# American Institute of Mathematical Sciences

March  2021, 6(1): 61-98. doi: 10.3934/puqr.2021004

## Stochastic ordering by g-expectations

 School of Physical and Mathematical Sciences, Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371

Email: lysel001@e.ntu.edu.sg, nprivault@ntu.edu.sg

Received  October 31, 2019 Accepted  January 06, 2021 Published  March 2021

Fund Project: This research is supported by the Ministry of Education, Singapore (Grant No. MOE2018-T1-001-201)

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity, and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.

Citation: Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004
##### References:
 [1] Alvarez, O., Lasry, J.-M. and Lions, P.-L., Convex viscosity solutions and state constraints, J. Math. Pures Appl., 1997, 16(3): 265-288. [2] Arnaudon, M., Breton, J.-C. and Privault, N., Convex ordering for random vectors using predictable representation, Potential Anal., 2008, 29(4): 327-349. doi: 10.1007/s11118-008-9100-x. [3] Azagra, D., Global and fine approximation of convex functions, Proc. Lond. Math. Soc., 2013, 107(4): 799-824. doi: 10.1112/plms/pds099. [4] Belzunce, F., Riquelme, C. M. and Mulero, J., An Introduction to Stochastic Orders, Academic Press, 2015. [5] Bergenthum, J. and Rüschendorf, L., Comparison of option prices in semimartingale models, Finance and Stochastics, 2006, 10(2): 229-249. [6] Bergenthum, J. and Rüschendorf, L., Comparison of semimartingales and Lévy processes, Ann. Probab., 2007, 35(1): 228-254. [7] Bian, B. and Guan, P., Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 2008, 15(1): 39-51. [8] Bian, B. and Guan, P., A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 2009, 177: 307-335. doi: 10.1007/s00222-009-0179-5. [9] Bismut, J. M., Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 1973, 44(2): 384-404. doi: 10.1016/0022-247X(73)90066-8. [10] Briand, P., Coquet, F., Hu, Y., Memin, J. and Peng, S., A converse comparison theorem for BSDEs and related properties of g-expectation, Electron. Comm. Probab., 2000, 5: 101-117. doi: 10.1214/ECP.v5-1025. [11] Chen, Z. and Epstein, L., Ambiguity, risk, and asset returns in continuous time, Econometrica, 2002, 70(4): 1403-1443. doi: 10.1111/1468-0262.00337. [12] Chen, Z. and Peng, S., A general downcrossing inequality for g-martingales, Statist. Probab. Lett., 2000, 46(2): 169-175. doi: 10.1016/S0167-7152(99)00102-9. [13] Chen, Z., Kulperger, R. and Jiang, L., Jensen’s inequality for g-expectation: part 2, C. R. Math. Acad. Sci. Paris, 2003, 337(12): 797-800. doi: 10.1016/j.crma.2003.09.037. [14] Chen, Z., Chen, T. and Davison, M., Choquet expectation and Peng’s g-expectation, Ann. Probab., 2005, 33(3): 1179-1199. doi: 10.1214/009117904000001053. [15] Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R., Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, 2005. [16] Douglas, J., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 1996, 6(3): 940-968. doi: 10.1214/aoap/1034968235. [17] El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1-71. doi: 10.1111/1467-9965.00022. [18] El Karoui, N., Jeanblanc, M. and Shreve, S., Robustness of the Black and Scholes formula, Math. Finance, 1998, 8(2): 93-126. doi: 10.1111/1467-9965.00047. [19] Epstein, L. G. and Ji, S., Ambiguous volatility and asset pricing in continuous time, The Review of Financial Studies, 2013, 26(7): 1740-1786. doi: 10.1093/rfs/hht018. [20] Epstein, L. G. and Ji, S., Ambiguous volatility, possibility and utility in continuous time, Journal of Mathematical Economics, 2014, 50: 269-282. doi: 10.1016/j.jmateco.2013.09.005. [21] Giga, Y., Goto, S., Ishii, H. and Sato, M.-H., Comparison principle and convexity properties for singular degenerate parabolic equations on unbounded domains, Indiana University Mathematics Journal, 1991, 40(2): 443-470. doi: 10.1512/iumj.1991.40.40023. [22] Grigorova, M., Stochastic dominance with respect to a capacity and risk measures, Statistics & Risk Modeling, 2014a, 31(3-4): 259-295. [23] Grigorova, M., Stochastic orderings with respect to a capacity and an application to a financial optimization problem, Statistics & Risk Modeling, 2014b, 31(2): 183-213. [24] Gushchin, A. A. and Mordecki, E., Bounds on option prices for semimartingale market models, Proceeding of the Steklov Institute of Mathematics, 2002, 273: 73-113. [25] Jakobsen, E. R. and Karlsen, K. H., Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 2002, 183(2): 497-525. doi: 10.1006/jdeq.2001.4136. [26] Jiang, Y., Luo, P., Wang, L. and Xiong, D., Utility maximization under g*-expectation, Stochastic Analysis and Applications, 2016, 34(4): 644-661. doi: 10.1080/07362994.2016.1165121. [27] Jouini, E. and Kallal, H., Arbitrage in securities markets with short-sales constraints, Mathematical finance, 1995, 5(3): 197-232. doi: 10.1111/j.1467-9965.1995.tb00065.x. [28] Klein, Th., Ma, Y. and Privault, N., Convex concentration inequalities via forward-backward stochastic calculus. Electron. J. Probab., 2006, 11: 486-512. [29] Ladyženskaja, O. A., Solonnikov, V. A. and Ural0ceva, N., Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Provid, R.I., 1968. [30] Lepeltier, J.-P. and San Martin, J., Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., 1997, 32: 425-430. doi: 10.1016/S0167-7152(96)00103-4. [31] Levy, H., Stochastic dominance: Investment decision making under uncertainty. Springer-Verlag, 2015. [32] Lions, P.-L. and Musiela, M., Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 2006, 342: 915-921. doi: 10.1016/j.crma.2006.02.014. [33] Ma, J. and Yong, J., Forward-backward stochastic differential equations and their applications, volume 1702 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. [34] Ma, J., Protter, P. and Yong, J., Solving forward-backward stochastic differential equations explicitly - a four step scheme, Probab. Theory Relat. Fields, 1994, 98: 339-359. doi: 10.1007/BF01192258. [35] Ma, Y. T. and Privault, N., Convex concentration for additive functionals of jump stochastic differential equations, Acta Math. Sin. (Engl. Ser.), 2013, 29: 1449-1458. doi: 10.1007/s10114-013-2635-9. [36] Mishura, Y. and Shevchenko, G., Theory and Statistical Applications of Stochastic Processes, John Wiley & Sons, 2017. [37] Müller, A. and Stoyan, D., Comparison methods for stochastic models and risks, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2002. [38] Pardoux É., Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, In: Stochastic Analysis and Related Topics VI, Progress in Probability, Birkhäuser, Boston, MA, 1998: 42. https://doi.org/10.1007/978-1-4612-2022-0_2. [39] Pardoux, É. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 1990, 14(1): 55-61. [40] Pardoux, É. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci., Springer, Berlin, 1992, 176: 200-217. [41] Peng, S., Backward SDE and related g-expectation. In: Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser., Longman, Harlow, 1997, 364: 141-159. [42] Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic Methods in Finance, Lecture Notes in Math., Springer, Berlin, 2004, 1856: 165-253. [43] Peng, S., Nonlinear expectations and stochastic calculus under uncertainty, Preprint arXiv: 1002.4546v1, 2010a. [44] Peng, S., Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians, Hindustan Book Agency, New Delhi, 2010b, I: 393-432. [45] Perrakis, S., Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research. Springer, 2019. [46] Rosazza-Gianin, E., Risk measures via g-expectations, Insurance Math. Econom., 2006, 39(1): 19-34. doi: 10.1016/j.insmatheco.2006.01.002. [47] Shaked, M. and Shanthikumar, G., Stochastic orders, Springer, 2007. [48] Sriboonchita, S., Wong, W. K., Dhompongsa, S. and Nguyen, H. T., Stochastic Dominance and Applications to Finance, Risk and Economics, Chapman & Hall/CRC, 2009. [49] Tian, D. and Jiang, L., Uncertainty orders on the sublinear expectation space, Open Mathematics, 2016, 14(1): 247-259. doi: 10.1515/math-2016-0023. [50] Zhang, J., Backward stochastic differential equations, In: Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86.

show all references

##### References:
 [1] Alvarez, O., Lasry, J.-M. and Lions, P.-L., Convex viscosity solutions and state constraints, J. Math. Pures Appl., 1997, 16(3): 265-288. [2] Arnaudon, M., Breton, J.-C. and Privault, N., Convex ordering for random vectors using predictable representation, Potential Anal., 2008, 29(4): 327-349. doi: 10.1007/s11118-008-9100-x. [3] Azagra, D., Global and fine approximation of convex functions, Proc. Lond. Math. Soc., 2013, 107(4): 799-824. doi: 10.1112/plms/pds099. [4] Belzunce, F., Riquelme, C. M. and Mulero, J., An Introduction to Stochastic Orders, Academic Press, 2015. [5] Bergenthum, J. and Rüschendorf, L., Comparison of option prices in semimartingale models, Finance and Stochastics, 2006, 10(2): 229-249. [6] Bergenthum, J. and Rüschendorf, L., Comparison of semimartingales and Lévy processes, Ann. Probab., 2007, 35(1): 228-254. [7] Bian, B. and Guan, P., Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 2008, 15(1): 39-51. [8] Bian, B. and Guan, P., A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 2009, 177: 307-335. doi: 10.1007/s00222-009-0179-5. [9] Bismut, J. M., Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 1973, 44(2): 384-404. doi: 10.1016/0022-247X(73)90066-8. [10] Briand, P., Coquet, F., Hu, Y., Memin, J. and Peng, S., A converse comparison theorem for BSDEs and related properties of g-expectation, Electron. Comm. Probab., 2000, 5: 101-117. doi: 10.1214/ECP.v5-1025. [11] Chen, Z. and Epstein, L., Ambiguity, risk, and asset returns in continuous time, Econometrica, 2002, 70(4): 1403-1443. doi: 10.1111/1468-0262.00337. [12] Chen, Z. and Peng, S., A general downcrossing inequality for g-martingales, Statist. Probab. Lett., 2000, 46(2): 169-175. doi: 10.1016/S0167-7152(99)00102-9. [13] Chen, Z., Kulperger, R. and Jiang, L., Jensen’s inequality for g-expectation: part 2, C. R. Math. Acad. Sci. Paris, 2003, 337(12): 797-800. doi: 10.1016/j.crma.2003.09.037. [14] Chen, Z., Chen, T. and Davison, M., Choquet expectation and Peng’s g-expectation, Ann. Probab., 2005, 33(3): 1179-1199. doi: 10.1214/009117904000001053. [15] Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R., Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, 2005. [16] Douglas, J., Ma, J. and Protter, P., Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 1996, 6(3): 940-968. doi: 10.1214/aoap/1034968235. [17] El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1-71. doi: 10.1111/1467-9965.00022. [18] El Karoui, N., Jeanblanc, M. and Shreve, S., Robustness of the Black and Scholes formula, Math. Finance, 1998, 8(2): 93-126. doi: 10.1111/1467-9965.00047. [19] Epstein, L. G. and Ji, S., Ambiguous volatility and asset pricing in continuous time, The Review of Financial Studies, 2013, 26(7): 1740-1786. doi: 10.1093/rfs/hht018. [20] Epstein, L. G. and Ji, S., Ambiguous volatility, possibility and utility in continuous time, Journal of Mathematical Economics, 2014, 50: 269-282. doi: 10.1016/j.jmateco.2013.09.005. [21] Giga, Y., Goto, S., Ishii, H. and Sato, M.-H., Comparison principle and convexity properties for singular degenerate parabolic equations on unbounded domains, Indiana University Mathematics Journal, 1991, 40(2): 443-470. doi: 10.1512/iumj.1991.40.40023. [22] Grigorova, M., Stochastic dominance with respect to a capacity and risk measures, Statistics & Risk Modeling, 2014a, 31(3-4): 259-295. [23] Grigorova, M., Stochastic orderings with respect to a capacity and an application to a financial optimization problem, Statistics & Risk Modeling, 2014b, 31(2): 183-213. [24] Gushchin, A. A. and Mordecki, E., Bounds on option prices for semimartingale market models, Proceeding of the Steklov Institute of Mathematics, 2002, 273: 73-113. [25] Jakobsen, E. R. and Karlsen, K. H., Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 2002, 183(2): 497-525. doi: 10.1006/jdeq.2001.4136. [26] Jiang, Y., Luo, P., Wang, L. and Xiong, D., Utility maximization under g*-expectation, Stochastic Analysis and Applications, 2016, 34(4): 644-661. doi: 10.1080/07362994.2016.1165121. [27] Jouini, E. and Kallal, H., Arbitrage in securities markets with short-sales constraints, Mathematical finance, 1995, 5(3): 197-232. doi: 10.1111/j.1467-9965.1995.tb00065.x. [28] Klein, Th., Ma, Y. and Privault, N., Convex concentration inequalities via forward-backward stochastic calculus. Electron. J. Probab., 2006, 11: 486-512. [29] Ladyženskaja, O. A., Solonnikov, V. A. and Ural0ceva, N., Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Provid, R.I., 1968. [30] Lepeltier, J.-P. and San Martin, J., Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett., 1997, 32: 425-430. doi: 10.1016/S0167-7152(96)00103-4. [31] Levy, H., Stochastic dominance: Investment decision making under uncertainty. Springer-Verlag, 2015. [32] Lions, P.-L. and Musiela, M., Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 2006, 342: 915-921. doi: 10.1016/j.crma.2006.02.014. [33] Ma, J. and Yong, J., Forward-backward stochastic differential equations and their applications, volume 1702 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. [34] Ma, J., Protter, P. and Yong, J., Solving forward-backward stochastic differential equations explicitly - a four step scheme, Probab. Theory Relat. Fields, 1994, 98: 339-359. doi: 10.1007/BF01192258. [35] Ma, Y. T. and Privault, N., Convex concentration for additive functionals of jump stochastic differential equations, Acta Math. Sin. (Engl. Ser.), 2013, 29: 1449-1458. doi: 10.1007/s10114-013-2635-9. [36] Mishura, Y. and Shevchenko, G., Theory and Statistical Applications of Stochastic Processes, John Wiley & Sons, 2017. [37] Müller, A. and Stoyan, D., Comparison methods for stochastic models and risks, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2002. [38] Pardoux É., Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, In: Stochastic Analysis and Related Topics VI, Progress in Probability, Birkhäuser, Boston, MA, 1998: 42. https://doi.org/10.1007/978-1-4612-2022-0_2. [39] Pardoux, É. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 1990, 14(1): 55-61. [40] Pardoux, É. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci., Springer, Berlin, 1992, 176: 200-217. [41] Peng, S., Backward SDE and related g-expectation. In: Backward Stochastic Differential Equations, Pitman Res. Notes Math. Ser., Longman, Harlow, 1997, 364: 141-159. [42] Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures. In: Stochastic Methods in Finance, Lecture Notes in Math., Springer, Berlin, 2004, 1856: 165-253. [43] Peng, S., Nonlinear expectations and stochastic calculus under uncertainty, Preprint arXiv: 1002.4546v1, 2010a. [44] Peng, S., Backward stochastic differential equation, nonlinear expectation and their applications. In: Proceedings of the International Congress of Mathematicians, Hindustan Book Agency, New Delhi, 2010b, I: 393-432. [45] Perrakis, S., Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research. Springer, 2019. [46] Rosazza-Gianin, E., Risk measures via g-expectations, Insurance Math. Econom., 2006, 39(1): 19-34. doi: 10.1016/j.insmatheco.2006.01.002. [47] Shaked, M. and Shanthikumar, G., Stochastic orders, Springer, 2007. [48] Sriboonchita, S., Wong, W. K., Dhompongsa, S. and Nguyen, H. T., Stochastic Dominance and Applications to Finance, Risk and Economics, Chapman & Hall/CRC, 2009. [49] Tian, D. and Jiang, L., Uncertainty orders on the sublinear expectation space, Open Mathematics, 2016, 14(1): 247-259. doi: 10.1515/math-2016-0023. [50] Zhang, J., Backward stochastic differential equations, In: Probability Theory and Stochastic Modelling, Springer, New York, 2017, 86.
 [1] Sel Ly, Nicolas Privault. $G$-expectation approach to stochastic ordering. Frontiers of Mathematical Finance, , () : -. doi: 10.3934/fmf.2021012 [2] Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115 [3] Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $G$-expectation framework. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 883-901. doi: 10.3934/dcdsb.2021072 [4] Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022011 [5] Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281 [6] Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Optimal control problems of forward-backward stochastic Volterra integral equations. Mathematical Control and Related Fields, 2015, 5 (3) : 613-649. doi: 10.3934/mcrf.2015.5.613 [7] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [8] Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157 [9] Mahmoud Abouagwa, Ji Li. G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1583-1606. doi: 10.3934/dcdsb.2019241 [10] Huijie Qiao, Jiang-Lun Wu. Path independence of the additive functionals for stochastic differential equations driven by G-lévy processes. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 101-118. doi: 10.3934/puqr.2022007 [11] Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565 [12] Fabio Paronetto. Elliptic approximation of forward-backward parabolic equations. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1017-1036. doi: 10.3934/cpaa.2020047 [13] Jie Xiong, Shuaiqi Zhang, Yi Zhuang. A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance. Mathematical Control and Related Fields, 2019, 9 (2) : 257-276. doi: 10.3934/mcrf.2019013 [14] 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 and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [15] Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002 [16] Flavia Smarrazzo, Alberto Tesei. Entropy solutions of forward-backward parabolic equations with Devonshire free energy. Networks and Heterogeneous Media, 2012, 7 (4) : 941-966. doi: 10.3934/nhm.2012.7.941 [17] Ying Liu, Yabing Sun, Weidong Zhao. Explicit multistep stochastic characteristic approximation methods for forward backward stochastic differential equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 773-795. doi: 10.3934/dcdss.2021044 [18] Jiongmin Yong. Forward-backward evolution equations and applications. Mathematical Control and Related Fields, 2016, 6 (4) : 653-704. doi: 10.3934/mcrf.2016019 [19] Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013 [20] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control and Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501

Impact Factor:

Article outline

[Back to Top]