Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples Read chapter Read first chapter

2019 | OriginalPaper | Chapter

On the Wellposedness of Some McKean Models with Moderated or Singular Diffusion Coefficient

Authors : Mireille Bossy, Jean-François Jabir

Published in: Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Méléard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Hölder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples
next chapter On the Uniqueness of Solutions to Quadratic BSDEs with Non-convex Generators
Appendix
Available only for authorised users
Literature
1.
Abergel, F., Tachet, R.: A nonlinear partial integro-differential equation from mathematical Finance. Discrete Continuous Dyn. Syst.-Series A (DCDS-A) 27(3), 907–917 (2010)MathSciNetCrossRef
2.
Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, 890–896 (1967)MathSciNetCrossRef
3.
Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.: Fokker-Planck-Kolmogorov Equations. American Mathematical Society (2015)
4.
Bossy, M., Jabir, J.-F., Talay, D.: On conditional McKean Lagrangian stochastic models. Probab. Theor. Relat. Fields 151(1–2), 319–351 (2011)MathSciNetCrossRef
5.
Bossy, M., Jabir, J.-F.: Lagrangian stochastic models with specular boundary condition. J. Funct. Anal. 268(6), 1309–1381 (2015)MathSciNetCrossRef
6.
Bossy, M., Espina, J., Morice, J., Paris, C., Rousseau, A.: Modeling the wind circulation around mills with a Lagrangian stochastic approach. SMAI-J. Comput. Math. 2, 177–214 (2016)MathSciNetCrossRef
7.
Bossy, M., Dupré, A., Drobinski, P., Violeau, L., Briard, C.: Stochastic Lagrangian approach for wind farm simulation (2018). hal.inria.fr
8.
Champagnat, N., Jabin, P.-E.: Strong solutions to stochastic differential equations with rough coefficients. To appear in Annals of Probability (2018)
9.
10.
Durbin, P.A., Speziale, C.G.: Realizability of second moment closure via stochastic analysis. J. Fluid Mech. 280, 395–407 (1994)CrossRef
11.
Evans, L.C.: Partial Differential Equations. American Mathematical Institute (1997)
12.
Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254(1), 109–153 (2008)MathSciNetCrossRef
13.
Funaki, T.: A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67(3), 331–348 (1984)MathSciNetCrossRef
14.
Guyon, J., Henry-Labordère, P.: The smile calibration problem solved. SSRN Electron. J. (2011)
15.
Gyöngy, I.: Mimicking the one-dimensional marginal distributions of processes having an Itô Differential. Probab. Theor. Relat. Fields 71, 501–516 (1986)CrossRef
16.
Jourdain, B., Méléard, S.: Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. H. Poincaré Probab. Statist. 34(6), 726–766 (1998)MathSciNetCrossRef
17.
Jourdain, B., Reygner, J.: Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation. Stochast. Partial Differ. Eq. Anal. Comput. 1(3), 455–506 (2013)MATH
18.
Jourdain, B., Zhou, A.: Existence of a calibrated regime switching local volatility model and new fake Brownian motions. Preprint (2016)
19.
Fournier, N., Jourdain, B.: Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes. Ann. Appl. Probab. 27(5), 2807–2861 (2017)MathSciNetCrossRef
20.
Kohatsu-Higa, A., Ogawa, S.: Weak rate of convergence for an Euler scheme of nonlinear SDE’s. Monte Carlo Meth. Appl. 3(4), 327–345 (1997)MathSciNetCrossRef
21.
Krylov, N.V.: Lecture on Elliptic and Parabolic Equations in Sobolev Spaces. American Mathematical Society (2008)
22.
Ladyženskaja, O.A., Solonnikov, V.A., Uralćeva, N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, coll. Translations of Mathematical Monographs (1968)
23.
Lions, J.-L.: Equations Différentielles Opérationnelles et Problèmes aux Limites. Grundlehren der mathematischen Wissenschaften (1961)
24.
Méléard, S.: Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In: Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics, vol. 1627, pp. 42–95 (1996)
25.
Méléard, S.: Monte-Carlo approximation for 2d Navier-Stokes equations with measure initial data. Probab. Theory Relat. Fields 121, 367–388 (2001)MathSciNetCrossRef
26.
Mishura, Y.S., Veretennikov, A.Y.: Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations. Preprint (2017)
27.
Oelschläger, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12(2), 458–497 (1984)MathSciNetCrossRef
28.
Oelschläger, K.: A law of large numbers for moderately interacting diffusion processes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 69(2), 279–322 (1985)MathSciNetCrossRef
29.
30.
Pope, S.B.: Turbulent Flows, 11th edn. Cambridge University Press (2011)
31.
Stroock, D., Varadhan, S.R.: Multidimensional Diffusion Processes. Springer-Verlag (1979)
32.
Sznitman, A.S.: A propagation of chaos result for Burgers’ equation. Probab. Theor. Relat. Fields 71(4), 581–613 (1986)MathSciNetCrossRef
33.
Sznitman, A.S.: Topics in Propagation of Chaos. In: École d’Été de Probabilités de Saint-Flour XIX-1989, pp. 165–251. In: Lecture Notes in Mathematics, 1464. Springer (1989)
34.
Vasquez, J.L.: The Porous Medium Equation. Oxford University Publications (2006)
35.
Veretennikov, A., Yu: On strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 111(3), 434–452 (1980)CrossRef
Metadata
Title
On the Wellposedness of Some McKean Models with Moderated or Singular Diffusion Coefficient
Authors
Mireille Bossy
Jean-François Jabir
Copyright Year
2019
Book
Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications

Print ISBN: 978-3-030-22284-0

Electronic ISBN: 978-3-030-22285-7

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-22285-7_2