November  2015, 35(11): 5273-5283. doi: 10.3934/dcds.2015.35.5273

On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case

1. 

Department of Mathematics, ETH-Zentrum, HG G 54.3, CH-8092 Zürich, Switzerland

2. 

IRMAR, Université Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex, France

3. 

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France

Received  March 2013 Revised  March 2014 Published  May 2015

In F. Delbaen, Y. Hu and A. Richou (Ann. Inst. Henri Poincaré Probab. Stat. 47(2):559--574, 2011), the authors proved that uniqueness of solution to quadratic BSDE with convex generator and unbounded terminal condition holds among solutions whose exponentials are $L^p$ with $p$ bigger than a constant $\gamma$ ($p>\gamma$). In this paper, we consider the critical case: $p=\gamma$. We prove that the uniqueness holds among solutions whose exponentials are $L^\gamma$ under the additional assumption that the generator is strongly convex. These exponential moments are natural as they are given by the existence theorem.
Citation: Freddy Delbaen, Ying Hu, Adrien Richou. On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5273-5283. doi: 10.3934/dcds.2015.35.5273
References:
[1]

P. Barrieu and N. El Karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs,, Ann. Probab., 41 (2013), 1831.  doi: 10.1214/12-AOP743.  Google Scholar

[2]

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

[3]

P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations,, Stochastic Process. Appl., 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

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P. Briand and R. Elie, A simple constructive approach to quadratic BSDEs with or without delay,, Stochastic Process. Appl., 123 (2013), 2921.  doi: 10.1016/j.spa.2013.02.013.  Google Scholar

[5]

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

[6]

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

[7]

J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs., , (2013).   Google Scholar

[8]

P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative,, J. Funct. Anal., 266 (2014), 1257.  doi: 10.1016/j.jfa.2013.12.004.  Google Scholar

[9]

P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure,, , (2015).  doi: 10.1080/17442508.2015.1013959.  Google Scholar

[10]

F. Delbaen, Y. Hu and A. Richou, On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions,, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 559.  doi: 10.1214/10-AIHP372.  Google Scholar

[11]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[12]

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

[13]

Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets,, Ann. Appl. Probab., 15 (2005), 1691.  doi: 10.1214/105051605000000188.  Google Scholar

[14]

Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations,, SIAM J. Control Optim., 42 (2003), 123.  doi: 10.1137/S0363012901391330.  Google Scholar

[15]

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

[16]

M. Kohlmann and S. Tang, Multidimensional backward stochastic Riccati equations and applications,, SIAM J. Control Optim., 41 (2003), 1696.  doi: 10.1137/S0363012900378760.  Google Scholar

[17]

F. Masiero and A. Richou, A note on the existence of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition,, Electron. J. Probab., 18 (2013).   Google Scholar

[18]

M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation,, Ann. Appl. Probab., 15 (2005), 2113.  doi: 10.1214/105051605000000395.  Google Scholar

[19]

M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem,, Finance Stoch., 13 (2009), 121.  doi: 10.1007/s00780-008-0079-3.  Google Scholar

[20]

M. A. Morlais, Utility maximization in a jump market model,, Stochastics, 81 (2009), 1.  doi: 10.1080/17442500802201425.  Google Scholar

[21]

M. A. Morlais, A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem,, Stochastic Process. Appl., 120 (2010), 1966.  doi: 10.1016/j.spa.2010.05.011.  Google Scholar

[22]

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

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and Their Applications (Charlotte, (1991), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[24]

Z. Qian and X. Y. Zhou, Existence of solutions to a class of indefinite stochastic Riccati equations,, SIAM J. Control Optim., 51 (2013), 221.  doi: 10.1137/120873777.  Google Scholar

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth,, Ann. Appl. Probab., 21 (2011), 1933.  doi: 10.1214/10-AAP744.  Google Scholar

[26]

A. Richou, Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition,, Stochastic Process. Appl., 122 (2012), 3173.  doi: 10.1016/j.spa.2012.05.015.  Google Scholar

[27]

R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Math. Finance, 10 (2000), 259.  doi: 10.1111/1467-9965.00093.  Google Scholar

[28]

S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations,, SIAM J. Control Optim., 42 (2003), 53.  doi: 10.1137/S0363012901387550.  Google Scholar

[29]

R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth,, Stochastic Process. Appl., 118 (2008), 503.  doi: 10.1016/j.spa.2007.05.009.  Google Scholar

show all references

References:
[1]

P. Barrieu and N. El Karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs,, Ann. Probab., 41 (2013), 1831.  doi: 10.1214/12-AOP743.  Google Scholar

[2]

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

[3]

P. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of backward stochastic differential equations,, Stochastic Process. Appl., 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

[4]

P. Briand and R. Elie, A simple constructive approach to quadratic BSDEs with or without delay,, Stochastic Process. Appl., 123 (2013), 2921.  doi: 10.1016/j.spa.2013.02.013.  Google Scholar

[5]

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

[6]

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

[7]

J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs., , (2013).   Google Scholar

[8]

P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative,, J. Funct. Anal., 266 (2014), 1257.  doi: 10.1016/j.jfa.2013.12.004.  Google Scholar

[9]

P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure,, , (2015).  doi: 10.1080/17442508.2015.1013959.  Google Scholar

[10]

F. Delbaen, Y. Hu and A. Richou, On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions,, Ann. Inst. Henri Poincaré Probab. Stat., 47 (2011), 559.  doi: 10.1214/10-AIHP372.  Google Scholar

[11]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[12]

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

[13]

Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets,, Ann. Appl. Probab., 15 (2005), 1691.  doi: 10.1214/105051605000000188.  Google Scholar

[14]

Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations,, SIAM J. Control Optim., 42 (2003), 123.  doi: 10.1137/S0363012901391330.  Google Scholar

[15]

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

[16]

M. Kohlmann and S. Tang, Multidimensional backward stochastic Riccati equations and applications,, SIAM J. Control Optim., 41 (2003), 1696.  doi: 10.1137/S0363012900378760.  Google Scholar

[17]

F. Masiero and A. Richou, A note on the existence of solutions to Markovian superquadratic BSDEs with an unbounded terminal condition,, Electron. J. Probab., 18 (2013).   Google Scholar

[18]

M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation,, Ann. Appl. Probab., 15 (2005), 2113.  doi: 10.1214/105051605000000395.  Google Scholar

[19]

M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem,, Finance Stoch., 13 (2009), 121.  doi: 10.1007/s00780-008-0079-3.  Google Scholar

[20]

M. A. Morlais, Utility maximization in a jump market model,, Stochastics, 81 (2009), 1.  doi: 10.1080/17442500802201425.  Google Scholar

[21]

M. A. Morlais, A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem,, Stochastic Process. Appl., 120 (2010), 1966.  doi: 10.1016/j.spa.2010.05.011.  Google Scholar

[22]

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

[23]

E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations,, in Stochastic Partial Differential Equations and Their Applications (Charlotte, (1991), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[24]

Z. Qian and X. Y. Zhou, Existence of solutions to a class of indefinite stochastic Riccati equations,, SIAM J. Control Optim., 51 (2013), 221.  doi: 10.1137/120873777.  Google Scholar

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth,, Ann. Appl. Probab., 21 (2011), 1933.  doi: 10.1214/10-AAP744.  Google Scholar

[26]

A. Richou, Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition,, Stochastic Process. Appl., 122 (2012), 3173.  doi: 10.1016/j.spa.2012.05.015.  Google Scholar

[27]

R. Rouge and N. El Karoui, Pricing via utility maximization and entropy,, Math. Finance, 10 (2000), 259.  doi: 10.1111/1467-9965.00093.  Google Scholar

[28]

S. Tang, General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations,, SIAM J. Control Optim., 42 (2003), 53.  doi: 10.1137/S0363012901387550.  Google Scholar

[29]

R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth,, Stochastic Process. Appl., 118 (2008), 503.  doi: 10.1016/j.spa.2007.05.009.  Google Scholar

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