Abstract polynomial processes

We propose a novel approach to polynomial processes, which allows us to analyse such processes on general state spaces. We do not need to specify polynomials explicitly but can work with a general sequence of graded vector spaces of functions on the state space. Elements of these graded vector spaces form the polynomials. By introducing a sequence of vector space complements, we obtain the sets of monomials. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of polynomials acting on a polynomial process to polynomials of the same or lower grade.

We study abstract polynomial processes under algebraic and topological assumptions on the polynomial actions, and establish an affine drift structure. Moreover, we characterize the covariance structure under similar but slightly stronger conditions. A crucial part in our analysis is the use of the (algebraic or topological) dual of the monomials of grade one, which serves as a linearization of the state space of the polynomial process. We provide several examples of polynomial processes that do not fall into the classical setting but are polynomial processes according to our definition and can be analyzed with the tools we provide here.