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June  2021, 41(6): 2543-2557. doi: 10.3934/dcds.2020374

Comparison theorem for diagonally quadratic BSDEs

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

Received  May 2019 Revised  September 2020 Published  June 2021 Early access  November 2020

Fund Project: Financial support from the Natural Sciences and Engineering Research Council of Canada, Grant RGPIN- 2017-04054.

The present work is devoted to study comparison and converse comparison theorems for diagonally quadratic BSDEs. We give sufficient and necessary conditions under which the comparison holds. Sufficient and necessary conditions for non-positive and non-negative solutions are presented.

Citation: Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
References:
[1]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.  Google Scholar

[2]

P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618.  doi: 10.1007/s00440-006-0497-0.  Google Scholar

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P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567.  doi: 10.1007/s00440-007-0093-y.  Google Scholar

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R. BuckdahnM. Quincampoix and A. Răşcanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab. Theory Related Fields, 116 (2000), 485-504.  doi: 10.1007/s004400050260.  Google Scholar

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C. Frei, Splitting multidimensional BSDEs and finding local equilibria, Stochastic Process. Appl., 124 (2014), 2654-2671.  doi: 10.1016/j.spa.2014.03.004.  Google Scholar

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C. Frei and G. dos Reis, A financial market with interacting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182.  doi: 10.1007/s11579-011-0039-0.  Google Scholar

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Y. Hu and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I, 343 (2006), 135-140.  doi: 10.1016/j.crma.2006.05.019.  Google Scholar

[8]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121.  doi: 10.1007/s00440-009-0202-1.  Google Scholar

[9]

Y. Hu and S. Tang, Multi-dimensional backward stochastic differential equations of diagonally quadratic generators, Stochastic Process. Appl., 126 (2016), 1066-1086.  doi: 10.1016/j.spa.2015.10.011.  Google Scholar

[10]

G. Jia and N. Zhang, Quadratic $g$-convexity, $C$-convexity and their relationships, Stochastic Process. Appl., 125 (2015), 2272-2294.  doi: 10.1016/j.spa.2014.12.012.  Google Scholar

[11]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.  Google Scholar

[12]

J. Ma and S. Yao, On quadratic $g$-Evaluations/Expectations and related analysis, Stoch. Anal. Appl., 28 (2010), 711-734.  doi: 10.1080/07362994.2010.482827.  Google Scholar

[13]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[14]

Y. Xu, Multidimensional dynamic risk measure via conditional $g$-expectation, Math. Finance., 26 (2016), 638-673.  doi: 10.1111/mafi.12062.  Google Scholar

show all references

References:
[1]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404.  doi: 10.1016/0022-247X(73)90066-8.  Google Scholar

[2]

P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields, 136 (2006), 604-618.  doi: 10.1007/s00440-006-0497-0.  Google Scholar

[3]

P. Briand and Y. Hu, Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Related Fields, 141 (2008), 543-567.  doi: 10.1007/s00440-007-0093-y.  Google Scholar

[4]

R. BuckdahnM. Quincampoix and A. Răşcanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab. Theory Related Fields, 116 (2000), 485-504.  doi: 10.1007/s004400050260.  Google Scholar

[5]

C. Frei, Splitting multidimensional BSDEs and finding local equilibria, Stochastic Process. Appl., 124 (2014), 2654-2671.  doi: 10.1016/j.spa.2014.03.004.  Google Scholar

[6]

C. Frei and G. dos Reis, A financial market with interacting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182.  doi: 10.1007/s11579-011-0039-0.  Google Scholar

[7]

Y. Hu and S. Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I, 343 (2006), 135-140.  doi: 10.1016/j.crma.2006.05.019.  Google Scholar

[8]

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147 (2010), 89-121.  doi: 10.1007/s00440-009-0202-1.  Google Scholar

[9]

Y. Hu and S. Tang, Multi-dimensional backward stochastic differential equations of diagonally quadratic generators, Stochastic Process. Appl., 126 (2016), 1066-1086.  doi: 10.1016/j.spa.2015.10.011.  Google Scholar

[10]

G. Jia and N. Zhang, Quadratic $g$-convexity, $C$-convexity and their relationships, Stochastic Process. Appl., 125 (2015), 2272-2294.  doi: 10.1016/j.spa.2014.12.012.  Google Scholar

[11]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.  doi: 10.1214/aop/1019160253.  Google Scholar

[12]

J. Ma and S. Yao, On quadratic $g$-Evaluations/Expectations and related analysis, Stoch. Anal. Appl., 28 (2010), 711-734.  doi: 10.1080/07362994.2010.482827.  Google Scholar

[13]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, System Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[14]

Y. Xu, Multidimensional dynamic risk measure via conditional $g$-expectation, Math. Finance., 26 (2016), 638-673.  doi: 10.1111/mafi.12062.  Google Scholar

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