Abstract
The paper is devoted to the well-posedness for nonlinear McKean-Vlasov type diffusions with coefficients depending on the median or, more generally, on the α-quantile of the underlying distribution. The median is not a continuous function on the space of probability measures equipped with the weak convergence. This is one reason why well-posedness of the SDE considered in the paper does not follow by standard arguments.
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Acknowledgements
In June 2012 on a conference in Bielefeld, after the author made the presentation of his theory of nonlinear Markov processes, Tom Kurtz asked him whether his methods would allow to get well-posedness for nonlinear McKean-Vlasov type diffusions with coefficients depending on the distribution via the median. The present paper is the answer to this question. The author is also grateful to Yu. Kondratiev and L. Bogachev for organizing this stimulating conference in Bielefeld, to Sigurd Assing and Astrid Hilbert for fruitful discussion and to the unknown referees for useful comments. Supported by the AFOSR grant FA9550-09-1-0664, by IPI RAN grants RFBR 11-01-12026 and 12-07-00115, and by the grant 4402 of the Ministry of Education and Science of Russia.
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Kolokoltsov, V.N. Nonlinear Diffusions and Stable-Like Processes with Coefficients Depending on the Median or VaR. Appl Math Optim 68, 85–98 (2013). https://doi.org/10.1007/s00245-013-9199-z
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DOI: https://doi.org/10.1007/s00245-013-9199-z