American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

A. Aman and M. N'zi, Backward stochastic nonlinear Volterra integral equation with local Lipschitz drift, Prob. Math. Stat., 25 (2005), 105-127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Berger and V. Mizel, Volterra equations with Itô integrals, I,II, J. Int. Equ., 2 (1980), 187-245, 319-337.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab., 22 (1994), 431-441. doi: 10.1214/aop/1176988866.  Google Scholar

[12]

T. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math., 20 (1994), 507-526.  Google Scholar

[13]

D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Sympos. Math. Statist. Prob., 3 (1956), 171-197.  Google Scholar

[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.  Google Scholar

[22]

J. Lasry and P. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638.  Google Scholar

[30]

P. Protter, Volterra equations driven by semimartingales, Ann. Prabab., 13 (1985), 519-530. doi: 10.1214/aop/1176993006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[38]

T. Wang and J. Yong, Comparison theorems for backward stochastic volterra integral equations,, Preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. Yong and X. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations," Springer-Verlag, New York, 1999.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Aman and M. N'zi, Backward stochastic nonlinear Volterra integral equation with local Lipschitz drift, Prob. Math. Stat., 25 (2005), 105-127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Berger and V. Mizel, Volterra equations with Itô integrals, I,II, J. Int. Equ., 2 (1980), 187-245, 319-337.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab., 22 (1994), 431-441. doi: 10.1214/aop/1176988866.  Google Scholar

[12]

T. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow J. Math., 20 (1994), 507-526.  Google Scholar

[13]

D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68. doi: 10.1080/17442500902723575.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Gärtner, On the Mckean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248. doi: 10.1002/mana.19881370116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Kac, Foundations of kinetic theory, Proc. 3rd Berkeley Sympos. Math. Statist. Prob., 3 (1956), 171-197.  Google Scholar

[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.  Google Scholar

[22]

J. Lasry and P. Lions, Mean field games, Japan J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

E. Pardoux and P. Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab., 18 (1990), 1635-1655. doi: 10.1214/aop/1176990638.  Google Scholar

[30]

P. Protter, Volterra equations driven by semimartingales, Ann. Prabab., 13 (1985), 519-530. doi: 10.1214/aop/1176993006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[38]

T. Wang and J. Yong, Comparison theorems for backward stochastic volterra integral equations,, Preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. Yong and X. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations," Springer-Verlag, New York, 1999.  Google Scholar

[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.  Google Scholar

[1]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[2]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[3]

Ludger Overbeck, Jasmin A. L. Röder. Path-dependent backward stochastic Volterra integral equations with jumps, differentiability and duality principle. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 4-. doi: 10.1186/s41546-018-0030-2
[4]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501
[5]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026
[6]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[7]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[8]

Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287
[9]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
[10]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[11]

Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3
[12]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432
[13]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006
[14]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[15]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[16]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
[17]

Tianxiao Wang, Yufeng Shi. Symmetrical solutions of backward stochastic Volterra integral equations and their applications. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 251-274. doi: 10.3934/dcdsb.2010.14.251
[18]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control & Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613
[19]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381
[20]

Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences