Probability measure-valued polynomial diffusions

Abstract

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming-Viot process is a particular example. The defining property of finite dimensional polynomial processes considered by Cuchiero et al. (2012); Filipović and Larsson (2016) is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

Ranked masses in two-parameter Fleming-Viot diffusions

“…A consequence of our calculations is that these Fleming-Viot processes are F -polynomial processes in the sense of [7]. We conjecture that they are polynomial processes in the sense of [4,5], but have thus far been unable to compute the generator on the polynomials considered there.…”

“…Our arguments are related to those that appear in the study of polynomial processes [3,6], which have recently drawn significant interest in mathematical finance for their balance of generality and computational tractability. Recall that a (classical) polynomial process is a Markov process on R d whose semigroup preserves, for each m, the set of polynomials of degree at most m. Recently there have also been efforts to extend the study of polynomial processes to the infinite-dimensional setting [4,5,7], where the appropriate notion of "polynomial" depends on the context. Jacobi diffusions and Wright-Fisher diffusions are two classical examples of polynomial processes, and the key step in our argument is to identify statistics of the Fleming-Viot processes constructed in [16,38] that evolve as Jacobi diffusions and Wright-Fisher diffusions.…”

Viscosity Solutions for Controlled McKean--Vlasov Jump-Diffusions

“…This derivative was used by Fleming & Viot [22] to study a martingale problem in populations dynamics. Also recently Cuchiero, Larsson & Svaluto-Ferro [15] provided several of its properties in the context of polynomial diffusions.…”

Polynomial Jump-Diffusion Models

“…Existence and uniqueness for a large class of polynomial diffusions on semialgebraic state spaces are developed by Filipović and Larsson (2016). Probability measure-valued polynomial diffusions are studied by Cuchiero et al (2019). The theoretical literature in the jump-diffusion case is less abundant.…”