Weak Existence And Uniqueness For Mckean–Vlasov Sdes With Common Noise

William R.P. Hammersley*, David Šiška, Łukasz Szpruch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

This paper concerns the McKean–Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its role in connecting weak solutions to McKean–Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise’s diffusion coefficient is nondegenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge–Kantorovich problem and representation of weak solutions via Girsanov transformations.

Original languageEnglish
Pages (from-to)527-555
Number of pages29
JournalAnnals of Probability
Volume49
Issue number2
Early online date17 Mar 2021
DOIs
Publication statusPublished - 31 Mar 2021

Keywords / Materials (for Non-textual outputs)

  • Girsanov transformations
  • mean-field equations
  • Stochastic McKean–Vlasov equations

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