We will discuss the notion of "Levy copula" which, just as the notion
of 'copula' characterize the dependence structure of a random vector,
characterizes the dependence among components of multidimensional Lévy
processes (Cont & Tankov 2003, Kallsen & Tankov 2006). We discuss the
dynamic analogue of Sklar's theorem for Levy porcesses and the
relation between the Levy copula and the copula of the law of the Levy
process. We give parametric examples of Levy copulas, and illustrate
how they can be used for constructing multidimensional infinitely
divisible distribution with given marginals.

(joint with F. Ledrappier)

Suppose T:X-->X is a map, and m is a T-invariant ergodic
measure. We say that m has a "generalized law of large numbers"is
there is a procedure which

(1) accepts as input the times n such that T^n(x) is in a set E (but
not x or E)
(2) gives as output the measure of E

and so that the procedure works for all E measurable,  for almost all
x in X.

If m(X)=1, then the strong law of large numbers gives us such a
procedure (take the limit of (1/n)[1_E(x)+1_E(Tx)+...+1_E(T^n x)]. But
if m(X) is infinite, this fails.

I will discuss alternative "generalized laws of large numbers" for a
natural class of dynamical systems arising from hyperbolic geometry.
No background in hyperbolic geometry is required.

This presentation is devoted to the development and analysis of
mathematical models to enhance understanding of extreme events in
financial markets.
This will be undertaken through two specific problems in the
mathematics of risk management:

* The analysis of asset-price dynamics in models that capture the
possibility of sudden, large changes in prices --- i.e., ``jumps";

* The development and application of tools from mathematical physics to
analyze market dynamics leading to a ``crash", and the corresponding
matching with tools from the Mathematical Finance.

Solutions to the equations arising in the corresponding mathematical
models will be analized.

Minimum-variance hedging in mathematical finance has a long
history, having been initiated by Markowitz in the 1950's for
one-period trading models, then extended to continuous-time
dynamic models by Duffie, Richardson, and Schweizer during
the 1990's, with continuing contributions by Hu, Lim and
Zhou during the past 10 years.
   In this talk we look at the problem of minimum-variance
hedging in a complete standard financial market,
subject to a convex constraint on the portfolio and an
almost-sure "insurance" constraint on the terminal wealth.
We apply the Rockafellar-Moreau approach for problems of
convex minimization. This gives a rational method for
(1) formulating an appropriate vector space of dual variables;
(2) synthesizing a convex-concave bifunction (the Lagrangian)
on the product of the spaces of primal and dual variables;
(3) synthesizing a concave dual functional on the space of
dual variables; (4) establishing existence of a maximizer
for the dual functional (the Lagrange multiplier); and
(5) synthesizing a set of Kuhn-Tucker optimality relations
which are equivalent to the primal and dual problems each
having a solution with equal optimal values.
    In the context of the problem of minimum-variance hedging with
a combination of portfolio and terminal-wealth constraints,
it turns out that the Lagrange multiplier comprises an Ito process
paired with a member of the (topological) dual of the space of
essentially bounded random variables measurable with respect
to the event sigma-algebra at close of trade. A characterization
of this dual space due to Yosida and Hewitt (1952) is used to
make the dual functional more tractable, and it is shown how
a variational analysis on the optimality of the Lagrange
multiplier leads to the construction of a portfolio which
(when paired with the Lagrange multiplier) satisfies the
Kuhn-Tucker relations, and hence is the optimal portfolio.