Abstracts

Polynomial preserving processes are defined as time-homogeneous Markov jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. These processes are attractive for financial modeling because of their tractability and robustness. In this work we study approximations of polynomial preserving processes with finite-state Markov processes via a moment-matching methodology. This approximation aims to exploit the defining property of polynomial preserving processes in order to reduce the complexity of the implementation of such models. More precisely, we study sufficient conditions for the existence of finite-state Markov processes that match the moments of a given polynomial preserving process. We first construct discrete time finite-state Markov processes that match moments of arbitrary order. This discrete time construction relies on the existence of long-run moments for the polynomial process and cubature methods over these moments. In the second part we give a characterization theorem for the existence of a continuous time finite-state Markov process that matches the moments of a given polynomial preserving process. This theorem illustrates the complexity of the problem in continuous time by combining algebraic and geometric considerations. We show the impossibility of constructing in general such a process for polynomial preserving diffusions, for high order moments and for sufficiently many points in the state space. We provide however a positive result by showing that the construction is possible when one considers finite-state Markov chains on lifted versions of the state space.