Abstract
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustrate the abstract results, we consider large particle systems with mean-field interaction and common noise.
Highlights
In this paper we study existence of probability measure valued jump-diffusions, whose dynamics is specified by means of a martingale problem
Processes taking values in spaces of probability measures play an important role in a number of applied contexts
In this paper we develop a simple device for embedding Pp(Rd), and other similar spaces, into compact spaces where the classical existence theory of martingale problems can be applied
Summary
In this paper we study existence of probability measure valued jump-diffusions, whose dynamics is specified by means of a martingale problem. L satisfies the positive maximum principle on X if and only if there exists a possibly killed solution to the martingale problem for (L, D, X ) for every initial condition μ ∈ X. This allows us to speak about possibly killed solutions to the martingale problem. Since L satisfies the positive maximum principle on X , Theorem 2.4 yields a possibly killed solution to the martingale problem for (L, D, X ) for any initial condition ν ∈ X. Xt ∈ Pw for all t ≥ 0, as claimed
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