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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 |
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, ().
|
[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. |
show all references
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, ().
|
[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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