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**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
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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,
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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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