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Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables
Stochastic ordering by g-expectations
School of Physical and Mathematical Sciences, Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 |
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.
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. Google Scholar |
[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. Google Scholar |
[5] |
Bergenthum, J. and Rüschendorf, L., Comparison of option prices in semimartingale models, Finance and Stochastics, 2006, 10(2): 229-249. Google Scholar |
[6] |
Bergenthum, J. and Rüschendorf, L., Comparison of semimartingales and Lévy processes, Ann. Probab., 2007, 35(1): 228-254. Google Scholar |
[7] |
Bian, B. and Guan, P., Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 2008, 15(1): 39-51. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[39] |
Pardoux, É. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 1990, 14(1): 55-61. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[43] |
Peng, S., Nonlinear expectations and stochastic calculus under uncertainty, Preprint arXiv: 1002.4546v1, 2010a. Google Scholar |
[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. Google Scholar |
[45] |
Perrakis, S., Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research. Springer, 2019. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[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. Google Scholar |
[5] |
Bergenthum, J. and Rüschendorf, L., Comparison of option prices in semimartingale models, Finance and Stochastics, 2006, 10(2): 229-249. Google Scholar |
[6] |
Bergenthum, J. and Rüschendorf, L., Comparison of semimartingales and Lévy processes, Ann. Probab., 2007, 35(1): 228-254. Google Scholar |
[7] |
Bian, B. and Guan, P., Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 2008, 15(1): 39-51. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[39] |
Pardoux, É. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems & Control Letters, 1990, 14(1): 55-61. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[43] |
Peng, S., Nonlinear expectations and stochastic calculus under uncertainty, Preprint arXiv: 1002.4546v1, 2010a. Google Scholar |
[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. Google Scholar |
[45] |
Perrakis, S., Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research. Springer, 2019. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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