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\centerline{\tt MATHEMATISCHES FORSCHUNGSINSTITUT OBEFRWOLFACH}
\vskip 15mm\relax
\centerline{\tt T\ a\ g\ u\ n\ g\ s\ b\ e\ r\ i\ c\ h\ t\ \ 38/1997}
\vskip 10mm plus 5mm minus 5mm
\centerline{\large Stochastic Analysis in Finance and Insurance}
\vskip 5mm plus 2mm minus 2mm
\centerline{\large 14.\,9.$\!$ -- 20.\,9.\,1997}
\vskip 10mm plus 5mm minus 5mm
\noindent
This meeting was organised jointly by
Darrell Duffie (Stanford), Paul Embrechts (Z\"urich)
and Hans F\"ollmer (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:
\smallskip
\item{--} incomplete financial markets, in particular stochastic
volatility, equilibrium analysis, stochastic optimisation
problems, and applications in insurance,
\smallskip
\item{--} hedging strategies in the presence of transaction costs
and constraints,
\smallskip
\item{--} financial versus actuarial pricing principles,
asset liability modelling, default risk, and insurance derivatives,
\smallskip
\item{--} new approaches to the modelling of asset price dynamics,
\smallskip
\item{--} stochastic dynamics of the term structure of interest
rates, in particular geometric aspects, interest rate derivatives,
and extremes,
\smallskip
\item{--} theoretical problems in stochastic analysis motivated by
applications in finance, in particular martingale inequalities,
backward stochastic differential equations and the structure of
Brownian filtrations.
\smallskip\noindent
The meeting had 49 participants.
\par\eject
\leftline{\large Abstracts}
%\vskip-\medskipamount
\abstract{Knut Aase}
{A new equilibrium asset pricing model
based on L\'evy 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 situation where the underlying random modelling
the spot prices of assets are exponentials of L\'evy 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 equity premium
puzzle. The \lq\lq survival\rq\rq\ hypothesis of Brown, Goetzmann
and Ross (1995) is also investigated within this model, giving a
very low crash probability of the market.}
\abstract{Ole E. Barndorff-Nielsen}
{Some thoughts on statistical modelling in finance}
{There are striking similarities between finance and turbulence as
regard 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 exhibit the type of (quasi) long-range
dependence or scaling/self\0similarity 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 L\'evy processes, was discussed.}
\abstract{Tomas Bj\"ork}
{Forward rate models and invariant manifolds}
{We investigate when the dynamics of a given forward rate model is
consistent with a given finitely parameterized family of forward
rate curves. Consistency, in this context, simply means that the
forward rate model actually is able to produce forward rate curves
belonging to the parameterized 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 necessary 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.\looseness=-1}
\abstract{Rainer Buckdahn}
{Viability for BSDE and associated PDE}
{Let $K\subset\re^N$ be a nonempty and closed set and let $F$ be a
progressively measurable $\re^N\5$-valued function such that the
BSDE
$$
Y_t=\xi+\int_t^TF(s,Y_s,Z_s)\,ds-\int_t^TZ_s\,dW_{\5s},
\qquad0\le t\le T,
$$
has a unique solution $(Y,Z)\in B^2$ for each
$\xi\in L^2(\Omega,{\cal F}_T^W\5,\Pa,\re^N)$, where $W$ is a
$d$-dimensional standard Brownian motion defined on
$(\Omega,{\cal F},\Pa)$. 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 (Ia\c si).}
\abstract{R\"udiger 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 asset price follows an SDE of the form
$dS_t=\sigma_t\4S_t\,dW_t$, where $\sigma_t$ is not adapted to the
filtration generated by the Brownian motion $W\5$.
Therefore these models are incomplete such that there are \lq\lq
many\rq\rq\ equivalent (local) martingale measures for $S$. We
show that for unbounded $\sigma_t$ and for a European call option
the quantities
$$
\overbar{C}_K=\sup\{\4\E^Q[(S_T-K)^+]\4\colon\4
\hbox{$Q$ equivalent martingale measure}\4\}
$$
and
$$
\underline{C\!}\4_K=\inf\{\4\E^Q[(S_T-K)^+]\4\colon\4
\hbox{$Q$ equivalent martingale measure}\4\}
$$
are given by
$\overbar{C}_K=S_0$ and
$\underline{C\!}\4_K=(S_0-K)^+\5$.\vadjust{\goodbreak} Hence it
follows from the 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
\lq\lq meaningful\rq\rq\ superhedging strategies under the additional
assumption that $\sigma_t$ is a bounded process. 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 characterisation
of superhedging strategies. The first part of the talk is based on
joint work with C.~Sin.}
\abstract{Marco Frittelli}
{Valuation principles in incomplete financial markets}
{We describe a general principle for the valuation problem in
incomplete markets that reconciles the \lq\lq utility\rq\rq\ and
\lq\lq martingale\rq\rq\ approaches. We provide a general
criterion for selecting one equivalent martingale measure 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.}
\abstract{H\'elyette Geman}
{Transaction clock, asset price dynamics and volatility estimates}
{Normality of asset returns is a central assumption in a number of
fundamental 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 distribution 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 historical volatility,
the implied volatility or the Garman--Klass volatility.
\looseness=-1}
\abstract{Hansueli Gerber}
{From ruin theory to option pricing}
{We examine the joint distribution of the time of ruin, the surplus
immediately 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
discounted 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 certain renewal equation. By replacing the
penalty at ruin with a payoff at exercise, these results can be
applied to pricing a perpetual American 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 formula for
determining the optimal boundary. (This is joint work with Elias
S.\,W. Shiu.)}
\abstract{Farshid Jamshidian}
{Libor and swap derivatives}
{A general model for arbitrage-free movements of term structure of
forward 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. Appropriate numeraires and measures are identified for
construction of such models from the specification of any volatility
function. For the lognormal 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.}
\abstract{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
$$
dS_t=S_{t-}(\mu\,dt+\sigma\,dW_t+\varphi(dN_t-\lambda\,dt)),
\qquad S_0=x,
$$
where $W$ is a Brownian motion and $N$ a Poisson process with
constant intensity $\lambda$.
In a first part (joint work with N. Bellamy) we study the set
$\cal Q$ of equivalent martingale measures and establish that
$$
\bigl\{\4\E_Q[\4e^{-rT}(S_T-K)^+]\,\colon\,Q\in{\cal Q}\4\bigr\}
=\mathopen{]}{\cal BS}(x),x\mathclose{[}\4,
$$
where ${\cal BS}$ is the Black--Scholes function, i.\,e.,
$$
{\cal
BS}(x)=
\E\bigl[\bigl(x\exp((r-\sigma^2/2)T+\sigma W_T)-K\bigr)^+\bigr],
\qquad x>0.
$$
We establish similar results for the values
$\E_Q[(S_T-K)^+\conditioned{\cal F}_t]$,
where
${\cal F}_t=\sigma(W_s,N_s;s\le t)=\sigma(S_s; s\le t)$,
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 \lq\lq
informed\rq\rq\ agent who knows $N_T$. For this agent the dynamics
of the asset's price is
$$\belowdisplayskip=0.3\belowdisplayshortskip
dS_t=S_{t-}\bigl([\mu+\varphi(\Gamma_t-\lambda)]\,dt
+\varphi\,dM_t^*+\sigma\,dW_t\bigr)
$$
where
$$\abovedisplayskip=0.5\belowdisplayshortskip
M_t^*=N_t+\int_{\strut0}^t(\Gamma_s-\lambda)\,ds
\quad\hbox{and}\quad
\Gamma_s={N_T-N_{s-}\over T-s}.
$$}
\abstract{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 Cvitani\v c 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.}
\abstract{Claudia Kl\"uppelberg}
{Extremal behaviour of term structure diffusion models}
{We investigate the extremal behaviour of diffusions given by the
SDE
$$
dX_t=\mu(X_t)\,dt+\sigma(X_t)\,dB_t,\qquad
\hbox{$t>0$, $X_0=0$,}
$$
where $\mu$ is the drift term, $\sigma$ the volatility and $B$
standard Brownian motion. Examples which have been considered as
term structure models include the Vasicek model, the
Cox--Ingersoll--Ross model and generalisations.
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.)}
\abstract{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 applied 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.}
\abstract{Dmitrii Kramkov/Walter Schachermayer}
{A growth condition for utility functions
and its relevance in duality theory}
{We consider the classical utility maximisation problem
$$
u(x)=\sup_H\E[U(x+(H\cdot S)_T],\qquad x\in\re_+,
$$
where $U\from\re_+\to\re$ is a utility function with
$U'(0)=\infty$, $U'(\infty)=0$, $S=(S_t)_{0\le t\le T}$ is a
semimartingale taking its values in $\re^d_+$, modelling the
discounted price process of $d$ stocks, and $H$ ranges through the
admissible predictable trading strategies. We analyse under which
assumptions the value function $u(x)$ again is a utility function.
Under the standard assumptions $u(x)<\infty$ and
${\cal M}^{\rm e}(S)\not=\emptyset$, where ${\cal M}^{\rm e}(S)$
denotes the equivalent local martingale measures for the process,
we find that a necessary and sufficient condition is the
requirement that $U(x)$ is {\it asymptotically elastic,} i.\,e.,
$$
\limsup_{x\to\infty}{U'(x)\cdot x\over U(x)}<1.
$$
Defining the Legendre transforms
$$
V(y)=\sup_{x>0}\{U(x)-xy\}
\qquad\hbox{and}\qquad
v(y)=\sup_{x>0}\{u(x)-xy\},
$$
we also find that a necessary and sufficient condition for the
duality formula
$$
v(y)=\inf_{Q\in{\cal M}^{\rm e}(S)}\E\Bigl[V\Bigl(y{dQ\over
d\mskip1mu\Pa}\Bigr)\Bigr]
$$
to hold true again is the asymptotic elasticity of $U$.}
\abstract{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\0replication costs for American securities with
transaction costs. Our concern is the limit theorem for the
superreplication costs as $h\downarrow0$. We prove the limit
theorem and show that the limit is described by the solution of a
certain \lq\lq supermartingale problem.\rq\rq\ Finally we define
\lq\lq supermartingale problem\rq\rq\ and discuss about it.}
\abstract{David Lando}
{Term structures of credit spreads
with incomplete accounting information}
{Two approaches to modelling default risk are unified in the
following sense: It is shown that a \lq\lq structural
model\rq\rq---in which the assets of a defaultable issuer of bonds
are modelled as a diffusion process and default is a first hitting
time of this diffusion of a given boundary---becomes a \lq\lq
reduced-form\rq\rq\ model---in which default is modelled 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 receive instead only periodic and imperfect
accounting reports.}
\abstract{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\vadjust{\goodbreak} and 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
$$\belowdisplayskip=\belowdisplayshortskip
\abovedisplayskip=\belowdisplayshortskip
_{m|n}(I\ddot a)^{(k)}_x
\qquad\hbox{and}\qquad
_{m|n}(D\overbar{A})_{
\vbox{\null\vskip-3pt\baselineskip=.4\baselineskip
\halign{&$\scriptstyle#$\cr
&\scriptscriptstyle\42\cr
x_1&x_2&x_3\cr\scriptscriptstyle\41\cr}}}
$$
seemed to amuse the audience.}
\abstract{Bernt \O ksendal}
{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 $f(X_T(\omega))$, where
$(X_t)_{0\le t\le T}$ is an It\^o diffusion and
$f\from\re\to\re$ is a bounded measurable function. Our results
apply to cases which are not covered by the Black--Scholes partial
differential equation approach or by the Clark--Ocone formula. The
talk is based on work from a current project with K.~Aase and
J.~Ub\o e.}
\abstract{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. Furthermore 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 minimisation, the cumulative cost process represents a
martingale under the given objective probability measure. }
\abstract{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 $\beta$
with $-2\le\beta<0$ the model generalises the standard model
(with Brownian motion) which corresponds to $\beta=0$. Moreover, as
$\beta$ converges to $0$, the models converge weakly to the
standard model. A hedging result is also presented. The models
rely on the Emery--Az\'ema martingales, whose development was
originally motivated by quantum probability. (Based on joint work
with Michael Dritschel.)}
\abstract{Uwe Schmock}
{Estimating the value of the WinCat coupons
of the Winterthur Insurance convertible bond}
{The three annual $2\slashedfrac14\4$\% 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.}
\abstract{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:\vadjust{\goodbreak} The $u$-value 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
$L^2$-framework, we explicitly obtain the financial transforms of
the variance principle and the standard deviation principle. The
resulting financial valuation rules involve the expectation under
the variance-optimal martingale measure and the intrinsic
financial risk of the payoff.}
\abstract{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 consists of the
fixed-rate annuities, variable annuities and equity-indexed
annuities. These deferred annuities are investment products with
(exotic) options which should be priced and reserved using modern
option-pricing theory.}
\abstract{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 \lq\lq delta\rq\rq\ $v_x(t,x)$ and \lq\lq
gamma\rq\rq\ $v_{xx}(t,x)$ near the barrier. In practice, these
large derivatives prevent traders from implementing the \lq\lq
delta-hedging\rq\rq\ strategy. To overcome this difficulty, there
are three possible approaches:
\smallskip
\item{(1)}Artificially increase the barrier, and price and hedge
the option as if the higher barrier were the contracted one;
\item{(2)}Impose a transaction cost in the model to cover the
close-out of the short position mandated by \lq\lq
delta-hedging\rq\rq\ when the option knocks out;
\item{(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.\looseness=-1
\vadjust{\nobreak}
\smallskip\noindent
It is shown that approaches (2) and (3) are equivalent, and (1)
is a first-order approximation to them.}
\abstract{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.~Cvitani\v c of
Columbia University.) The second example was a model with
proportional transaction costs. I used the utility maximisation
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.)}
\abstract{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 stochastic integrals arising in mathematical
finance.}
\abstract{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.}
\abstract{Marc Yor}
{Weakly and strongly Brownian filtrations}
{In this lecture, I presented some recent results due to
B.~Tsirel'son, the most striking being:
{\it 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 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
${\cal F}_{L+}$ and ${\cal F}_L$, where $L$ is the end of a
predictable set. The answer is: Given any such $L$,
${\cal F}_{L+}$ differs from ${\cal F}_L$ 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 S\'eminaire
XXXII, Lecture Notes in Mathematics, Springer-Verlag (1998).}
\bigskip\bigskip\noindent
{\titlebold Berichterstatter:} Uwe Schmock (Z\"urich)
\bigskip
\def~{\lower3pt\hbox{\~{}\kern1pt}}
\noindent For a \TeX-version of the report see\hfil\break
{\tt http://www.math.ethz.ch/~schmock}
\vfill\break
\leftline{\large E-Mail Addresses}
\bigskip
\tabskip=0pt
\halign to\hsize{#\quad\hfil&#\hfil\tabskip=\centering\cr
Knut K. Aase&knut.aase@nhh.no\cr
Philippe Artzner&artzner@math.u-strasbg.fr\cr
O.\,E. Barndorff-Nielsen&atsoebn@mi.aau.dk\cr
Hans-Jochen Bartels&bartels@math.uni-mannheim.de\cr
Tomas Bj\"ork&fintb@hhs.se\cr
Rainer Buckdahn&rainer.buckdahn@univ-brest.fr\cr
Hans B\"uhlmann&hbuhl@math.ethz.ch\cr
Darrell Duffie&duffie@baht.stanford.edu\cr
Freddy Delbaen&delbaen@math.ethz.ch\cr
Ernst Eberlein&eberlein@bachelier.mathematik.uni-freiburg.de\cr
Paul Embrechts&embrechts@math.ethz.ch\cr
Hans F\"ollmer&foellmer@mathematik.hu-berlin.de\cr
R\"udiger Frey&frey@math.ethz.ch\cr
Marco Frittelli&marco.frittelli@unimi.it\cr
H\'elyette Geman&p\_\kern1pt geman@edu.essec.fr\cr
Hansueli Gerber&hgerber@hec.unil.ch\cr
Christian Hipp&christian.hipp@wiwi.uni-karlsruhe.de\cr
Jean Jacod&jj@ccr.jussieu.fr\cr
Farshid Jamshidian&farshid@sgc.com\cr
Stefan Jaschke&jaschke@mathematik.hu-berlin.de\cr
M. Jeanblanc-Picque&jeanbl@lami.univ-evry.fr\cr
Yuri Kabanov&kabanov@vega.univ-fcomte.fr\cr
Claudia Kl\"uppelberg&cklu@mathematik.tu-muenchen.de\cr
Ralf Korn&korn@mat.mathematik.uni-mainz.de\cr
Dmitrii Kramkov&kramkov@ipsun.ras.ru\cr
Uwe K\"uchler&kuechler@mathematik.hu-berlin.de\cr
Shigeo Kusuoka&kusuoka@ms.u-tokyo.ac.jp\cr
Damien Lamberton&dlamb@math.univ-mlv.fr\cr
David Lando&dlando@math.ku.dk\cr
Peter Leukert&leukert@mathematik.hu-berlin.de\cr
Kristian Miltersen&krm@busieco.ou.dk\cr
Ragnar Norberg&ragnar@math.ku.dk\cr
Bernt \O ksendal&oksendal@math.uio.no\cr
Dietmar Pfeifer&pfeifer@math.uni-hamburg.de\cr
Eckhard Platen&eckhard@orac.anu.edu.au\cr
Philip Protter&protter@math.purdue.edu\cr
W.\,J. Runggaldier&runggal@math.unipd.it\cr
Ludger R\"uschendorf&ruschen@buffon.mathematik.uni-freiburg.de\cr
Walter Schachermayer&wschach@stat1.bwl.UniVie.ac.at\cr
Uwe Schmock&schmock@math.ethz.ch\cr
Thomas Sch\"ockel&schoeckel@mathematik.hu-berlin.de\cr
Martin Schweizer&mschweiz@math.tu-berlin.de\cr
Elias S.\,W. Shiu&eshiu@stat.uiowa.edu\cr
Steven E. Shreve&shreve@cmu.edu\cr
Dieter Sondermann&sonderma@finasto.uni-bonn.de\cr
Mete Soner&mete+@cmu.edu and sonermet@boun.edu.tr\cr
Christophe Stricker&stricker@math.univ-fcomte.fr\cr
Nizar Touzi&touzi@ceremade.dauphine.fr\cr
Marc Yor&---\cr
}
\bigskip\bigskip
\leftline{\large World Wide Web Addresses}
\bigskip
\tabskip=0pt
\halign to\hsize{#\quad\hfil&#\hfil\tabskip=\centering\cr
Knut K. Aase&http://www.nhh.no\cr
Philippe Artzner&http://www.u-strasbg.fr\cr
O.\,E. Barndorff-Nielsen&http://www.mi.aau.dk/~atsoebn\cr
Hans-Jochen Bartels&http://www.math.uni-mannheim.de/Bartels.html\cr
Tomas Bj\"ork&http://www.hhs.se/secfi\cr
Rainer Buckdahn&http://www.univ-brest.fr\cr
Hans B\"uhlmann&http://www.math.ethz.ch\cr
Darrell Duffie&http://www-leland.stanford.edu/~duffie\cr
Freddy Delbaen&http://www.math.ethz.ch\cr
Ernst Eberlein&http://zeus.mathematik.uni-freiburg.de\cr
Paul Embrechts&http://www.math.ethz.ch\cr
Hans F\"ollmer&http://www.mathematik.hu-berlin.de\cr
R\"udiger Frey&http://www.math.ethz.ch/~frey\cr
Marco Frittelli&http://www.unimi.it\cr
H\'elyette Geman&http://babel.essec.fr:8008/domsite/cv.nsf/\cr
\noalign{\nobreak}
&\phantom{http://}WebCv/Helyette+Geman\cr
Hansueli Gerber&http://www.hec.unil.ch/annuaire/hgerber\cr
Christian Hipp&http://www.uni-karlsruhe.de/~ivw\cr
Jean Jacod&http://www.proba.jussieu.fr\cr
Farshid Jamshidian&http://www.swap.com\cr
Stefan Jaschke&http://kuo.mathematik.hu-berlin.de/~jaschke\cr
M. Jeanblanc-Picque&
http://www.univ-evry.fr/labos/lami/maths\cr
Yuri Kabanov&http://www.univ-fcomte.fr/\cr
Claudia Kl\"uppelberg&
http://www-m4.mathematik.tu-muenchen.de/m4\cr
Ralf Korn&
http://www.mathematik.uni-mainz.de/\cr
\noalign{\nobreak}
&\phantom{http://}Stochastik/Arbeitsgruppe/korn.html\cr
Dmitrii Kramkov&http://www.ras.ru\cr
Uwe K\"uchler&http://www.mathematik.hu-berlin.de\cr
Shigeo Kusuoka&http://liaison.ms.u-tokyo.ac.jp/Faculty.html\cr
Damien Lamberton&http://www.univ-mlv.fr\cr
David Lando&http://www.math.ku.dk/~dlando\cr
Peter Leukert&http://www.mathematik.hu-berlin.de\cr
Kristian Miltersen&http://www.busieco.ou.dk/man/faculty\cr
Ragnar Norberg&http://www.math.ku.dk/~ragnar\cr
Bernt \O ksendal&http://www.math.uio.no\cr
Dietmar Pfeifer&http://www.math.uni-hamburg.de/home/pfeifer\cr
Eckhard Platen&http://wwwmaths.anu.edu.au\cr
Philip Protter&http://www.math.purdue.edu/~protter\cr
W.\,J. Runggaldier
&http://www.math.unipd.it/people/faculty/\cr
\noalign{\nobreak}
&\phantom{http://}runggaldier.html\cr
Ludger R\"uschendorf&http://zeus.mathematik.uni-freiburg.de\cr
Walter Schachermayer&http://ito.bwl.univie.ac.at/~wschach\cr
Uwe Schmock&http://www.math.ethz.ch/~schmock\cr
Thomas Sch\"ockel&http://www.mathematik.hu-berlin.de\cr
Martin Schweizer&http://www.math.tu-berlin.de/stoch\cr
Elias S.\,W. Shiu&http://www.stat.uiowa.edu/~eshiu\cr
Steven E. Shreve&http://www.math.cmu.edu/math/\cr
\noalign{\nobreak}
&\phantom{http://}people/shreve.html\cr
Dieter Sondermann&http://www.finasto.uni-bonn.de\cr
Mete Soner&http://www.math.cmu.edu/math/\cr
\noalign{\nobreak}
&\phantom{http://}people/soner.html\cr
Christophe Stricker&http://www.univ-fcomte.fr\cr
Nizar Touzi&http://www.ceremade.dauphine.fr\cr
Marc Yor&http://www.proba.jussieu.fr\cr
}
\vfil\end