# American Institute of Mathematical Sciences

September  2013, 18(7): 1929-1967. doi: 10.3934/dcdsb.2013.18.1929

## Mean-field backward stochastic Volterra integral equations

 1 Institute for Financial Studies and School of Mathematics, Shandong University, Jinan, Shandong 250100 2 Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100 3 Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received  September 2012 Revised  March 2013 Published  May 2013

Mean-field backward stochastic Volterra integral equations (MF-BSVIEs, for short) are introduced and studied. Well-posedness of MF-BSVIEs in the sense of introduced adapted M-solutions is established. Two duality principles between linear mean-field (forward) stochastic Volterra integral equations (MF-FSVIEs, for short) and MF-BSVIEs are obtained. A Pontryagin's type maximum principle is established for an optimal control of MF-FSVIEs.
Citation: Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929
##### References:
 [1] N. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Control Optim., 46 (2007), 356-378. doi: 10.1137/050645944. [2] N. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stoch. Proc. Appl., 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X. [3] A. Aman and M. N'zi, Backward stochastic nonlinear Volterra integral equation with local Lipschitz drift, Prob. Math. Stat., 25 (2005), 105-127. [4] D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356. doi: 10.1007/s00245-010-9123-8. [5] V. Anh, W. Grecksch and J. Yong, Regularity of backward stochastic Volterra integral equations in Hilbert spaces, Stoch. Anal. Appl., 29 (2011), 146-168. doi: 10.1080/07362994.2011.532046. [6] M. Berger and V. Mizel, Volterra equations with Itô integrals, I,II, J. Int. Equ., 2 (1980), 187-245, 319-337. [7] V. Borkar and K. Kumar, McKean-Vlasov limit in portfolio optimization, Stoch. Anal. Appl., 28 (2010), 884-906. doi: 10.1080/07362994.2010.482836. [8] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216. doi: 10.1007/s00245-011-9136-y. [9] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: a limit approach, Ann. Probab., 37 (2009), 1524-1565. doi: 10.1214/08-AOP442. [10] R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. Appl., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002. [11] T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab., 22 (1994), 431-441. doi: 10.1214/aop/1176988866. [12] T. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math., 20 (1994), 507-526. [13] D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575. [14] D. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), 29-85. doi: 10.1007/BF01010922. [15] D. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446. [16] J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. [17] C. Graham, McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets, Stoch. Proc. Appl., 40 (1992), 69-82. doi: 10.1016/0304-4149(92)90138-G. [18] Y. Hu and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Math. Acad. Sci. Paris, 343 (2006), 135-140. doi: 10.1016/j.crma.2006.05.019. [19] M. Huang, R. Malhamé and P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Comm. Inform. Systems, 6 (2006), 221-252. [20] M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Sympos. Math. Statist. Prob., 3 (1956), 171-197. [21] P. Kotelenez and T. Kurtz, Macroscopic limit for stochastic partial differential equations of McKean-Vlasov type, Prob. Theory Rel. Fields, 146 (2010), 189-222. doi: 10.1007/s00440-008-0188-0. [22] J. Lasry and P. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. [23] J. Lin, Adapted solution of a backward stochastic nonlinear Volterra integral equation, Stoch. Anal. Appl., 20 (2002), 165-183. doi: 10.1081/SAP-120002426. [24] N. Mahmudov and M. McKibben, On a class of backward McKean-Vlasov stochastic equations in Hilbert space: existence and convergence properties, Dynamic Systems Appl., 16 (2007), 643-664. [25] H. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA, 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907. [26] T. Meyer-Brandis, B. Oksendal and X. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666. doi: 10.1080/17442508.2011.651619. [27] J. Park, P. Balasubramaniam and Y. Kang, Controllability of McKean-Vlasov stochastic integrodifferential evolution equation in Hilbert spaces, Numer. Funct. Anal. Optim., 29 (2008), 1328-1346. doi: 10.1080/01630560802580679. [28] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [29] E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638. [30] P. Protter, Volterra equations driven by semimartingales, Ann. Prabab., 13 (1985), 519-530. doi: 10.1214/aop/1176993006. [31] Y. Ren, On solutions of backward stochastic Volterra integral equations with jumps in Hilbert spaces, J. Optim. Theory Appl., 144 (2010), 319-333. doi: 10.1007/s10957-009-9596-2. [32] M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, J. Austral. Math. Soc., Ser. A, 43 (1987), 246-256. doi: 10.1017/S1446788700029384. [33] Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations, J. Korean Math. Soc., 49 (2012), 1301-1321. doi: 10.4134/JKMS.2012.49.6.1301. [34] A. Sznitman, "Topics in Propagation of Chaos," Ecôle de Probabilites de Saint Flour, XIX-1989. Lecture Notes in Math, 1464, Springer, Berlin, 1989, 165-251. doi: 10.1007/BFb0085169. [35] T. Wang, $L^p$solutions of backward stochastic Volterra integral equations, Acta Math. Sinica, 28 (2012), 1875-1882. doi: 10.1007/s10114-012-9738-6. [36] T. Wang and Y. Shi, Symmetrical solutions of backward stochastic Volterra integral equations and applications, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 251-274. doi: 10.3934/dcdsb.2010.14.251. [37] T. Wang and Y. Shi, A class of time inconsistent risk measures and backward stochastic Volterra integral equations, Risk and Decision Analysis, 4 (2013), 17-24. [38] T. Wang and J. Yong, Comparison theorems for backward stochastic volterra integral equations, Preprint, arXiv:1208.2064. [39] Z. Wang and X. Zhang, Non-Lipschitz backward stochastic Volterra type equations with jumps, Stoch. Dyn., 7 (2007), 479-496. doi: 10.1142/S0219493707002128. [40] A. Veretennikov, "On Ergodic Measures for McKean-Vlasov Stochastic Equations," From Stochastic Calculus to Mathematical Finance, Springer, Berline, 2006, 623-633. doi: 10.1007/3-540-31186-6_29. [41] J. Yong, Backward stochastic Volterra integral equations and some related problems, Stochastic Proc. Appl., 116 (2006), 779-795. doi: 10.1016/j.spa.2006.01.005. [42] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation, Probab. Theory Relat. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6. [43] J. Yong and X. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations," Springer-Verlag, New York, 1999. [44] X. Zhang, Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, J. Funct. Anal., 258 (2010), 1361-1425. doi: 10.1016/j.jfa.2009.11.006.

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##### References:
 [1] N. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Control Optim., 46 (2007), 356-378. doi: 10.1137/050645944. [2] N. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stoch. Proc. Appl., 60 (1995), 65-85. doi: 10.1016/0304-4149(95)00050-X. [3] A. Aman and M. N'zi, Backward stochastic nonlinear Volterra integral equation with local Lipschitz drift, Prob. Math. Stat., 25 (2005), 105-127. [4] D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356. doi: 10.1007/s00245-010-9123-8. [5] V. Anh, W. Grecksch and J. Yong, Regularity of backward stochastic Volterra integral equations in Hilbert spaces, Stoch. Anal. Appl., 29 (2011), 146-168. doi: 10.1080/07362994.2011.532046. [6] M. Berger and V. Mizel, Volterra equations with Itô integrals, I,II, J. Int. Equ., 2 (1980), 187-245, 319-337. [7] V. Borkar and K. Kumar, McKean-Vlasov limit in portfolio optimization, Stoch. Anal. Appl., 28 (2010), 884-906. doi: 10.1080/07362994.2010.482836. [8] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216. doi: 10.1007/s00245-011-9136-y. [9] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: a limit approach, Ann. Probab., 37 (2009), 1524-1565. doi: 10.1214/08-AOP442. [10] R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. Appl., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002. [11] T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab., 22 (1994), 431-441. doi: 10.1214/aop/1176988866. [12] T. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math., 20 (1994), 507-526. [13] D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575. [14] D. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Statist. Phys., 31 (1983), 29-85. doi: 10.1007/BF01010922. [15] D. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308. doi: 10.1080/17442508708833446. [16] J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116. [17] C. Graham, McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets, Stoch. Proc. Appl., 40 (1992), 69-82. doi: 10.1016/0304-4149(92)90138-G. [18] Y. Hu and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Math. Acad. Sci. Paris, 343 (2006), 135-140. doi: 10.1016/j.crma.2006.05.019. [19] M. Huang, R. Malhamé and P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Comm. Inform. Systems, 6 (2006), 221-252. [20] M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Sympos. Math. Statist. Prob., 3 (1956), 171-197. [21] P. Kotelenez and T. Kurtz, Macroscopic limit for stochastic partial differential equations of McKean-Vlasov type, Prob. Theory Rel. Fields, 146 (2010), 189-222. doi: 10.1007/s00440-008-0188-0. [22] J. Lasry and P. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. [23] J. Lin, Adapted solution of a backward stochastic nonlinear Volterra integral equation, Stoch. Anal. Appl., 20 (2002), 165-183. doi: 10.1081/SAP-120002426. [24] N. Mahmudov and M. McKibben, On a class of backward McKean-Vlasov stochastic equations in Hilbert space: existence and convergence properties, Dynamic Systems Appl., 16 (2007), 643-664. [25] H. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA, 56 (1966), 1907-1911. doi: 10.1073/pnas.56.6.1907. [26] T. Meyer-Brandis, B. Oksendal and X. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666. doi: 10.1080/17442508.2011.651619. [27] J. Park, P. Balasubramaniam and Y. Kang, Controllability of McKean-Vlasov stochastic integrodifferential evolution equation in Hilbert spaces, Numer. Funct. Anal. Optim., 29 (2008), 1328-1346. doi: 10.1080/01630560802580679. [28] E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [29] E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638. [30] P. Protter, Volterra equations driven by semimartingales, Ann. Prabab., 13 (1985), 519-530. doi: 10.1214/aop/1176993006. [31] Y. Ren, On solutions of backward stochastic Volterra integral equations with jumps in Hilbert spaces, J. Optim. Theory Appl., 144 (2010), 319-333. doi: 10.1007/s10957-009-9596-2. [32] M. Scheutzow, Uniqueness and non-uniqueness of solutions of Vlasov-McKean equations, J. Austral. Math. Soc., Ser. A, 43 (1987), 246-256. doi: 10.1017/S1446788700029384. [33] Y. Shi and T. Wang, Solvability of general backward stochastic Volterra integral equations, J. Korean Math. Soc., 49 (2012), 1301-1321. doi: 10.4134/JKMS.2012.49.6.1301. [34] A. Sznitman, "Topics in Propagation of Chaos," Ecôle de Probabilites de Saint Flour, XIX-1989. Lecture Notes in Math, 1464, Springer, Berlin, 1989, 165-251. doi: 10.1007/BFb0085169. [35] T. Wang, $L^p$solutions of backward stochastic Volterra integral equations, Acta Math. Sinica, 28 (2012), 1875-1882. doi: 10.1007/s10114-012-9738-6. [36] T. Wang and Y. Shi, Symmetrical solutions of backward stochastic Volterra integral equations and applications, Discrete Contin. Dyn. Syst., Ser. B, 14 (2010), 251-274. doi: 10.3934/dcdsb.2010.14.251. [37] T. Wang and Y. Shi, A class of time inconsistent risk measures and backward stochastic Volterra integral equations, Risk and Decision Analysis, 4 (2013), 17-24. [38] T. Wang and J. Yong, Comparison theorems for backward stochastic volterra integral equations, Preprint, arXiv:1208.2064. [39] Z. Wang and X. Zhang, Non-Lipschitz backward stochastic Volterra type equations with jumps, Stoch. Dyn., 7 (2007), 479-496. doi: 10.1142/S0219493707002128. [40] A. Veretennikov, "On Ergodic Measures for McKean-Vlasov Stochastic Equations," From Stochastic Calculus to Mathematical Finance, Springer, Berline, 2006, 623-633. doi: 10.1007/3-540-31186-6_29. [41] J. Yong, Backward stochastic Volterra integral equations and some related problems, Stochastic Proc. Appl., 116 (2006), 779-795. doi: 10.1016/j.spa.2006.01.005. [42] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equation, Probab. Theory Relat. Fields, 142 (2008), 21-77. doi: 10.1007/s00440-007-0098-6. [43] J. Yong and X. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations," Springer-Verlag, New York, 1999. [44] X. Zhang, Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, J. Funct. Anal., 258 (2010), 1361-1425. doi: 10.1016/j.jfa.2009.11.006.
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