doi: 10.3934/fmf.2021012
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$ G $-expectation approach to stochastic ordering

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

* Corresponding author: Nicolas Privault

Received  December 2020 Revised  August 2021 Early access May 2022

This paper studies stochastic ordering under nonlinear expectations $ {\mathcal E}_{\mathcal{G}} $ generated by solutions of $ G $-Backward Stochastic Differential Equations ($ G $-BSDEs) defined on $ G $-expectation spaces. We derive sufficient conditions for the convex, increasing convex, and monotonic $ G $-stochastic orderings of $ G $-diffusion processes at terminal time. Our approach relies on comparison properties for $ G $-Forward-Backward Stochastic Differential Equations ($ G $-FBSDEs) and on relevant extensions of convexity, monotonicity and continuous dependence properties for the solutions of associated Hamilton-Jacobi-Bellman (HJB) equations. Applications of $ G $-stochastic ordering to contingent claim superhedging price comparison under ambiguous coefficients are provided.

Citation: Sel Ly, Nicolas Privault. $ G $-expectation approach to stochastic ordering. Frontiers of Mathematical Finance, doi: 10.3934/fmf.2021012
References:
[1]

O. AlvarezJ.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9), 76 (1997), 265-288.  doi: 10.1016/S0021-7824(97)89952-7.

[2]

M. ArnaudonJ.-C. Breton and N. Privault, Convex ordering for random vectors using predictable representation, Potential Anal., 29 (2008), 327-349.  doi: 10.1007/s11118-008-9100-x.

[3]

F. Belzunce, C. Martínez-Riquelme and J. Mulero, An Introduction to Stochastic Orders, Academic Press, 2016. doi: 10.1016/B978-0-12-803768-3.00001-6.

[4]

J. Bergenthum and L. Rüschendorf, Comparison of option prices in semimartingale models, Finance Stoch., 10 (2006), 222-249.  doi: 10.1007/s00780-006-0001-9.

[5]

J. Bergenthum and L. Rüschendorf, Comparison of semimartingales and Lévy processes, Ann. Probab., 35 (2007), 228-254.  doi: 10.1214/009117906000000386.

[6]

F. BiaginiJ. Mancin and T. M. Brandis, Robust mean-variance hedging via $G$-expectation, Stochastic Process. Appl., 129 (2019), 1287-1325.  doi: 10.1016/j.spa.2018.04.007.

[7]

B. Bian and P. Guan, Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 15 (2008), 39-51.  doi: 10.4310/MAA.2008.v15.n1.a5.

[8]

B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 177 (2009), 307-335.  doi: 10.1007/s00222-009-0179-5.

[9]

S. N. Cohen, S. L. Ji and S. Peng, Sublinear expectations and martingales in discrete time, preprint, arXiv: 1104.5390v1, 2011.

[10]

M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks: Measures, Orders and Models, Wiley, 2005. doi: 10.1002/0470016450.

[11]

J. Douglas Jr.J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.  doi: 10.1214/aoap/1034968235.

[12]

N. El KarouiM. Jeanblanc-Picqué and S. E. Shreve, Robustness of the Black and Scholes formula, Math. Finance, 8 (1998), 93-126.  doi: 10.1111/1467-9965.00047.

[13]

F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.

[14]

Y. GigaS. GotoH. Ishii and M.-H. Sato, Comparison principle and convexity properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470.  doi: 10.1512/iumj.1991.40.40023.

[15]

D. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[16]

M. Grigorova, Stochastic dominance with respect to a capacity and risk measures, Stat. Risk Model., 31 (2014), 259-295.  doi: 10.1515/strm-2014-1167.

[17]

M. Grigorova, Stochastic orderings with respect to a capacity and an application to a financial optimization problem, Stat. Risk Model., 31 (2014), 183-213.  doi: 10.1515/strm-2013-1151.

[18]

A. A. Gushchin and È. Mordecki, Bounds on option prices for semimartingale market models, Proc. Steklov Inst. Math., 237 (2002), 73-113. 

[19]

M. Hu and S. Ji, A note on pricing of contingent claims under G-expectation, preprint, arXiv: 1303.4274, 2013.

[20]

M. HuS. JiS. Peng and Y. Song, Backward stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 759-784.  doi: 10.1016/j.spa.2013.09.010.

[21]

M. HuS. JiS. Peng and Y. Song, Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 1170-1195.  doi: 10.1016/j.spa.2013.10.009.

[22]

M. HuS. Ji and S. Yang, A stochastic recursive optimal control problem under the $G$-expectation framework, Appl. Math. Optim., 70 (2014), 253-278.  doi: 10.1007/s00245-014-9242-8.

[23]

Z. Hu and Q. Zhou, Convergences of random variables under sublinear expectations, Chinese Ann. Math. Ser. B, 40 (2019), 39-54.  doi: 10.1007/s11401-018-0116-2.

[24]

E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 183 (2002), 497-525.  doi: 10.1006/jdeq.2001.4136.

[25]

Th. KleinY. Ma and N. Privault, Convex concentration inequalities via forward-backward stochastic calculus, Electron. J. Probab., 11 (2006), 486-512.  doi: 10.1214/EJP.v11-332.

[26]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR, Izv., 20 (1983), 459-492. 

[27]

H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, Springer, Cham, 2016. doi: 10.1007/978-3-319-21708-6.

[28]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921.  doi: 10.1016/j.crma.2006.02.014.

[29]

S. Ly and N. Privault, Stochastic orderings by $g$-expectations, Probability, Uncertainty and Quantitative Risk, 6 (2021), 61-98.  doi: 10.3934/puqr.2021004.

[30]

Y. Mishura and G. Shevchenko, Theory and Statistical Applications of Stochastic Processes, John Wiley & Sons, 2017. doi: 10.1002/9781119441601.

[31]

A. M{ü}ller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 2002.

[32]

S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, In Stochastic Methods in Finance, volume 1856 of Lecture Notes in Math., pages 165–253. Springer, Berlin, 2004. doi: 10.1007/978-3-540-44644-6_4.

[33]

S. Peng, $G$-Brownian motion and dynamic risk measure under volatility uncertainty, preprint, arXiv: 0711.2834, 2007.

[34]

S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, In Stochastic Analysis and Applications, volume 2 of Abel Symp., pages 541–567. Springer, Berlin, 2007. doi: 10.1007/978-3-540-70847-6_25.

[35]

S. Peng, A new central limit theorem under sublinear expectations, preprint, arXiv: 0803.2656, 2008.

[36]

S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty, volume 95 of Probability Theory and Stochastic Modelling, Springer, Berlin, 2019. doi: 10.1007/978-3-662-59903-7.

[37]

S. Peng, S. Yang and J. Yao, Improving Value-at-Risk prediction under model uncertainty, Journal of Financial Econometrics, 2020. arXiv: 1805.03890.. doi: 10.1093/jjfinec/nbaa022.

[38]

S. Perrakis, Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research, Springer, 2019. doi: 10.1007/978-3-030-11590-6.

[39]

M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, 2007. doi: 10.1007/978-0-387-34675-5.

[40]

Y. Song, Normal approximation by Stein's method under sublinear expectations, Stochastic Process. Appl., 130 (2020), 2838-2850.  doi: 10.1016/j.spa.2019.08.005.

[41]

Y. Song, Stein's method for law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 6 (2021), 199–212. arXiv: 1904.04674, 2019. doi: 10.3934/puqr.2021010.

[42]

S. Sriboonchita, W.-K. Wong, S. Dhompongsa and H. T. Nguyen, Stochastic Dominance and Applications to Finance, Risk and Economics, Chapman & Hall/CRC, 2009. doi: 10.1201/9781420082678.

[43]

D. Tian and L. Jiang, Uncertainty orders on the sublinear expectation space, Open Math., 14 (2016), 247-259.  doi: 10.1515/math-2016-0023.

[44]

N. Touzi, Stochastic Control Problems, Viscosity Solutions and Application to Finance, Scuola Normale Superiore, 2004.

[45]

J. Urbas, Lecture on second order linear partial differential equations, In Instructional Workshop on Analysis and Geometry, Part 1, volume 34 of Proceedings of the Centre for Mathematics and its Applications, pages 39–75, Canberra, 1996. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University.

[46]

J. Vorbrink, Financial markets with volatility uncertainty, J. Math. Econom., 53 (2014), 64-78.  doi: 10.1016/j.jmateco.2014.05.008.

[47]

Y. Xu, Backward stochastic differential equations under super linear $G$-expectation and associated Hamilton-Jacobi-Bellman equations, preprint, arXiv: 1009.1042, 2010.

[48]

D. Zhang and Z. Chen, Stability theorem for stochastic differential equations driven by $G$-Brownian motion, An. ŞtiinŞ. Univ. "Ovidius" ConstanŞa Ser. Mat., 19 (2011), 205–221.

[49]

J. Zhang, Backward Stochastic Differential Equations, volume 86 of Probability Theory and Stochastic Modelling, Springer, New York, 2017.

show all references

References:
[1]

O. AlvarezJ.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl. (9), 76 (1997), 265-288.  doi: 10.1016/S0021-7824(97)89952-7.

[2]

M. ArnaudonJ.-C. Breton and N. Privault, Convex ordering for random vectors using predictable representation, Potential Anal., 29 (2008), 327-349.  doi: 10.1007/s11118-008-9100-x.

[3]

F. Belzunce, C. Martínez-Riquelme and J. Mulero, An Introduction to Stochastic Orders, Academic Press, 2016. doi: 10.1016/B978-0-12-803768-3.00001-6.

[4]

J. Bergenthum and L. Rüschendorf, Comparison of option prices in semimartingale models, Finance Stoch., 10 (2006), 222-249.  doi: 10.1007/s00780-006-0001-9.

[5]

J. Bergenthum and L. Rüschendorf, Comparison of semimartingales and Lévy processes, Ann. Probab., 35 (2007), 228-254.  doi: 10.1214/009117906000000386.

[6]

F. BiaginiJ. Mancin and T. M. Brandis, Robust mean-variance hedging via $G$-expectation, Stochastic Process. Appl., 129 (2019), 1287-1325.  doi: 10.1016/j.spa.2018.04.007.

[7]

B. Bian and P. Guan, Convexity preserving for fully nonlinear parabolic integro-differential equations, Methods Appl. Anal., 15 (2008), 39-51.  doi: 10.4310/MAA.2008.v15.n1.a5.

[8]

B. Bian and P. Guan, A microscopic convexity principle for nonlinear partial differential equations, Invent. Math., 177 (2009), 307-335.  doi: 10.1007/s00222-009-0179-5.

[9]

S. N. Cohen, S. L. Ji and S. Peng, Sublinear expectations and martingales in discrete time, preprint, arXiv: 1104.5390v1, 2011.

[10]

M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks: Measures, Orders and Models, Wiley, 2005. doi: 10.1002/0470016450.

[11]

J. Douglas Jr.J. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968.  doi: 10.1214/aoap/1034968235.

[12]

N. El KarouiM. Jeanblanc-Picqué and S. E. Shreve, Robustness of the Black and Scholes formula, Math. Finance, 8 (1998), 93-126.  doi: 10.1111/1467-9965.00047.

[13]

F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 119 (2009), 3356-3382.  doi: 10.1016/j.spa.2009.05.010.

[14]

Y. GigaS. GotoH. Ishii and M.-H. Sato, Comparison principle and convexity properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470.  doi: 10.1512/iumj.1991.40.40023.

[15]

D. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[16]

M. Grigorova, Stochastic dominance with respect to a capacity and risk measures, Stat. Risk Model., 31 (2014), 259-295.  doi: 10.1515/strm-2014-1167.

[17]

M. Grigorova, Stochastic orderings with respect to a capacity and an application to a financial optimization problem, Stat. Risk Model., 31 (2014), 183-213.  doi: 10.1515/strm-2013-1151.

[18]

A. A. Gushchin and È. Mordecki, Bounds on option prices for semimartingale market models, Proc. Steklov Inst. Math., 237 (2002), 73-113. 

[19]

M. Hu and S. Ji, A note on pricing of contingent claims under G-expectation, preprint, arXiv: 1303.4274, 2013.

[20]

M. HuS. JiS. Peng and Y. Song, Backward stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 759-784.  doi: 10.1016/j.spa.2013.09.010.

[21]

M. HuS. JiS. Peng and Y. Song, Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 1170-1195.  doi: 10.1016/j.spa.2013.10.009.

[22]

M. HuS. Ji and S. Yang, A stochastic recursive optimal control problem under the $G$-expectation framework, Appl. Math. Optim., 70 (2014), 253-278.  doi: 10.1007/s00245-014-9242-8.

[23]

Z. Hu and Q. Zhou, Convergences of random variables under sublinear expectations, Chinese Ann. Math. Ser. B, 40 (2019), 39-54.  doi: 10.1007/s11401-018-0116-2.

[24]

E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations, J. Differential Equations, 183 (2002), 497-525.  doi: 10.1006/jdeq.2001.4136.

[25]

Th. KleinY. Ma and N. Privault, Convex concentration inequalities via forward-backward stochastic calculus, Electron. J. Probab., 11 (2006), 486-512.  doi: 10.1214/EJP.v11-332.

[26]

N. V. Krylov, Boundedly nonhomogeneous elliptic and parabolic equations, Math. USSR, Izv., 20 (1983), 459-492. 

[27]

H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, Springer, Cham, 2016. doi: 10.1007/978-3-319-21708-6.

[28]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921.  doi: 10.1016/j.crma.2006.02.014.

[29]

S. Ly and N. Privault, Stochastic orderings by $g$-expectations, Probability, Uncertainty and Quantitative Risk, 6 (2021), 61-98.  doi: 10.3934/puqr.2021004.

[30]

Y. Mishura and G. Shevchenko, Theory and Statistical Applications of Stochastic Processes, John Wiley & Sons, 2017. doi: 10.1002/9781119441601.

[31]

A. M{ü}ller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 2002.

[32]

S. Peng, Nonlinear expectations, nonlinear evaluations and risk measures, In Stochastic Methods in Finance, volume 1856 of Lecture Notes in Math., pages 165–253. Springer, Berlin, 2004. doi: 10.1007/978-3-540-44644-6_4.

[33]

S. Peng, $G$-Brownian motion and dynamic risk measure under volatility uncertainty, preprint, arXiv: 0711.2834, 2007.

[34]

S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type, In Stochastic Analysis and Applications, volume 2 of Abel Symp., pages 541–567. Springer, Berlin, 2007. doi: 10.1007/978-3-540-70847-6_25.

[35]

S. Peng, A new central limit theorem under sublinear expectations, preprint, arXiv: 0803.2656, 2008.

[36]

S. Peng, Nonlinear Expectations and Stochastic Calculus under Uncertainty, volume 95 of Probability Theory and Stochastic Modelling, Springer, Berlin, 2019. doi: 10.1007/978-3-662-59903-7.

[37]

S. Peng, S. Yang and J. Yao, Improving Value-at-Risk prediction under model uncertainty, Journal of Financial Econometrics, 2020. arXiv: 1805.03890.. doi: 10.1093/jjfinec/nbaa022.

[38]

S. Perrakis, Stochastic Dominance Option Pricing: An Alternative Approach to Option Market Research, Springer, 2019. doi: 10.1007/978-3-030-11590-6.

[39]

M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, 2007. doi: 10.1007/978-0-387-34675-5.

[40]

Y. Song, Normal approximation by Stein's method under sublinear expectations, Stochastic Process. Appl., 130 (2020), 2838-2850.  doi: 10.1016/j.spa.2019.08.005.

[41]

Y. Song, Stein's method for law of large numbers under sublinear expectations, Probab. Uncertain. Quant. Risk, 6 (2021), 199–212. arXiv: 1904.04674, 2019. doi: 10.3934/puqr.2021010.

[42]

S. Sriboonchita, W.-K. Wong, S. Dhompongsa and H. T. Nguyen, Stochastic Dominance and Applications to Finance, Risk and Economics, Chapman & Hall/CRC, 2009. doi: 10.1201/9781420082678.

[43]

D. Tian and L. Jiang, Uncertainty orders on the sublinear expectation space, Open Math., 14 (2016), 247-259.  doi: 10.1515/math-2016-0023.

[44]

N. Touzi, Stochastic Control Problems, Viscosity Solutions and Application to Finance, Scuola Normale Superiore, 2004.

[45]

J. Urbas, Lecture on second order linear partial differential equations, In Instructional Workshop on Analysis and Geometry, Part 1, volume 34 of Proceedings of the Centre for Mathematics and its Applications, pages 39–75, Canberra, 1996. Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University.

[46]

J. Vorbrink, Financial markets with volatility uncertainty, J. Math. Econom., 53 (2014), 64-78.  doi: 10.1016/j.jmateco.2014.05.008.

[47]

Y. Xu, Backward stochastic differential equations under super linear $G$-expectation and associated Hamilton-Jacobi-Bellman equations, preprint, arXiv: 1009.1042, 2010.

[48]

D. Zhang and Z. Chen, Stability theorem for stochastic differential equations driven by $G$-Brownian motion, An. ŞtiinŞ. Univ. "Ovidius" ConstanŞa Ser. Mat., 19 (2011), 205–221.

[49]

J. Zhang, Backward Stochastic Differential Equations, volume 86 of Probability Theory and Stochastic Modelling, Springer, New York, 2017.

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