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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 thepricing and hedging of financial risks. We will start with the classical dualapproach for financial markets, which al- lows to rewrite super-hedgingproblems in terms of optimal control problems in standard form. Based on this,we shall then consider hedging and pricing prob- lems under utility or riskminimization criteria. This approach will turn out to be powerful wheneverlinear (or essentially linear) problems are considered, but not adapted to moregeneral settings with non-linear dynamics (e.g. large investor models, highfrequency trading with market impact features, mixed finance/insurance issues).In the second part of this lecture, we will develop on a new approach for riskcontrol problems based on a stochastic target formulation. We will see howflexible 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 expectedloss pricing problems can be embeded into this framework, for a very largeclass of financial models. We shall finally consider a simple example ofoptimal book liquidation in which the control is a continuous non-decreasingprocess, as an illustration of possible practical developments in optimaltrading under risk constraint.These lectures are organized in small chapters,each of them being focused on a particular aspect.

Portfolio management under risk contraints - Lectures given at MITACS-PIMS-UBC Summer School in Risk Management and Risk Sharing