  The aim of these lectures at MITACS-PIMS-UBC Summer School in Risk Man-
agement and Risk Sharing is to discuss risk controlled approaches for the
pricing and hedging of financial risks. We will start with the classical dual
approach for financial markets, which al- lows to rewrite super-hedging
problems in terms of optimal control problems in standard form. Based on this,
we shall then consider hedging and pricing prob- lems under utility or risk
minimization criteria. This approach will turn out to be powerful whenever
linear (or essentially linear) problems are considered, but not adapted to more
general settings with non-linear dynamics (e.g. large investor models, high
frequency trading with market impact features, mixed finance/insurance issues).
In the second part of this lecture, we will develop on a new approach for risk
control problems based on a stochastic target formulation. We will see how
flexible this approach is and how it allows to characterize very easily super-
hedging prices in term of suitable Hamilton-Jacobi-Bellman type partial differ-
ential equations (PDEs). We will then see how quantile hedging and expected
loss pricing problems can be embeded into this framework, for a very large
class of financial models. We shall finally consider a simple example of
optimal book liquidation in which the control is a continuous non-decreasing
process, as an illustration of possible practical developments in optimal
trading under risk constraint.These lectures are organized in small chapters,
each of them being focused on a particular aspect.
