We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.
References
[1] David Baños et al., The bismut–elworthy–li formula for mean-field stochastic differential equations, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 54 (2018), no. 1, 220–233.[1] David Baños et al., The bismut–elworthy–li formula for mean-field stochastic differential equations, Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 54 (2018), no. 1, 220–233.
[2] David Baños, Thilo Meyer-Brandis, Frank Proske, and Sindre Duedahl, Computing Deltas Without Derivatives, Finance and Stochastics 21 (2017), no. 2, 509–549.[2] David Baños, Thilo Meyer-Brandis, Frank Proske, and Sindre Duedahl, Computing Deltas Without Derivatives, Finance and Stochastics 21 (2017), no. 2, 509–549.
[4] Rainer Buckdahn, Boualem Djehiche, Juan Li, and Shige Peng, Mean-field backward stochastic differential equations: a limit approach, The Annals of Probability 37 (2009), no. 4, 1524–1565. 1176.60042 10.1214/08-AOP442 euclid.aop/1248182147[4] Rainer Buckdahn, Boualem Djehiche, Juan Li, and Shige Peng, Mean-field backward stochastic differential equations: a limit approach, The Annals of Probability 37 (2009), no. 4, 1524–1565. 1176.60042 10.1214/08-AOP442 euclid.aop/1248182147
[5] Rainer Buckdahn, Juan Li, and Shige Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications 119 (2009), no. 10, 3133–3154. 1183.60022 10.1016/j.spa.2009.05.002[5] Rainer Buckdahn, Juan Li, and Shige Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Processes and their Applications 119 (2009), no. 10, 3133–3154. 1183.60022 10.1016/j.spa.2009.05.002
[6] Rainer Buckdahn, Juan Li, Shige Peng, and Catherine Rainer, Mean-field stochastic differential equations and associated PDEs, The Annals of Probability 45 (2017), no. 2, 824–878. 06797081 10.1214/15-AOP1076 euclid.aop/1490947309[6] Rainer Buckdahn, Juan Li, Shige Peng, and Catherine Rainer, Mean-field stochastic differential equations and associated PDEs, The Annals of Probability 45 (2017), no. 2, 824–878. 06797081 10.1214/15-AOP1076 euclid.aop/1490947309
[7] Pierre Cardaliaguet, Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France, Available on the website of Collège de France ( https://www.ceremade.dauphine.fr/~cardalia/MFG100629.pdf), 2013. MR3616319[7] Pierre Cardaliaguet, Notes on Mean Field Games (from P.-L. Lions’ lectures at Collège de France, Available on the website of Collège de France ( https://www.ceremade.dauphine.fr/~cardalia/MFG100629.pdf), 2013. MR3616319
[9] René Carmona and François Delarue, The master equation for large population equilibriums, Stochastic Analysis and Applications 2014, Springer, 2014, pp. 77–128.[9] René Carmona and François Delarue, The master equation for large population equilibriums, Stochastic Analysis and Applications 2014, Springer, 2014, pp. 77–128.
[10] René Carmona and François Delarue, Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics, The Annals of Probability 43 (2015), no. 5, 2647–2700. MR3395471 10.1214/14-AOP946 euclid.aop/1441792295[10] René Carmona and François Delarue, Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics, The Annals of Probability 43 (2015), no. 5, 2647–2700. MR3395471 10.1214/14-AOP946 euclid.aop/1441792295
[11] René Carmona, François Delarue, and Aimé Lachapelle, Control of McKean–Vlasov dynamics versus mean field games, Mathematics and Financial Economics (2013), 1–36.[11] René Carmona, François Delarue, and Aimé Lachapelle, Control of McKean–Vlasov dynamics versus mean field games, Mathematics and Financial Economics (2013), 1–36.
[12] René Carmona, Jean-Pierre Fouque, Seyyed Mostafa Mousavi, and Li-Hsien Sun, Systemic risk and stochastic games with delay, Journal of Optimization Theory and Applications (2016), 1–34. 06985467 10.1007/s10957-018-1267-8[12] René Carmona, Jean-Pierre Fouque, Seyyed Mostafa Mousavi, and Li-Hsien Sun, Systemic risk and stochastic games with delay, Journal of Optimization Theory and Applications (2016), 1–34. 06985467 10.1007/s10957-018-1267-8
[13] Rene Carmona, Jean-Pierre Fouque, and Li-Hsien Sun, Mean field games and systemic risk, Communications in Mathematical Sciences 13 (2015), no. 4, 911–933. 1337.91031 10.4310/CMS.2015.v13.n4.a4[13] Rene Carmona, Jean-Pierre Fouque, and Li-Hsien Sun, Mean field games and systemic risk, Communications in Mathematical Sciences 13 (2015), no. 4, 911–933. 1337.91031 10.4310/CMS.2015.v13.n4.a4
[14] René Carmona and Daniel Lacker, A probabilistic weak formulation of mean field games and applications, The Annals of Applied Probability 25 (2015), no. 3, 1189–1231. 1332.60100 10.1214/14-AAP1020 euclid.aoap/1427124127[14] René Carmona and Daniel Lacker, A probabilistic weak formulation of mean field games and applications, The Annals of Applied Probability 25 (2015), no. 3, 1189–1231. 1332.60100 10.1214/14-AAP1020 euclid.aoap/1427124127
[15] A. S. Cherny, On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations, Theory of Probability & Its Applications 46 (2002), no. 3, 406–419.[15] A. S. Cherny, On the Uniqueness in Law and the Pathwise Uniqueness for Stochastic Differential Equations, Theory of Probability & Its Applications 46 (2002), no. 3, 406–419.
[17] Giulia Di Nunno, Bernt Øksendal, and Frank Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, first ed., Universitext, Springer-Verlag Berlin Heidelberg, 2009.[17] Giulia Di Nunno, Bernt Øksendal, and Frank Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, first ed., Universitext, Springer-Verlag Berlin Heidelberg, 2009.
[18] Nathalie Eisenbaum, Integration with respect to local time, Potential Analysis 13 (2000), no. 4, 303–328. MR1804175 10.1023/A:1026440719120[18] Nathalie Eisenbaum, Integration with respect to local time, Potential Analysis 13 (2000), no. 4, 303–328. MR1804175 10.1023/A:1026440719120
[19] Jean-Pierre Fouque and Tomoyuki Ichiba, Stability in a model of inter-bank lending, SIAM Journal on Financial Mathematics 4 (2013), 784–803. 1295.91099 10.1137/110841096[19] Jean-Pierre Fouque and Tomoyuki Ichiba, Stability in a model of inter-bank lending, SIAM Journal on Financial Mathematics 4 (2013), 784–803. 1295.91099 10.1137/110841096
[21] Josselin Garnier, George Papanicolaou, and Tzu-Wei Yang, Large deviations for a mean field model of systemic risk, SIAM Journal on Financial Mathematics 4 (2013), no. 1, 151–184. 1283.60044 10.1137/12087387X[21] Josselin Garnier, George Papanicolaou, and Tzu-Wei Yang, Large deviations for a mean field model of systemic risk, SIAM Journal on Financial Mathematics 4 (2013), no. 1, 151–184. 1283.60044 10.1137/12087387X
[22] G Stephen Jones, Fundamental inequalities for discrete and discontinuous functional equations, Journal of the Society for Industrial and Applied Mathematics 12 (1964), no. 1, 43–57. 0154.05702 10.1137/0112004[22] G Stephen Jones, Fundamental inequalities for discrete and discontinuous functional equations, Journal of the Society for Industrial and Applied Mathematics 12 (1964), no. 1, 43–57. 0154.05702 10.1137/0112004
[23] Benjamin Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations, ESAIM: Probability and Statistics 1 (1997), 339–355. 0929.60062 10.1051/ps:1997113[23] Benjamin Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations, ESAIM: Probability and Statistics 1 (1997), 339–355. 0929.60062 10.1051/ps:1997113
[24] Benjamin Jourdain, Sylvie Méléard, and Wojbor A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA, Latin American Journal of Probability (2008), 1–29.[24] Benjamin Jourdain, Sylvie Méléard, and Wojbor A. Woyczynski, Nonlinear SDEs driven by Lévy processes and related PDEs, ALEA, Latin American Journal of Probability (2008), 1–29.
[25] M. Kac, Foundations of Kinetic Theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics (Berkeley, Calif.), University of California Press, 1956, pp. 171–197. MR0084985[25] M. Kac, Foundations of Kinetic Theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics (Berkeley, Calif.), University of California Press, 1956, pp. 171–197. MR0084985
[26] Gopinath Kallianpur and Jie Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monograph Series 26 (1995), iii–342.[26] Gopinath Kallianpur and Jie Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces, Lecture Notes-Monograph Series 26 (1995), iii–342.
[27] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer New York, 1991. 0734.60060[27] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer New York, 1991. 0734.60060
[28] Oliver Kley, Claudia Klüppelberg, and Lukas Reichel, Systemic risk through contagion in a core-periphery structured banking network, Banach Center Publications 104 (2015), no. 1, 133–149.[28] Oliver Kley, Claudia Klüppelberg, and Lukas Reichel, Systemic risk through contagion in a core-periphery structured banking network, Banach Center Publications 104 (2015), no. 1, 133–149.
[30] Giovanni Leoni, A first course in Sobolev spaces, vol. 105, American Mathematical Society Providence, RI, 2009. 1180.46001[30] Giovanni Leoni, A first course in Sobolev spaces, vol. 105, American Mathematical Society Providence, RI, 2009. 1180.46001
[31] Juan Li and Hui Min, Weak Solutions of Mean-Field Stochastic Differential Equations and Application to Zero-Sum Stochastic Differential Games, SIAM Journal on Control and Optimization 54 (2016), no. 3, 1826–1858. MR3517577 1346.60085 10.1137/15M1015583[31] Juan Li and Hui Min, Weak Solutions of Mean-Field Stochastic Differential Equations and Application to Zero-Sum Stochastic Differential Games, SIAM Journal on Control and Optimization 54 (2016), no. 3, 1826–1858. MR3517577 1346.60085 10.1137/15M1015583
[32] Xuerong Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Processes and their Applications 58 (1995), no. 2, 281–292. 0835.60049 10.1016/0304-4149(95)00024-2[32] Xuerong Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Processes and their Applications 58 (1995), no. 2, 281–292. 0835.60049 10.1016/0304-4149(95)00024-2
[33] H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc Natl Acad Sci U S A 56 (1966), no. 6, 1907–1911. 0149.13501 10.1073/pnas.56.6.1907[33] H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc Natl Acad Sci U S A 56 (1966), no. 6, 1907–1911. 0149.13501 10.1073/pnas.56.6.1907
[34] Olivier Menoukeu-Pamen, Thilo Meyer-Brandis, Torstein Nilssen, Frank Proske, and Tusheng Zhang, A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s, Mathematische Annalen 357 (2013), no. 2, 761–799. 1282.60057 10.1007/s00208-013-0916-3[34] Olivier Menoukeu-Pamen, Thilo Meyer-Brandis, Torstein Nilssen, Frank Proske, and Tusheng Zhang, A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s, Mathematische Annalen 357 (2013), no. 2, 761–799. 1282.60057 10.1007/s00208-013-0916-3
[35] Thilo Meyer-Brandis and Frank Proske, On the existence and explicit representability of strong solutions of Lévy noise driven SDE’s with irregular coefficients, Commun. Math. Sci. 4 (2006), no. 1, 129–154.[35] Thilo Meyer-Brandis and Frank Proske, On the existence and explicit representability of strong solutions of Lévy noise driven SDE’s with irregular coefficients, Commun. Math. Sci. 4 (2006), no. 1, 129–154.
[36] Thilo Meyer-Brandis and Frank Proske, Construction of strong solutions of SDE’s via Malliavin calculus, Journal of Functional Analysis 258 (2010), no. 11, 3922–3953. 1195.60082 10.1016/j.jfa.2009.11.010[36] Thilo Meyer-Brandis and Frank Proske, Construction of strong solutions of SDE’s via Malliavin calculus, Journal of Functional Analysis 258 (2010), no. 11, 3922–3953. 1195.60082 10.1016/j.jfa.2009.11.010
[37] Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, arXiv:1603.02212, March 2016. 1603.02212[37] Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, arXiv:1603.02212, March 2016. 1603.02212
[40] W.P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, Springer New York, 1989. 0692.46022[40] W.P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, Springer New York, 1989. 0692.46022
