Matrix processes: simulation and modeling of dependency in finance.

Summary The first part of this thesis is devoted to the simulation of stochastic differential equations defined on the nexus of positive symmetric matrices. We present new high order discretization schemes for this type of stochastic differential equations, and study their weak convergence. We are particularly interested in the Wishart process, often used in financial modeling. For this process we propose both a law-accurate scheme and high order discretizations. To date, this method is the only one that can be used whatever the parameters involved in the definition of these models. We show, moreover, how we can reduce the algorithmic complexity of these methods and we verify the theoretical results on numerical implementations. In the second part, we are interested in processes with values in the space of correlation matrices. We propose a new class of stochastic differential equations defined in this space. This model can be considered as an extension of the Wright-Fisher model (or Jacobi process) to the space of correlation matrices. We study the weak and strong existence of solutions. Then, we explain the links with Wishart processes and multi-allele Wright-Fisher processes. We show the ergodic character of the model and give Girsanov representations that can be used in finance. For practical use, we explain two high order discretization schemes. This part concludes with numerical results illustrating the convergence behavior of these schemes. The last part of this thesis is devoted to the use of these processes for multi-dimensional modeling issues in finance. An important modeling issue, which is still difficult to address, is the identification of a type of model that allows to calibrate both the options market on an index and on its components. We propose, here, two types of models: one with local correlation and the other with stochastic correlation. In both cases, we explain which procedure should be adopted to obtain a good calibration of the market data.