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Weak Existence and Uniqueness for McKean-Vlasov SDEs with Common Noise | | William R.P. Hammersley
; David Šiška
; Łukasz Szpruch
; | | Date: |
2 Aug 2019 | | Abstract: | This paper concerns the McKean-Vlasov stochastic differential equation (SDE)
with common noise, a distribution dependent SDE with a conditional
non-linearity. Such equations describe the limiting behaviour of a
representative particle in a mean-field interacting system driven by correlated
noises as the particle number tends to infinity. 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
r^ole 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 for the
approximation process, 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 noise’s diffusion coefficient is non-degenerate
and the drift is regular in the total variation distance. This seems sharp when
one considers finite-dimensional noise. The proof relies on a suitably tailored
cost function in the Monge-Kantorovich problem and extends a remarkable
technique based on Girsanov transformations previously employed in the case of
uncorrelated noises to the common noise setting. | | Source: | arXiv, 1908.0955 | | Services: | Forum | Review | PDF | Favorites | |
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