MULTIVARIATE EXTREMES, MAX-STABLE PROCESS ESTIMATION AND
DYNAMIC FINANCIAL MODELING

ABSTRACT: Studies have shown time series data from finance, insurance
and environment etc. are fat tailed and clustered when extremal events
occur. In order to characterize such extremal processes, max-stable
processes or min-stable processes have been proposed since the 1980s and
some probabilistic properties have been obtained, but the applications are
very limited due to lack of efficient statistical estimation methods.

In this work, some probabilistic properties of the processes are proved
and a series of estimation procedures to estimate the underlying
max-stable processes are proposed, i.e. multivariate maxima of moving
maxima processes. The first proposed method is purely probabilistic. It is
designed for the time series with only one signature pattern, which can be
regarded as a clustering pattern. It gives true parameter values if the
model is correct. The second proposed method is a two step estimating
method. At the first step, the method gives exact parameter values within
each signature pattern, then it estimates the proportions of different
signature patterns in the process. Consistency and asymptotic properties
for the estimators are proved. The third proposed method is a generalized
version of the second one but is not tied with the data, i.e. the data are
not assumed to follow the model exactly. It is practically applicable.
Three variants of the third method are proposed. They are designed to
provide more specific estimators for special cases of the model, such as
symmetric, monotone and asymmetric data structure respectively. All the
estimators have been proved to be consistent and asymptotically normal.

To date, the exceedance over threshold approach which uses a
generalized Pareto distribution(GPD) has been advocated. Assuming
the population distribution belongs to the multivariate domains of
attraction of multivariate extreme value distributions we develop
threshold methods to estimate the parameters of the underlying
max-stable process from the observed data. All previously
developed six methods have their corresponding version under
threshold methods.

How to manage a portfolio efficiently, with the highest expected
return for a given level of risk, or equivalently, the least risk
for a given level of expected return, is a key to the success or
failure of a financial system. As an application of max-stable processes,
financial time series data are standardized and transformed. The new time
series are modelled as max-stable processes. The VaR ( Value at Risk ),
maximal possible losses of portfolios under given confidence level,
of portfolios are calculated and portfolio optimizations under VaR
constraints are then studied.

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TUESDAY, APRIL 23, 2002
NEW WEST 212
3:30pm -- 4:30pm
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