuri KABANOV, Masaaki KIJIMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Multiparameter Generalizations of the Dalang–Morton– Willinger Theorem Yuri KABANOV, Yuliya MISHURA, Ludmila SAKHNO . . . . . . . . . . . 333 A Didactic Note on Affine Stochastic Volatility Models Jan KALLSEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Uniform Optimal Transmission of Gaussian Messages Pavel K. KATYSHEV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 A Note on the Brownian Motion Kiyoshi KAWAZU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 Continuous Time Volatility Modelling: COGARCH versus Ornstein–Uhlenbeck Models ¨ Claudia KLUPPELBERG, Alexander LINDNER, Ross MALLER . . . . . . 393 Contents XIII Tail Distributions of Supremum and Quadratic Variation of Local Martingales Robert LIPTSER, Alexander NOVIKOV . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Stochastic Differential Equations: A Wiener Chaos Approach Sergey LOTOTSKY and Boris ROZOVSKII . . . . . . . . . . . . . . . . . . . . . . . . 433 A Martingale Equation of Exponential Type Michael MANIA, Revaz TEVZADZE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 On Local Martingale and its Supremum: Harmonic Functions and beyond. ´ Marc YOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Jan OB3LOJ, On the Fundamental Solution of the Kolmogorov–Shiryaev Equation Goran PESKIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Explicit Solution to an Irreversible Investment Model with a Stochastic Production Capacity Huyˆen PHAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 Gittins Type Index Theorem for Randomly Evolving Graphs Ernst PRESMAN, Isaac SONIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 On the Existence of Optimal Portfolios for the Utility Maximization Problem in Discrete Time Financial Market Models ´ Mikl´ os RASONYI, L 3 ukasz STETTNER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 The Optimal Stopping of a Markov Chain and Recursive Solution of Poisson and Bellman Equations Isaac M. SONIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 On Lower Bounds for Mixing Coefficients of Markov Diffusions A.Yu. VERETENNIKOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Albert SHIRYAEV Albert Shiryaev, outstanding Russian mathematician, celebrated his 70th birthday on October 12, 2004. The authors of this biographic note, his former students and collaborators, have the pleasure and honour to recollect briefly several facts of the exciting biography of this great man whose personality influenced them so deeply. Albert’s choice of a mathematical career was not immediate or obvious. In view of his interests during his school years, he could equally well have become a diplomat, as his father was, or a rocket engineer as a number of his relatives were. Or even a ballet dancer or soccer player: Albert played right-wing in a local team. However, after attending the mathematical evening school at Moscow State University, he decided – definitely – mathematics. Graduating with a Gold Medal, Albert was admitted to the celebrated mechmat, the Faculty of Mechanics and Mathematics, without taking exams, just after an interview. In the 1950s and 1960s this famous faculty was at the zenith of its glory: rarely in history have so many brilliant mathematicians, professors and students – real stars and superstars – been concentrated in one place, at the five central levels of the impressive university building dominating the Moscow skyline. One of the most prestigious chairs, and the true heart of the faculty, was Probability Theory and Mathematical Statistics, headed by A.N. Kolmogorov. This was Albert’s final choice after a trial year at the chair of Differential Equations. In a notice signed by A.N. Kolmogorov, then the dean of the faculty, we read: “Starting from the fourth year A. Shiryaev, supervised by R.L. Dobrushin, studied probability theory. His subject was nonhomogeneous composite Markov chains. He obtained an estimate for the variance of the sum of random variables forming a composite Markov chain, which is a substantial step towards proving a central limit theorem for such chains. This year A. Shiryaev has shown that the limiting distribution, if it exists, is necessarily infinitely divisible”. Besides mathematics, what was Albert’s favourite activity? Sport, of course. He switched to downhill skiing, rather exotic at that time, and it XVI Albert Shiryaev became a lifetime passion. Considering the limited facilities available in Central...
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