Mean-field backward stochastic Volterra integral equations

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

Mean-field backward stochastic Volterra integral equations

Yufeng Shi ^1, , Tianxiao Wang ^2, and Jiongmin Yong ^3,

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

Abstract
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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.

Keywords: Mean-field stochastic Volterra integral equation, maximum principle., mean-field backward stochastic Volterra integral equation, duality principle.

Mathematics Subject Classification: 60H20, 93E20, 35Q8.

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.

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, 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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