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.

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