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McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets

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  • Graham, Carl

Abstract

We consider a 'nonlinear' McKean-Vlasov Ito-Skorohod SDE, and develop a L1 contraction scheme so as to get good results on the non-compensated jumps. We prove existence and uniqueness results under natural Lipschitz assumptions. We show that a wide class of nonlinear martingale problems, giving most diffusions with discrete jump sets, can be represented by SDEs satisfying our L1 assumptions, but not more classical L2 ones. We use this on a probabilistic model for a chromatographic tube. We finish by a propagation of chaos result on sample-paths.

Suggested Citation

  • Graham, Carl, 1992. "McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 69-82, February.
    • Handle: RePEc:eee:spapps:v:40:y:1992:i:1:p:69-82
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    Cited by:

    1. Tomoyuki Ichiba & Michael Ludkovski & Andrey Sarantsev, 2019. "Dynamic contagion in a banking system with births and defaults," Annals of Finance, Springer, vol. 15(4), pages 489-538, December.
    2. Bayraktar, Erhan & Wu, Ruoyu, 2021. "Mean field interaction on random graphs with dynamically changing multi-color edges," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 197-244.
    3. Graham, Carl, 2011. "Convergence of multi-class systems of fixed possibly infinite sizes," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 31-35, January.
    4. Yulin Song, 2020. "Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps," Journal of Theoretical Probability, Springer, vol. 33(1), pages 201-238, March.
    5. Tugaut, Julian, 2013. "Self-stabilizing processes in multi-wells landscape in Rd-convergence," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1780-1801.
    6. Detering, Nils & Fouque, Jean-Pierre & Ichiba, Tomoyuki, 2020. "Directed chain stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2519-2551.
    7. Benazzoli, Chiara & Campi, Luciano & Di Persio, Luca, 2020. "Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6927-6964.
    8. E. Löcherbach, 2020. "Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2280-2314, December.
    9. Jun Moon & Wonhee Kim, 2020. "Explicit Characterization of Feedback Nash Equilibria for Indefinite, Linear-Quadratic, Mean-Field-Type Stochastic Zero-Sum Differential Games with Jump-Diffusion Models," Mathematics, MDPI, vol. 8(10), pages 1-23, September.
    10. Erny, Xavier, 2022. "Well-posedness and propagation of chaos for McKean–Vlasov equations with jumps and locally Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 192-214.
    11. Benazzoli, Chiara & Campi, Luciano & Di Persio, Luca, 2019. "ε-Nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.
    12. Cecchin, Alekos & Pelino, Guglielmo, 2019. "Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4510-4555.
    13. Dai Pra, Paolo & Formentin, Marco & Pelino, Guglielmo, 2021. "A hierarchical mean field model of interacting spins," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 287-338.

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