← ψ ψ ←

MATHEMATISCHES FORSCHUNGSINSTITUT OBEFRWOLFACH

T a g u n g s b e r i c h t 38/1997

Stochastic Analysis in Finance and Insurance

14. 9. – 20. 9. 1997

This meeting was organised jointly by Darrell Duffie (Stanford), Paul

Embrechts (Zürich) and Hans Föllmer (Berlin). In 28 talks and many
informal discussions, it covered a wide range of problems in finance and
insurance which involve advanced methods of stochastic analysis. Key
topics included:
– incomplete financial markets, in particular stochastic volatility,
equilibrium analysis, stochastic optimisation problems, and appli-
cations in insurance,
– hedging strategies in the presence of transaction costs and con-
straints,
– financial versus actuarial pricing principles, asset liability mod-
elling, default risk, and insurance derivatives,
– new approaches to the modelling of asset price dynamics,
– stochastic dynamics of the term structure of interest rates, in par-
ticular geometric aspects, interest rate derivatives, and extremes,
– theoretical problems in stochastic analysis motivated by applica-
tions in finance, in particular martingale inequalities, backward
stochastic differential equations and the structure of Brownian fil-
trations.
The meeting had 49 participants.

1
Abstracts

Knut Aase
A new equilibrium asset pricing model based on Lévy processes
The talk presented some security market pricing results in the setting of
a security market equilibrium in continuous time. The model consists
in relaxing the distributional assumptions of asset returns to a situa-
tion where the underlying random modelling the spot prices of assets
are exponentials of Lévy processes, the latter having normal inverse
Gaussian marginals, and where the aggregate consumption is inverse
Gaussian. Normal inverse Gaussian distributions have proved to fit
stock return remarkably well in empirical investigations. Within this
framework we demonstrate that contingent claims can be priced in a
preference-free manner, a concept defined in the paper. Our results
can be compared to those emerging from stochastic volatility models,
although these two approaches are very different. Equilibrium equity
premiums are derived and calibrated to the data in the Mehra and
Prescott (1985) study. The model gives a possible resolution of the eq-
uity premium puzzle. The “survival” hypothesis of Brown, Goetzmann
and Ross (1995) is also investigated within this model, giving a very
low crash probability of the market.

Ole E. Barndorff-Nielsen
Some thoughts on statistical modelling in finance
There are striking similarities between finance and turbulence as re-
gard to some of the most essential empirical features that relate to
logarithmic asset prices on the one hand and streamwise velocities on
the other. After a discussion of these similarities, the talk concentrated
on the problems of constructing tractable stochastic processes that ex-
hibit the type of (quasi) long-range dependence or scaling/selfsimilarity
behaviour observed in both of the two fields of study. In particular, a
method of setting up selfsimilar processes, with second order stationary
increments and driven by bivariate Lévy processes, was discussed.

Tomas Björk
Forward rate models and invariant manifolds
We investigate when the dynamics of a given forward rate model is con-
sistent with a given finitely parameterized family of forward rate curves.

2
Consistency, in this context, simply means that the forward rate model
actually is able to produce forward rate curves belonging to the pa-
rameterized family. Mathematically this leads to the question when a
finite-dimensional manifold in C-space is invariant under the action of
the (C-valued) infinite-dimensional forward rate process. We give nec-
essary and sufficient conditions for consistency, and apply the results to
some concrete examples. We also propose a new parameterized family
and give conditions for the existence of a consistent forward rate model.

Rainer Buckdahn
Viability for BSDE and associated PDE
Let K ⊂ RN be a nonempty and closed set and let F be a progressively
measurable RN -valued function such that the BSDE
Z T Z T
Yt = ξ + F (s, Ys , Zs ) ds − Zs dWs , 0 ≤ t ≤ T,
t t

has a unique solution (Y, Z) ∈ B 2 for each ξ ∈ L2 (Ω, FTW , P, RN ), where

W is a d-dimensional standard Brownian motion defined on (Ω, F, P).
The talk studies a minimal assumption on F , under which the BSDE
enjoys the viability property with respect to K. The talk is based on a
joint work with Marc Quincampoix (Brest) and Aurel Rascanu (Iaşi).

Rüdiger Frey
Superreplication under stochastic volatility
Stochastic volatility models have been developed in order to cope with
the well-known empirical deficiencies of the standard Black–Scholes
model of geometric Brownian motion. In this class of models the as-
set price follows an SDE of the form dSt = σt St dWt , where σt is not
adapted to the filtration generated by the Brownian motion W. There-
fore these models are incomplete such that there are “many” equivalent
(local) martingale measures for S. We show that for unbounded σt and
for a European call option the quantities

CK = sup{EQ [(ST − K)+ ]: Q equivalent martingale measure}

and

CK = inf{EQ [(ST − K)+ ]: Q equivalent martingale measure}

are given by CK = S0 and CK = (S0 − K)+. Hence it follows from the

3
optional decomposition theorems of Delbaen, El-Karoui and Quenez or
Kramkov that there is no nontrivial super- or subreplication strategy
for the options. We go on and determine “meaningful” superhedging
strategies under the additional assumption that σt is a bounded pro-
cess. In both cases the principal tool is the result that every continuous
local martingale can be represented as time-changed Brownian motion.
We close by discussing the relation of our results to the PDE charac-
terisation of superhedging strategies. The first part of the talk is based
on joint work with C. Sin.

Marco Frittelli
Valuation principles in incomplete financial markets
We describe a general principle for the valuation problem in incomplete
markets that reconciles the “utility” and “martingale” approaches. We
provide a general criterion for selecting one equivalent martingale mea-
sure that requires minimising an appropriate functional which depends
on investors utility. We give sufficient conditions for the existence of the
martingale measure that minimises this functional. We then show that
most existing financial criteria for pricing in incomplete markets are
particular cases of our approach. The results are derived by applying
duality theory and Legendre transforms.

Hélyette Geman
Transaction clock, asset price dynamics and volatility estimates
Normality of asset returns is a central assumption in a number of fun-
damental problems in finance such as portfolio theory, the capital asset
pricing model or the Black–Scholes option pricing. In the measurement
of value at risk, the tails of the distribution obviously play a key role. In
order to address the issue of non-normality of asset returns, which has
been documented in a vast number of empirical studies in finance, this
paper proposes to represent as in Clark (1973) the price process as a
subordinated process. At variance with Clark however, no a priori dis-
tribution is imposed on the subordinator. Using the number of trades
as the stochastic clock, a quasi perfect normality of returns is exhibited
on a high frequency data base of S&P 500 future contracts. Moreover,
we are able to construct an activity-related volatility which reveals to
be a better estimator of the volatility to incorporate in the Black–
Scholes formula (in a stochastic volatility framework) than the histor-
ical volatility, the implied volatility or the Garman–Klass volatility.

4
Hansueli Gerber
From ruin theory to option pricing
We examine the joint distribution of the time of ruin, the surplus im-
mediately before ruin, and the deficit at ruin. The time of ruin is
analysed in terms of its Laplace transform, which can naturally be
interpreted as discounting. We show how to calculate an expected dis-
counted penalty, which is due at ruin, and may depend on the deficit at
ruin and the surplus immediately before ruin. The expected discounted
penalty, considered as a function of the initial surplus, satisfies a cer-
tain renewal equation. By replacing the penalty at ruin with a payoff
at exercise, these results can be applied to pricing a perpetual Amer-
ican put option on a stock, where the logarithm of the stock price is
a shifted compound Poisson process. Because of the stationary nature
of the perpetual option, its optimal option-exercise boundary does not
vary with respect to the time variable. We have derived an explicit for-
mula for determining the optimal boundary. (This is joint work with
Elias S. W. Shiu.)

Farshid Jamshidian
Libor and swap derivatives
A general model for arbitrage-free movements of term structure of for-
ward Libor and swap rates is presented within the framework of a finite-
dimensional securities market model, and applied to evaluate common
swap derivatives such as European and Bermudian swaptions. Appro-
priate numeraires and measures are identified for construction of such
models from the specification of any volatility function. For the log-
normal case the construction is explicit. This is of special importance
in practice as it corresponds to the way cap and European swaptions
are evaluated in the market place.

Monique Jeanblanc
Incomplete markets, range of prices, informed agent
We study an incomplete market where two assets are traded: a riskless
asset with constant interest rate r and a risky asset whose dynamics is

dSt = St− (µ dt + σ dWt + ϕ(dNt − λ dt)), S0 = x,

where W is a Brownian motion and N a Poisson process with constant

intensity λ. In a first part (joint work with N. Bellamy) we study the

5
set Q of equivalent martingale measures and establish that
EQ [e−rT (ST − K)+ ] : Q ∈ Q = ]BS(x), x[,
© ª

where BS is the Black–Scholes function, i. e.,

¢+ ¤
BS(x) = E x exp((r − σ 2 /2)T + σWT ) − K
£¡
, x > 0.
We establish similar results for the values EQ [(ST − K)+ | Ft ], where
Ft = σ(Ws , Ns ; s ≤ t) = σ(Ss ; s ≤ t), and for American and Asian
options. In a second part (work in progress with R. Elliott) we address
the problem of range of prices/optimisation for an “informed” agent
who knows NT . For this agent the dynamics of the asset’s price is
dSt = St− [µ + ϕ(Γt − λ)] dt + ϕ dMt∗ + σ dWt
¡ ¢

where
Z t
NT − Ns−
Mt∗ = Nt + (Γs − λ) ds and Γs = .
0 T −s

Yuri Kabanov
Hedging and liquidation under transaction costs
We study a problem of initial endowment needed to hedge a contingent
claim in various currencies (or other assets). Being inspired by the
recent papers by Cvitanič and Karatzas, we derive a duality description
for this set and apply the result to a problem of optimal control with
a terminal functional. The main message of the talk is that a partial
ordering induced by the solvency cone provides a convenient tool and
elucidates many aspects of the theory of markets with transaction costs.

Claudia Klüppelberg
Extremal behaviour of term structure diffusion models
We investigate the extremal behaviour of diffusions given by the SDE
dXt = µ(Xt ) dt + σ(Xt ) dBt , t > 0, X0 = 0,
where µ is the drift term, σ the volatility and B standard Brownian mo-
tion. Examples which have been considered as term structure models
include the Vasicek model, the Cox–Ingersoll–Ross model and general-
isations. Under suitable conditions the extremes of X have the same
asymptotic behaviour as the extremes of i. i. d. random variables with
a well-specified distribution function, which we derive for the above
examples. (This is joint work with Milan Borkovec.)

6
Ralf Korn
Some applications of optimal impulse control in mathematical finance
Applications of optimal stochastic control in the idealised situation of
continuous trading typically result in optimal trading strategies that
require trading at every time instant. However, under imperfections of
real security markets (such as the occurrence of transaction costs) it
is impossible to follow such strategies. The appropriate mathematical
setting that is able to cope with the presence of transaction costs is
given by the impulse control framework. We show how it can be ap-
plied to three different problems of mathematical finance: the optimal
cash management in equity index tracking, portfolio selection under
transaction costs and the optimal control of the exchange rate. To all
these problems different solution methods (such as an optimal stopping
method, the quasi-variational inequalities approach, and an asymptotic
analysis of the problem) are given.

Dmitrii Kramkov/Walter Schachermayer

A growth condition for utility functions and its relevance in duality
theory
We consider the classical utility maximisation problem

u(x) = sup E[U (x + (H · S)T ], x ∈ R+ ,

H

where U : R+ → R is a utility function with U ′ (0) = ∞, U ′ (∞) = 0,

S = (St )0≤t≤T is a semimartingale taking its values in Rd+ , modelling
the discounted price process of d stocks, and H ranges through the
admissible predictable trading strategies. We analyse under which as-
sumptions the value function u(x) again is a utility function. Under
the standard assumptions u(x) < ∞ and Me (S) 6= ∅, where Me (S)
denotes the equivalent local martingale measures for the process, we
find that a necessary and sufficient condition is the requirement that
U (x) is asymptotically elastic, i. e.,

U ′ (x) · x
lim sup < 1.
x→∞ U (x)

Defining the Legendre transforms

V (y) = sup{U (x) − xy} and v(y) = sup{u(x) − xy},

x>0 x>0

7
we also find that a necessary and sufficient condition for the duality
formula
h ³ dQ ´i
v(y) = infe E V y
Q∈M (S) dP

to hold true again is the asymptotic elasticity of U .

Shigeo Kusuoka
Replication costs for American securities with transaction costs
We first think of a discrete-time complete stochastic finance market
with time unit h, and introduce transaction costs. We also define super-
replication costs for American securities with transaction costs. Our
concern is the limit theorem for the superreplication costs as h ↓ 0.
We prove the limit theorem and show that the limit is described by
the solution of a certain “supermartingale problem.” Finally we define
“supermartingale problem” and discuss about it.

David Lando
Term structures of credit spreads with incomplete accounting infor-
mation
Two approaches to modelling default risk are unified in the follow-
ing sense: It is shown that a “structural model”—in which the as-
sets of a defaultable issuer of bonds are modelled as a diffusion pro-
cess and default is a first hitting time of this diffusion of a given
boundary—becomes a “reduced-form” model—in which default is mod-
elled through a stochastic intensity—if the assets in the structural
model are observed with noise. As an application of this we study the
implications of imperfect accounting information for the term structure
of credit spreads. Leland’s 1994 model is extended by an assumption
that bond investors cannot observe the issuer’s assets directly and re-
ceive instead only periodic and imperfect accounting reports.

Ragnar Norberg
Topics in insurance mathematics
This talk reviews some selected basic areas of insurance mathematics
and discusses their relations—factual and potential—to mathematical
finance. Special emphasis is laid on life insurance mathematics and

8
the probability of ruin. Some pieces of technical progress are reported,
in particular on a Poisson-driven Ornstein–Uhlenbeck process and its
applications to insurance and finance. A brief introduction to actuarial
notation like
(k)
m|n (Iä)x and m|n (DA)x1 x
2
2 x3
1

seemed to amuse the audience.

Bernt Øksendal
The Wick product and the Donsker delta function: How to hedge a
discontinuous claim
We use the white noise calculus, including the Wick product and the
Donsker delta function, to find explicit formulae for the replicating
portfolios in a Black–Scholes market for a class of contingent T -claims,
including claims of the form f (XT (ω)), where (Xt )0≤t≤T is an Itô dif-
fusion and f : R → R is a bounded measurable function. Our results
apply to cases which are not covered by the Black–Scholes partial dif-
ferential equation approach or by the Clark–Ocone formula. The talk
is based on work from a current project with K. Aase and J. Ubøe.

Eckhard Platen
Modelling the dynamics of financial markets
The talk described an approach to the modelling of financial markets.
Starting from two working principles, a non-linear stochastic volatility
dynamics and a short-term forward rate dynamics were derived. The
drift of a stock was specified in a linear mean-reverting way. Further-
more a notion of market risk as an average of squared returns or cost
increments was introduced. Then the dynamics of stochastic volatility
and short-term forward rate followed from the minimisation of market
risk. Many stylised empirical facts about these market characteristics
can be explained by the result. The minimisation of an analogous
market risk for a mixed derivatives and insurance market resulted in
prices for contingent claims that are based on the minimal equivalent
martingale measure. The approach naturally allows the inclusion of
transaction costs and constraints. As typical for local risk minimisa-
tion, the cumulative cost process represents a martingale under the
given objective probability measure.

9
Philip Protter
Complete markets with a discontinuous price process
We propose a parametrised family of financial market models. These
models have jumps in the price process yet are complete with equivalent
martingale measures and no arbitrage. For each β with −2 ≤ β < 0 the
model generalises the standard model (with Brownian motion) which
corresponds to β = 0. Moreover, as β converges to 0, the models con-
verge weakly to the standard model. A hedging result is also presented.
The models rely on the Emery–Azéma martingales, whose development
was originally motivated by quantum probability. (Based on joint work
with Michael Dritschel.)

Uwe Schmock
Estimating the value of the WinCat coupons of the Winterthur Insur-
ance convertible bond
The three annual 2 1/4 % interest coupons of the Winterthur Insurance
convertible bond (face value Chf 4 700) will only be paid out if during
their corresponding observation periods no major storm or hail storm
on one single day damages more than 6 000 motor vehicles insured with
Winterthur Insurance. Data for events, where storm or hail damaged
more than 1 000 insured vehicles, are available for the last ten years.
Using a constant-parameter model, the estimated discounted value of
the three Wincat coupons together is Chf 263.29. A conservative
evaluation, which accounts for the standard deviation of the estimate,
gives a coupon value of Chf 238.25. However, fitting a model, which
admits a trend in the expected number of events per observation period,
leads to substantially higher knock-out probabilities of the coupons.
The estimated discounted value of the coupons drops to Chf 214.44;
a conservative evaluation as above leads to substantially lower values.
Hence, the model uncertainty is in this case substantially higher than
the standard deviations of the used estimators.

Martin Schweizer
From actuarial to financial valuation principles
A valuation principle is a mapping that assigns a number (value) to a
random variable (payoff). We construct a transformation on valuation
principles by embedding them in a financial environment. Given an a
priori valuation rule u, we define the associated a posteriori valuation
rule h on payoffs as follows by an indifference argument: The u-value

10
of optimally investing in the financial market alone should equal the
u-value of first selling the payoff at its h-value and then choosing an
investment strategy that is optimal inclusive of the payoff. In an L2 -
framework, we explicitly obtain the financial transforms of the variance
principle and the standard deviation principle. The resulting finan-
cial valuation rules involve the expectation under the variance-optimal
martingale measure and the intrinsic financial risk of the payoff.

Elias S. W. Shiu
Deferred annuities: Equity-indexed annuities
The purpose of the talk is to point out applications of modern financial
theory to the life insurance business. It explains the various options
granted by an insurance company in its assets and liabilities. Such
options need to be priced and reserved properly. A dominant segment
of the U. S. life insurance business is the deferred annuities, which con-
sists of the fixed-rate annuities, variable annuities and equity-indexed
annuities. These deferred annuities are investment products with (ex-
otic) options which should be priced and reserved using modern option-
pricing theory.

Steven E. Shreve
Hedging under portfolio constraints
Consider a European call which knocks-out (falls to zero value) if the
underlying stock crosses a barrier B prior to expiration. We assume
B exceeds the strike price. The classical Black–Scholes value v(t, x)
at time t if the stock price is x has large negative “delta” vx (t, x) and
“gamma” vxx (t, x) near the barrier. In practice, these large derivatives
prevent traders from implementing the “delta-hedging” strategy. To
overcome this difficulty, there are three possible approaches:
(1) Artificially increase the barrier, and price and hedge the option as
if the higher barrier were the contracted one;
(2) Impose a transaction cost in the model to cover the close-out of
the short position mandated by “delta-hedging” when the option
knocks out;
(3) Price and hedge the option subject to a constraint that the ratio of
the value of the stock shorted by the hedging portfolio to the total
value of the hedging portfolio cannot exceed a prespecified bound.
It is shown that approaches (2) and (3) are equivalent, and (1) is a
first-order approximation to them.

11
Mete Soner
Option pricing in incomplete markets
In this talk, I consider two different examples of incomplete markets
and outlined two approaches to pricing. In the first example, I used
the approach of superreplication to price a European call option with
portfolio constraints. I showed that the minimal price is the Black–
Scholes price of an adjusted claim. (This is joint work with N. Broadie
and J. Cvitanič of Columbia University.) The second example was a
model with proportional transaction costs. I used the utility maximisa-
tion approach of Hodges–Neuberger and Davis–Panas–Zariphopoulou
and asymptotic analysis to derive a nonlinear Black–Scholes equation.
(This is joint work with G. Barles of University of Tours.)

Christophe Stricker
Some inequalities in mathematical finance
This talk is based on two joint papers with T. Choulli and L. Krawczyk.
We give some extensions of the well-known Doob and Burkholder–
Davis–Gundy inequalities to more general processes than martingales.
Such an extension is crucial for the closedness of some spaces of stochas-
tic integrals arising in mathematical finance.

Nizar Touzi
Closed form solution to the super-replication problem under stochastic
volatility, portfolio constraints and transaction costs
We study the problem of finding the minimal initial amount which
allows to hedge a European-type contingent claim. We use a previously
known dual representation of the minimal price as a supremum of the
prices in some corresponding shadow markets. Although the Hamilton–
Jacobi–Bellman equation is not satisfied by the value function of the
dual problem, an explicit closed-form solution is derived using only the
supersolution property.

Marc Yor
Weakly and strongly Brownian filtrations
In this lecture, I presented some recent results due to B. Tsirel’son, the
most striking being: The filtration of Walsh’s Brownian motion with
at least three rays is a weakly Brownian, but not a strongly Brownian
filtration. More explicitly: Although all martingales in this filtration

12
can be written as stochastic integrals with respect to a given Brownian
motion, the filtration is not the natural filtration of a Brownian motion.
The method used helped to solve two other open problems, one about
the minimum of three harmonic measures for Brownian motion, the
other one about the difference between FL+ and FL , where L is the
end of a predictable set. The answer is: Given any such L, FL+ differs
from FL by at most the adjunction of one set (M. Barlow’s conjecture).
The results of B. Tsirel’son should appear in GATA and also in a
presentation by Barlow–Emery–Knight–Song–Yor in Séminaire XXXII,
Lecture Notes in Mathematics, Springer-Verlag (1998).

Berichterstatter: Uwe Schmock (Zürich)

For a TEX-version of the report see

http://www.math.ethz.ch/~ schmock

13
E-Mail Addresses
Knut K. Aase knut.aase@nhh.no
Philippe Artzner artzner@math.u-strasbg.fr
O. E. Barndorff-Nielsen atsoebn@mi.aau.dk
Hans-Jochen Bartels bartels@math.uni-mannheim.de
Tomas Björk fintb@hhs.se
Rainer Buckdahn rainer.buckdahn@univ-brest.fr
Hans Bühlmann hbuhl@math.ethz.ch
Darrell Duffie duffie@baht.stanford.edu
Freddy Delbaen delbaen@math.ethz.ch
Ernst Eberlein eberlein@bachelier.mathematik.uni-freiburg.de
Paul Embrechts embrechts@math.ethz.ch
Hans Föllmer foellmer@mathematik.hu-berlin.de
Rüdiger Frey frey@math.ethz.ch
Marco Frittelli marco.frittelli@unimi.it
Hélyette Geman p geman@edu.essec.fr
Hansueli Gerber hgerber@hec.unil.ch
Christian Hipp christian.hipp@wiwi.uni-karlsruhe.de
Jean Jacod jj@ccr.jussieu.fr
Farshid Jamshidian farshid@sgc.com
Stefan Jaschke jaschke@mathematik.hu-berlin.de
M. Jeanblanc-Picque jeanbl@lami.univ-evry.fr
Yuri Kabanov kabanov@vega.univ-fcomte.fr
Claudia Klüppelberg cklu@mathematik.tu-muenchen.de
Ralf Korn korn@mat.mathematik.uni-mainz.de
Dmitrii Kramkov kramkov@ipsun.ras.ru
Uwe Küchler kuechler@mathematik.hu-berlin.de
Shigeo Kusuoka kusuoka@ms.u-tokyo.ac.jp
Damien Lamberton dlamb@math.univ-mlv.fr
David Lando dlando@math.ku.dk
Peter Leukert leukert@mathematik.hu-berlin.de
Kristian Miltersen krm@busieco.ou.dk
Ragnar Norberg ragnar@math.ku.dk
Bernt Øksendal oksendal@math.uio.no
Dietmar Pfeifer pfeifer@math.uni-hamburg.de
Eckhard Platen eckhard@orac.anu.edu.au
Philip Protter protter@math.purdue.edu
W. J. Runggaldier runggal@math.unipd.it
Ludger Rüschendorf ruschen@buffon.mathematik.uni-freiburg.de

14
Walter Schachermayer wschach@stat1.bwl.UniVie.ac.at
Uwe Schmock schmock@math.ethz.ch
Thomas Schöckel schoeckel@mathematik.hu-berlin.de
Martin Schweizer mschweiz@math.tu-berlin.de
Elias S. W. Shiu eshiu@stat.uiowa.edu
Steven E. Shreve shreve@cmu.edu
Dieter Sondermann sonderma@finasto.uni-bonn.de
Mete Soner mete+@cmu.edu and sonermet@boun.edu.tr
Christophe Stricker stricker@math.univ-fcomte.fr
Nizar Touzi touzi@ceremade.dauphine.fr
Marc Yor —

World Wide Web Addresses

Knut K. Aase http://www.nhh.no

Philippe Artzner http://www.u-strasbg.fr
O. E. Barndorff-Nielsen http://www.mi.aau.dk/˜ atsoebn
Hans-Jochen Bartels http://www.math.uni-mannheim.de/Bartels.html
Tomas Björk http://www.hhs.se/secfi
Rainer Buckdahn http://www.univ-brest.fr
Hans Bühlmann http://www.math.ethz.ch
Darrell Duffie http://www-leland.stanford.edu/˜ duffie
Freddy Delbaen http://www.math.ethz.ch
Ernst Eberlein http://zeus.mathematik.uni-freiburg.de
Paul Embrechts http://www.math.ethz.ch
Hans Föllmer http://www.mathematik.hu-berlin.de
Rüdiger Frey http://www.math.ethz.ch/˜ frey
Marco Frittelli http://www.unimi.it
Hélyette Geman http://babel.essec.fr:8008/domsite/cv.nsf/
WebCv/Helyette+Geman
Hansueli Gerber http://www.hec.unil.ch/annuaire/hgerber
Christian Hipp http://www.uni-karlsruhe.de/˜ ivw
Jean Jacod http://www.proba.jussieu.fr
Farshid Jamshidian http://www.swap.com
Stefan Jaschke http://kuo.mathematik.hu-berlin.de/˜ jaschke
M. Jeanblanc-Picque http://www.univ-evry.fr/labos/lami/maths
Yuri Kabanov http://www.univ-fcomte.fr/
Claudia Klüppelberg http://www-m4.mathematik.tu-muenchen.de/m4

15
Ralf Korn http://www.mathematik.uni-mainz.de/
Stochastik/Arbeitsgruppe/korn.html
Dmitrii Kramkov http://www.ras.ru
Uwe Küchler http://www.mathematik.hu-berlin.de
Shigeo Kusuoka http://liaison.ms.u-tokyo.ac.jp/Faculty.html
Damien Lamberton http://www.univ-mlv.fr
David Lando http://www.math.ku.dk/˜ dlando
Peter Leukert http://www.mathematik.hu-berlin.de
Kristian Miltersen http://www.busieco.ou.dk/man/faculty
Ragnar Norberg http://www.math.ku.dk/˜ ragnar
Bernt Øksendal http://www.math.uio.no
Dietmar Pfeifer http://www.math.uni-hamburg.de/home/pfeifer
Eckhard Platen http://wwwmaths.anu.edu.au
Philip Protter http://www.math.purdue.edu/˜ protter
W. J. Runggaldier http://www.math.unipd.it/people/faculty/
runggaldier.html
Ludger Rüschendorf http://zeus.mathematik.uni-freiburg.de
Walter Schachermayer http://ito.bwl.univie.ac.at/˜ wschach
Uwe Schmock http://www.math.ethz.ch/˜ schmock
Thomas Schöckel http://www.mathematik.hu-berlin.de
Martin Schweizer http://www.math.tu-berlin.de/stoch
Elias S. W. Shiu http://www.stat.uiowa.edu/˜ eshiu
Steven E. Shreve http://www.math.cmu.edu/math/
people/shreve.html
Dieter Sondermann http://www.finasto.uni-bonn.de
Mete Soner http://www.math.cmu.edu/math/
people/soner.html
Christophe Stricker http://www.univ-fcomte.fr
Nizar Touzi http://www.ceremade.dauphine.fr
Marc Yor http://www.proba.jussieu.fr

16