A contribution in stochastic control applied to finance and insurance

The aim of this thesis is to investigate some solutions to the pricing of contingent claims in incomplete markets. We first consider the stochastic targetintroduced by Soner and Touzi (2002) for the general super-replication problem, and extended by Bouchard, Elie and Touzi (2009) in order to deal with more general approaches. We first generalize the work of Bouchard et al to a framework where the discusions are subject to jumps. In our particular settings, we need to consider a control taking the form of unbounded maps, which has non-trivial impacts on the derivation of the associated PDE. Our second contribution consists in establishing a version of stochastic target problems which is robust to model uncertainty. We provide, in a general setup, a relaxed geometric dynamic programming principle for this problem and derive, for the case of a controlled SDE, the corresponding dynamic programming equation in the sense of viscosity solutions. We consider an example of partial hedging under Knightian uncertainty. Finally, we focus on the problem of pricing hybrid claims. More specifically, we intend to give a sufficient condition for a (very popular) pricing rule, combining actuarial diversification with arbitrage free replication arguments, to hold.