Three essays on modeling dependence between financial assets.

Summary This thesis deals with two aspects of dependence between financial assets. The first part concerns the dependence between random vectors. The first chapter consists in a comparison of algorithms computing the optimal transport application for the quadratic cost between two probabilities on R^n, possibly continuous. These algorithms allow to compute couplings with an extreme dependence property, called maximum correlation coupling, which appear naturally in the definition of multivariate risk measures. The second chapter proposes a definition of the extreme dependence between random vectors based on the notion of covariogram. The extreme couplings are characterized as couplings of maximum correlation with linear modification of one of the multivariate marginals. A numerical method to compute these couplings is provided, and applications to dependence stress-testing for portfolio allocation and valuation of European options on several underlyings are detailed. The last part describes the spatial dependence between two Markovian diffusions, coupled using a correlation function depending on the state of the two diffusions. A forward integrated Kolmogorov PDE links the family of spatial copulas of the diffusion to the correlation function. We then study the problem of the spatial dependence attainable by two Brownian motions, and our results show that some classical copulas are not able to describe the stationary dependence between coupled Brownian motions.