|
STOCHASTIC PROCESSES, STOCHASTIC
CALCULUS AND APPLICATIONS
Rome, September 19-20, 2002
Abstracts |
“The Iterated Proportional Fitting
algorithm and its Bayesian version”
The celebrated IPF algorithm has the purpose of constructing a joint distribution with some specified marginals. We will discuss its asymptotic behaviour in a particularly simple example where pairwise compatible marginals are specified that do not come from a joint. We will then turn our attention to similar problems for the Bayesian IPF, which is a stochastic algorithm of Gibbs' sampler type recently proposed in the literature.
Francesca Biagini, University of
Bologna [mailto:biagini@dm.unibo.it]
“Minimal variance hedging for fractional Brownian motion”
Because of its properties (persistence/antipersistence and
self-simi-larity), the fractional Brownian motion ($fBm$) has been suggested as
a useful mathematical tool in many applications, including finance. For
example, these features of $fBm$ seem to appear in the log-returns of stocks,
in weather derivative models and in
electricity prices in a liberated electricity market. In view of this it is of
interest to develop a powerful calculus for $fBm$. Unfortunately, $fBm$ is not
a semimartingale nor a Markov process (unless $H_i=\frac{1}{2}$ for all $i$),
so these theories cannot be applied to $fBm$. However, if $H_i>\frac{1}{2}$
then the paths have zero quadratic variation and it is therefore possible to
define a {\em pathwise integral}, which will obey Stratonovich type (i.e.
``deterministic'') integration rules. Typically the expectation of such
integrals is not 0 and it is known that
the use of these integrals in finance will give markets with {\em arbitrage},
even in the most basic cases. In fact, this unpleasant situation (from a
modelling point of view) occurs whenever we use an integration theory with
Stratonovich integration rules in the generation of wealth from a portfolio.
A Wick-It\^o fractional calculus was subsequently introduced in a white noise setting and applied to finance
by {\O}ksendal and Hu, who proved that the resulting market is free of
arbitrage.
In this paper, we discuss the extension to the multi-dimensional case of
the Wick-It\^o integral with respect to fractional Brownian motion and apply
this approach to study the problem of minimal variance hedging in a (possibly
incomplete) market driven by $m$-dimensional $fBm$. The mean-variance optimal
strategy is obtained by projecting the option value on a suitable space of
stochastic integrals with respect to the $fBm$, which represents the attainable
claims. For classical Brownian motions (and semimartingales) this problem has
been studied by many researchers.
It turns out that for $fBm$ this problem is even harder than in the
classical case and in this paper we concentrate on a special case in order to
get more specific results.
Here we prove first a multi-dimensional It\^o type isometry for such
integrals, which is used in the proof of the multi-dimensional It\^o formula.
These results are then applied in order
to provide a necessary and sufficient characterization of the optimal strategy
as a solution of a differential
equation involving Malliavin derivatives.
Emilio
De Santis, University of Rome
“La Sapienza” [mailto:desantis@mat.uniroma1.it]
In this note we consider a simple {\it change-point} model, described as
follows. Let $U$ be a random time (the change-point) and let $\{N_t \}_{t \geq
0}$ be a simple counting process. Let $T_1 \leq T_2 \leq \dots $ be the arrival
times of $\{N_t \}_{t \geq 0}$ and denote by $h_t$ an "history" of
the type \begin{equation} \label{f1} h_t \equiv \{ T_1 = t_1, \dots , T_k = t_k
, T_{k+1} >t\} \end{equation} with $0 \leq t_1 < t_2 < \dots < t_k
< t $. We assume that $\{N_t \}_{t \geq 0}$ admits an intensity, described
by the condition \begin{equation} \label{int1} \lim_{\Delta t \to 0^+}\frac{1}{\Delta t} P( T_{k+1} <
t + \Delta t \, | \, h_t ; \, U \leq t) = \lambda_A (k) \end{equation}
\begin{equation} \label{int2} \lim_{\Delta
t \to 0^+} \frac{1}{\Delta t} P( T_{k+1} < t + \Delta t \, | \, h_t ;
\, U > t) = \lambda_B (k) \end{equation}
i.e., given the observation of the history $h_t$, $\lambda_A$ would be
the intensity, conditional on the
knowledge that time $t$ is After the
change-point $U$ and $\lambda_B$ would be the intensity,
conditional on the knowledge that time
$t$ is Before $U$. We think of the case when the random variable $U $ is not
observable and denote its density by $g_U$. Then, in this sense, $\{N_t \}_{t
\leq 0 } $ is a conditional {\it pure-birth } process and its intensity is
given by \begin{equation} \label{int3} \mu (t | h_t)= \lambda_A (k | h_t) P(U \leq t \, | \, h_t ) + \lambda_B (k | h_t) P(U > t \, | \,
h_t ). \end{equation}
Assume \begin{equation} \label{assu} \lambda_A (0) - \lambda_B (0)
\geq \lambda_A (1) - \lambda_B (1) \geq
\dots \end{equation} and compare two different observed histories
\begin{equation} \label{ist1} h'_t \equiv \{ T_1 = t'_1, \dots , T_k = t'_k ,
T_{k+1} >t\} \end{equation} \begin{equation} \label{ist2} h''_t \equiv \{
T_1 = t''_1, \dots , T_k = t''_k , T_{k+1} >t\} \end{equation} containing
the same number $k$ of arrivals in the same interval $[0,t]$. We write $ h'_t
\succeq h''_t $ if $t'_l \leq t''_l$,
for $l = 1,2, \dots , k$. Then: \begin{proposition} \label{p1} If $ h'_t
\succeq h''_t $ then $\mu (t | h'_t)
\geq \mu (t | h''_t)$.
\end{proposition}
Finally we briefly interpret the meaning of Proposition~\ref{p1} in the
frame of applications to the fields of statistical mechanics and reliability.
Rita
Giuliano Antonini, University of Pisa [mailto:giuliano@dm.unipi.it]
“Una Legge Forte dei
Grandi Numeri
per il Teorema Limite Centrale Quasi Certo generalizzato”
Nel presente
articolo viene studiato il comportamento
asintotico delle variabili aleatorie \lq\lq pesate"
$${{1}\over{\phi(n)}}\sum_{k=1}^n\big(\phi(k)-\phi(k-1)\big)X_k, \leqno(1.1)$$
dove $(X_k)_{k\geq 1}$ \`e una successione di variabili aleatorie di quadrato
integrabile e $\phi$ \`e una successione
crescente all'infinito. I risultati per le variabili aleatorie (1.1) sono noti
in letteratura con il nome di {\it Almost Sure Central Limit Theorems} (ASCLT).
Per una bibliografia estensiva si veda [1]. Il risultato principale del lavoro
che \`e uno dei pi\`u generali riguardanti l'ASCLT. Esso viene ottenuto per
mezzo di una nuova Legge Forte dei Grandi Numeri, dimostrata in ipotesi di
trascurabilit\`a asintotica delle covarianze. Vale la pena di osservare che il
risultato copre anche le situazioni di {\it quasi ortogonalit\`a} studiate da
M. Weber. Infine tra le applicazioni viene dimostrato un nuovo risultato di
tipo {\it intersettivo}, che
generalizza quello recentemente ottenuto in [2]. [Joint work with \sc Luca Pratelli\rm]
References
[1] Berkes I., Cs\'aki Endre (2001).
{\it A universal result in almost sure central limit theory}, Stoch. Proc.
Appl. 94, p. 105-134.
[2] Giuliano
Antonini R., Weber M. (2001). {\it The intersective ASCLT}, to appear.
Paul Glasserman, Columbia University, New York [mailto:pg20@columbia.edu]
“Importance Sampling in Finance”
Monte Carlo simulation is widely used in finance for the pricing of derivative securities and for risk management. Importance sampling is a technique for reducing the variance of simulation estimates through a change of measure that increases the probability of "important" outcomes. This talk discusses the application of importance sampling to finance. One application changes the drift of an underlying Brownian motion in the pricing of path-dependent options; the drift is selected through an optimization problem that depends on the form of the option payoff. A second application increases the probability of large changes in market prices to estimate the tail of a portfolio loss distribution. We discuss various notions of asymptotic optimality for importance sampling estimators and distinguish light-tailed and heavy-tailed settings.
Paolo
Guasoni, University
of Pisa [mailto:guasoni@dm.unipi.it]
“Excursion Theory and the Martingale Hypothesis”
We study the problem of testing the martingale-hypothesis from time-series
observations of a continuous asset process. The test considered is based on
It\^o's Law of Brownian Excursions, and thus is independent of the continuous
process driving the asset price. This feature provides results which are
insensitive to misspecifications of diffusion models.
We address the problems of estimation to market data, and show results
on long time series of stock market indices. Other applications of Excursion
Theory in Finance will also be discussed.
Claudia Klueppelberg, University of Muenchen [mailto:cklu@mathematik.tu-muenchen.de]
“Extreme behaviour of stochastic
processes
with applications to risk management”
Extreme value theory has been taylored for the mathematical/probabilistic
modelling of rare events. Meanwhile it has become an important tool in
quantitative risk management. For univariate iid data it is a standard
application of this theory to estimate downside risk measures as the
Value-at-Risk, which are based on a very low quantile. It is well-known,
however, that financial data often show a dependence structure which can be
modelled by diffusion models or (G)ARCH
models. Such models capture heavy-tailedness and clustering in the
extremes which affects the quantile estimation. We describe the extremal
behaviour of such volatility models.
Thomas G. Kurtz, University of Wisconsin, Madison [mailto:kurtz@math.wisc.edu]
“Stochastic equations for spatial birth and
death processes”
Spatial birth and death processes of the type to be considered were
first studied systematically by Preston in the 1970s. They began to play a central role in spatial statistics a few
years later when Ripley recognized that important classes of spatial point
processes (Gibbs distributions) could be obtained as stationary distributions
of birth and death processes and that this characterization provided a
computationally feasible method of simulation of the spatial point process (Markov
chain Monte Carlo). In the mathematical
arena, the infinite population versions provide a class of nonlattice
interacting particle systems.
Stochastic equations driven by Poisson random measures will be
formulated for a large class of birth and death processes. The relationship to the corresponding
martingale problem will be discussed.
Conditions for existence and uniqueness and ergodicity (spatial and
temporal) in the infinite population setting will be given. Application of Baddeley's time invariance
estimation methods will be illustrated.
Domenico Marinucci,
University of Rome “La Sapienza” [mailto:marinucc@scec.eco.uniroma1.it]
“Nonparametric model checks for long
memory regression”
We investigate the asymptotic behaviour of the marked empirical process
for regression residuals in the presence of long range dependence. We extend to
the long memory case an approach developed by Stute (1997), Stute et al.
(1998,1999), and we implement test of functional form for time series
regression.
“Epidemic-style
information dissemination”
This talk is concerned with epidemic-style, or gossip-based information
dissemination, according to which each participant of a group propagates
information by "gossiping" to, or "infecting", a few random
selected group members. In a first part of the talk we show how classical
results on epidemic spread, and on the connectivity of random graphs, can
characterise the performance of such information dissemination techniques. In
particular we investigate the robustness of the Erdos-Renyi law for the connectivity
of random graphs. In a second part, we consider the problem of designing
"good" random graphs for information dissemination. This leads us to
consider models of graph growth, and graph rebalancing.
Roberto Monte, University of Rome-Tor Vergata [mailto:monte@sefemeq.uniroma2.it]
“Insider
Trading in Continuous Time with Infinite Horizon”
We consider a market, which lives forever, where two assets
are traded in continuous time: a \emph{riskless asset} and a \emph{risky} one.
The riskless asset pays a constant interest rate $r>0$. The risky asset,
with price $P(t)$, pays a dividend flow, $D(t)$, driven by the stochastic
differential equation \begin{equation} dD\left( t\right) =\left( \Pi \left(
t\right) -\alpha _{D}D\left( t\right)\right) \,dt+\sigma _{D}\,dw(t).
\label{our-dividend-process-I} \end{equation} where $\Pi \left( t\right) $ is a
state variable which decays in time according to \begin{equation} d\Pi \left(
t\right) =-\alpha _{\Pi }\Pi \left( t\right) \,dt+\sigma _{\Pi}\,dw(t).
\label{our-information-process-I} \end{equation} Here $w\left( t\right) \equiv
\left( w_{1}\left( t\right) ,w_{2}\left(t\right) \right)^{\top }$ is a Wiener processes
with independent entries, $\alpha _{D}$, $\alpha _{\Pi }$ are constant
nonnegative parameters, and $\sigma _{D}\equiv (\sigma _{D,1},\sigma _{D,2})$,
$\sigma_{\Pi}\equiv(\sigma_{\Pi ,1},\sigma _{\Pi ,2})$ are constant matrices.
The above model for dividend rewards in continuous time is widely exploited in
literature (see e.g. Campbell and Kyle \cite{Campbell-Kyle-1993} (1993) and
Wang \cite{Wang-1993} (1993)).
There are three types of agents in the market: a
\emph{representative noise trader}, an \emph{insider trader}, and \emph{market
makers}. The representative noise trader, or \emph{liquidity trader}, models
the aggregate effect of all agents who trade in the market with price-inelastic
demand. His order flow, $U(t)$, follows an Ornstein-Uhlenbeck process,
\begin{equation} dU(t)=-\alpha _{U}U(t)+\sigma _{U}\,dw_{U}(t),
\label{our-noise-trader-order-flow-I} \end{equation} where $\alpha _{U},\sigma
_{U}$ are positive parameters, constant in time, and $w_{U}(t)$ is a Wiener
process independent of the entries of $w(t)$. The insider trader, or
\emph{informed trader}, enjoys some private information which allows him to
know the exact value of the parameters $-\alpha_{\Pi}$ and $\sigma_{\Pi}$ in
Equation (\ref{our-dividend-process-I}). In addition, he observes $U(t)$. The
insider trades in the market aiming to exploit all information available to
himself, on account of the feedback effect of his trade on the price $P\left(
t\right)$. The insider trader is \emph{risky averse}. He maximizes the expected
value of his exponential intertemporal utility over an infinite time horizon,
given at a certain time instant his wealth and the state of the economy. Hence
the insider trader's maximization problem becomes \begin{equation} \sup \left\{
\mathbf{E}_{t,m,Y}\left[ \int_{t}^{+\infty }-e^{-\left( \rho s+\psi c(s)\right)
}\,ds\right] \right\} , \label{informed-trader-utility-max.} \end{equation}
where $\rho >0$ and $\psi $ are constant parameters, $c(t)$ is the
consumption rate, and $\mathbf{E}_{t,m,Y}\left[ \cdot \right] $ is the
conditional expectation operator given the time instant $t$, the state vector
of the economy $Y$, and the informed trader's wealth $m$. The market makers
ignore the exact value of the parameters in (\ref{our-dividend-process-I}).
They are \emph{risky neutral} and clear the market by settting the price of the
risky asset equal to the conditional expected present value of the future
dividend stream, given the public information (i.e. the dividend time series),
and the aggregate order flow information. That is to say, \begin{equation}
P(t)=\mathbf{E}\left[ \int_{t}^{+\infty
}e^{-r(s-t)}D(s)\,ds|\mathcal{F}_{t}\right],
\label{market-maker-max.-cond.} \end{equation} where
$(\mathcal{F}_{t})_{t\geq 0}$ is the $\sigma $-field generated by the dividend
process and by the aggregate order flow. The market makers observe only the
aggregate order flow and they cannot distingush the insider trader's order
flow. Following the seminal papers of Kyle \cite{Kyle-1985} (1985) and Back
\cite{Back-1992} (1992), we study the equilibrium which arises from an utility
maximization condition (\ref{informed-trader-utility-max.}) and a market
efficiency condition (\ref{market-maker-max.-cond.}). [Joint
work with \sc Barbara Trivellato\rm, Politecnico di Torino]
References
\bibitem{Back-1992} K. Back: Insider Tra\-ding in
Continuous Time, \emph{The Review of Financial Studies}, \textbf{5}, 3, 387-409
(1992).
\bibitem{Campbell-Kyle-1993} J.Y. Campbell, A.S. Kyle:
Smart Money, Noise Trading and Stock Price Behaviour, \emph{Review of Economic
Studies}, \textbf{60}, 1-34 (1993).
\bibitem{Kyle-1985} A.S. Kyle: Continuous Auctions and
Insider Trading, \emph{E\-co\-no\-me\-tri\-ca}, \textbf{53}, 1315-1335 (1985).
\bibitem{Wang-1993}
J. Wang: A Model of Intertemporal Asset Prices Under Asymmetric
Information, \emph{Review of Economic Studies}, \textbf{60}, 249-282 (1993)
Sergio Scarlatti, University of
Chieti-G. D'Annunzio [mailto:scarlatt@sci.unich.it]
“A folk theorem for minority games”
In a minority game players must choose one of two
alternatives, for example one of two rooms (or one of two assets). Players who
find themselves in the less crowded room gain a positive pay-off. We consider
the case when players repeat the game over and over in time. This repeated game
has been investigated in recent years by people working on different scientific
topics (economics, physics,...). We show that a suitable version of the
"folk theorem" holds for this type of game.
Gianfausto Salvadori, University of
Lecce [mailto:gianfausto.salvadori@unile.it]
“A generalized Pareto intensity-duration
model
of storm rainfall exploiting 2-copulas”
Stochastic models of rainfall, usually based on Poisson arrivals of
rectangular pulses, generally assume exponential marginal distributions for
both storm duration and average rainfall intensity, and the statistical
independence between these variables. However, the advent of stochastic
multifractals made it clear that rainfall statistical properties are better
characterized by heavy tailed Pareto-like distributions, and also the
independence between duration and intensity turned out to be a non realistic assumption.
In this work an improved intensity-duration model is considered, which
describes the dependence between these variables by means of a suitable
2-Copula, and introduces Generalized Pareto marginals for both the storm
duration and the average storm intensity. Several theoretical results are
derived, and a case study is illustrated. [Joint work with C. De Michele, DIIAR (Sezione Idraulica),
Politecnico di Milano]
Adamo
Uboldi, University of Rome
“La Sapienza” and IAC-CNR, Rome [mailto:uboldi@iac.rm.cnr.it]
“The CIR model for Italian interest
rates”
In problems arising from Public Debt Management, interest rates play a
crucial role. In order to study the behavior and to forecast the evolution of
the short rate $r_t$ we implement the classical model proposed by
Cox-Ingersoll-Ross (CIR) with a mean-reverting structure: $$ dr_t=k(\mu -r_t)dt+\sqrt{r_t}dW_t. $$
Our analysis is set in the years 1999-2000, using the data from the
Datastream historical archive for the Italian bonds secondary market. We first
illustrate our statistical analysis:
1) Transcription errors and illiquidity of certain bonds can heavily
influence the implementation of the term structure. By applying the Chauvenet
principle, we propose a method to reject this kind of data.
2) The distribution between real and theoretical values is not normal,
showing large skewness and large kurtosis typical of "fat tail"
distributions.
After exposition of the results, we then perform two comparisons and
discuss our financial observations:
1) We quote the result obtained by Barone-Cuoco-Zautzik (BCZ) for the
years 1984-1989.
The term structure obtained by our analysis has a different behavior,
mainly due to the significant changes in the macro-economical framework and to
the different expectations of investors: for example, the implied volatility in
our period is one third of the previous one, which means higher stability of
the market.
We then prove that recursive rejection of data performed by Barone et
alii can significantly influence the results.
2) We implement the so called "parsimonious model" proposed by
Nelson-Siegel (NS) for the forward rate $f_t$: $$
f_t(\tau)=\beta_1+\beta_2e^{-\lambda \tau}+\beta_3 \lambda \tau e^{-\lambda
\tau}. $$
We finally analyze the results obtained: although there is a serious
lack of data for the short term period in the database, the CIR model shows a
quite good agreement with medium-long term outputs given by the NS method.We
investigate the asymptotic behaviour of the marked empirical process for
regression residuals in the presence of long range dependence. We extend to the
long memory case an approach developed by Stute (1997), Stute et al.
(1998,1999), and we implement test of functional form for time series
regression. [Joint work with Luca Torosantucci,
IAC-CNR, Rome]