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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

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