Reduced-form framework under model uncertainty and generalized Feynman-Kac formula in the G-setting

The thesis dealing with topics under model uncertainty consists of two main parts. In the first part, we introduce a reduced-form framework in the presence of multiple default times under model uncertainty. In particular, we define a sublinear conditional operator with respect to a family of possibly non-dominated priors for a filtration progressively enlarged by multiple ordered defaults. Moreover, we analyze the properties of this sublinear conditional expectation as a pricing instrument and consider an application to insurance market modeling with non-linear affine intensities. In the second part of this thesis, we prove a Feynman-Kac formula under volatility uncertainty which allows to take into account a discounting factor. In the first part, we generalize the results of a reduced-form framework under model uncertainty for a single default time in order to consider multiple ordered default times. The construction of these default times is based on a generalization of the Cox model under model uncertainty. Within this setting, we progressively enlarge a reference filtration by N ordered default times and define the sublinear expectation with respect to the enlarged filtration and a set of possibly non-dominated probability measures. We derive a weak dynamic programming principle for the operator and use it for the valuation of credit portfolio derivatives under model uncertainty. Moreover, we analyze the properties of the operator as a pricing instrument under model uncertainty. First, we derive some robust superhedging duality results for payment streams, which allow to interpret the operator as a pricing instrument in the context of superhedging. Second, we use the operator to price a contingent claim such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind”. Moreover, we provide some conditions which guarantee the existence of a modification of the operator which has quasi-sure càdlàg paths. Finally, we conclude this part by an application to insurance market modeling. For this purpose, we extend the reduced-form framework under model uncertainty for a single default time to include intensities following a non-linear affine process under parameter uncertainty. This allows to introduce a longevity bond under model uncertainty in a way consistent with the classical case under a single prior and to compute its valuation numerically. In the second part, we focus on volatility uncertainty and, more specifically on the G-expectation setting. In this setting, we provide a generalization of a Feynman-Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the current result in the literature, where the Lipschitz continuity of some functionals is assumed, which is not necessarily satisfied in our setting. Thus, we establish for the first time a relation between non-linear PDEs and G-conditional expectation of a discounted payoff. To do so, we introduce a family of fully non-linear PDEs identified by a regularizing parameter with terminal condition φ at time T > 0, and obtain the G-conditional expectation of a discounted payoff as the limit of the solutions of such a family of PDEs when the regularity parameter goes to zero. Using a stability result, we can prove that such a limit is a viscosity solution of the limit PDE. Therefore, we are able to show that the G-conditional expectation of the discounted payoff is a solution of the PDE. In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.