Exact and high-order discretization schemes for Wishart processes and their affine extensions

This work deals with the simulation of Wishart processes and affine diffusions on positive semidefinite matrices. To do so, we focus on the splitting of the infinitesimal generator in order to use composition techniques as did Ninomiya and Victoir [Appl. Math. Finance 15 (2008) 107.121] or Alfonsi [Math. Comp. 79 (2010) 209.237]. Doing so, we have found a remarkable splitting for Wishart processes that enables us to sample exactly Wishart distributions without any restriction on the parameters. It is related but extends existing exact simulation methods based on Bartlett.s decomposition. Moreover, we can construct high-order discretization schemes for Wishart processes and second-order schemes for general affine diffusions. These schemes are, in practice, faster than the exact simulation to sample entire paths. Numerical results on their convergence are given.