Abstract (EN)

This PhD dissertation presents three research topics. The first two topics are related to the domain of robust finance and the last is related to a numerical method applied in risk management of insurance companies. In the first part, we focus on the problem of super-replication duality for American options in discrete time financial models. We con- sider the robust framework with a family of non-dominated probability measures and the trading strategies are dynamic on the stocks and static on the options. We use two differ- ent ways to obtain the pricing-hedging duality. The first insight is that we can reformulate American options as European options on an enlarged space. The second insight is that by considering a fictitious extensions of the market on which all the assets are traded dynamically. We then show that the general results apply in two important examples of the robust framework. In the second part, we consider the problem of super-replication and utility maximization with proportional transaction cost in discrete time financial market with model uncertainty. Our key technique is to convert the original problem to a frictionless problem on an enlarged space by using a randomization technique to get her with the minimax theorem. For the super-replication problem, we obtain the duality results well-known in the classical dominated context. For the utility maximization problem, we are able to prove the existence of the optimal strategy and the convex duality theorem in our context with transaction costs. In the third part, we present a numerical method based on a sparse grid approximation to compute the loss distribution of the balance sheet of an insurance company. We compare the new numerical method with the traditional nested simulation approach and review the convergence of both methods to estimate the risk indicators under consideration.