Abstract
Recently Buckdahn et al. (Mean-field stochastic differential equations and associated PDEs, arXiv:1407.1215, 2014) studied a mean-field stochastic differential equation (SDE), whose coefficients depend on both the solution process and also its law, and whose solution process \({(X_s^{t,x,P_\xi},X_s^{t,\xi}=X_s^{t,x,P_\xi}|_{x=\xi})}\), \({s\in[t,T], (t,x)\in[0,T]\times \mathbb{R}^{d}, \xi\in L^2(\mathcal{F}_{t},\mathbb{R}^{d})}\), admits the flow property. This flow property is the key for the study of the associated nonlocal partial differential equation (PDE). In this work we extend these studies in a non-trivial manner to mean-field SDEs which, in addition to the driving Brownian motion, are governed by a compensated Poisson random measure. We show that under suitable regularity assumptions on the coefficients of the SDE, the solution \({X^{t,x,P_\xi}}\) is twice differentiable with respect to x and its law. We establish the associated nonlocal integral-PDE, and we show that \({V(t,x,P_\xi)=E[\Phi(X_T^{t,x,P_\xi},P_{X_T^{t,\xi}})]}\) is the unique classical solution \({V:[0,T]\times \mathbb{R}^{d}\times \mathcal{P}_{2}(\mathbb{R}^{d})\rightarrow\mathbb{R}}\) of this nonlocal integral-PDE with terminal condition \({\Phi}\).
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Barles G., Buckdahn R., Pardoux E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1–2), 57–83 (1997)
Bass R.F.: Stochastic differential equations with jumps. Prob. Surv. 1, 1–19 (2004)
Björk T., Kabanov Y., Runggaldier W.: Bond market structure in the presence of marked point processes. Math. Finance 7(2), 211–239 (1997)
Bossy M., Talay D.: A stochastic particle method for the McKean–Vlasov and the Burgers equation. Math. Comput. Am. Math. Soc. 66(217), 157–192 (1997)
Buckdahn R., Djehiche B., Li J., Peng S.: Mean-field backward stochastic differential equations: a limit approach. Ann. Prob. 37(4), 1524–1565 (2009)
Buckdahn R., Li J., Peng S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119(10), 3133–3154 (2009)
Buckdahn, R., Li, J., Peng, S., Rainer, C.: Mean-field stochastic differential equations and associated PDEs. arXiv:1407.1215 (2014)
Cardaliaguet, P.: Notes on Mean Field Games (from P.L. Lions’ lectures at Collège de France). https://www.ceremade.dauphine.fr/~cardalia/MFG100629
Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. arXiv:1305.7015 (2013)
Carmona, R., Delarue, F.: Forward–backward stochastic differential equations and controlled McKean Vlasov dynamics. arXiv:1303.5835 (2013)
Carmona, R., Delarue, F.: The master equation for large population equilibriums. arXiv:1404.4694 (2014)
Chan, T.: Dynamics of the McKean–Vlasov equation. Ann. Probab. 431–441 (1994)
Dawson D.A., Gärtner D.A.: Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247–308 (1987)
Hao, T., Li, J.: Backward stochastic differential equations coupled with value function and related optimal control problems. Abstr. Appl. Anal. 2014, 1–17 (2014). doi:10.1155/2014/262713
Kac, M.: Foundations of kinetic theory. In: Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, pp. 171–197 (1956)
Kloeden P.E., Lorenz T.: Stochastic differential equations with nonlocal sample dependence. Stoch. Anal. Appl. 28, 937–945 (2010)
Kotelenez P.: A class of quasilinear stochastic partial differential equations of McKean–Vlasov type with mass conservation. Probab. Theory Relat. Fields 102(2), 159–188 (1995)
Kotelenez P.M., Kurtz T.G.: Macroscopic limit for stochastic partial differential equations of McKean–Vlasov type. Theory Relat. Fields 146, 189222 (2010)
Lasry J.M., Lions P.L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)
Li J., Peng S.: Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations. Nonlinear Anal. 70, 1776–1796 (2009)
Li J., Wei Q.M.: L P estimates for fully coupled FBSDEs with jumps. Stoch. Process. Appl. 124, 1582–1611 (2014)
Lions, P.L.: Cours au Collège de France : Théorie des jeu à champs moyens. http://www.college-de-france.fr/default/EN/all/equ[1]der/audiovideo.jsp (2013)
McKean H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. 56, 1907–1911 (1966)
Overbeck, L.: Superprocesses and McKean–Vlasov equations with creation of mass (1995) (preprint)
Pra P.D., Hollander F.D.: McKean–Vlasov limit for interacting random processes in random media. J. Stat. Phys. 84(3–4), 735–772 (1996)
Sznitman A.S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56(3), 311–336 (1984)
Sznitman A.S.: Topics in propagation of chaos. Lect. Notes Math. 1464, 165–251 (1991)
Tang S.J., Li X.J.: Necessary conditions for optimal control of stochastic with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1999)
Wu Z.: Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration. J. Aust. Math. Soc. 74, 249–266 (2003)
Yong, J.M.: A linear-quadratic optimal control problem for mean-field stochastic differential equations. arXiv:1110.1564 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
The work has been supported by the NSF of People’s Republic of China (Nos. 11071144, 11171187, 11222110), Shandong Province (Nos. BS2011SF010, JQ201202), Program for New Century Excellent Talents in University (No. NCET-12-0331), 111 Project (No. B12023).
Rights and permissions
About this article
Cite this article
Hao, T., Li, J. Mean-field SDEs with jumps and nonlocal integral-PDEs. Nonlinear Differ. Equ. Appl. 23, 17 (2016). https://doi.org/10.1007/s00030-016-0366-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-016-0366-1