In this project our goal is to deal with a class of hedging and pricing problems which
arise in modern Mathematical Finance.
These problems are not only interesting from
the applications point of view
but also provide a good source for new mathematical questions which require
new tools in the area of probability theory.
We will focus on three main topics:

(i) Hedging with Friction.

(ii) Robust Hedging.

(iii) Numerical Schemes.


All the above topics are related to the theory of
pricing and hedging of derivative securities.
In the last 35 years there was great progress in this direction.
By now there is quite a good understanding of pricing and hedging
in frictionless financial markets with a known probabilistic structure.
This understanding was achieved by developing the machinery of stochastic calculus,
stochastic control, martingale theory,
hypothesis testing, etc.

In real market conditions,
it is very difficult to provide
a correct probabilistic model for the behavior
of stock prices.
Furthermore, trading of assets is subject to transaction costs,
i.e. there is a gap between an ask price and the bid price.
These two facts
raise the natural question
of understanding hedging in markets with friction and model uncertainty.

Usually, when dealing with complex models of financial markets, explicit
formulas for option prices and the corresponding super--replication strategies
are not available. This is the motivation to study numerical schemes
for several stochastic control problems which are related to hedging under volatility uncertainty.
In the current project we are interested not only in providing the algorithms of numerical schemes,
but also in implementing them.

In summary,
the proposed questions are not only crucial for the understanding
of pricing and
hedging in financial markets, but also a great source of new mathematical problems.
These problems require new tools and also attract the attention of world class mathematicians.