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, 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), 1831-1863. doi: 10.1214/12-AOP743.  Google Scholar

[2]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404. 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), 109-129. 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), 2921-2939. 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), 604-618. 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), 543-567. doi: 10.1007/s00440-007-0093-y.  Google Scholar

[7]

J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs. arXiv:1307.5741, 2013. Google Scholar

[8]

P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative, J. Funct. Anal., 266 (2014), 1257-1285. doi: 10.1016/j.jfa.2013.12.004.  Google Scholar

[9]

P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, arXiv:1309.6716, 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), 559-574. 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), 1-71. 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), 161-182. 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), 1691-1712. doi: 10.1214/105051605000000188.  Google Scholar

[14]

Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 123-137. doi: 10.1137/S0363012901391330.  Google Scholar

[15]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. 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), 1696-1721. 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), 15pp.  Google Scholar

[18]

M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15 (2005), 2113-2143. 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), 121-150. doi: 10.1007/s00780-008-0079-3.  Google Scholar

[20]

M. A. Morlais, Utility maximization in a jump market model, Stochastics, 81 (2009), 1-27. 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), 1966-1995. 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), 55-61. 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, NC, 1991) (eds. B. L. Rozovskii and R. B. Sowers), Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992, 200-217. 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), 221-229. doi: 10.1137/120873777.  Google Scholar

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, Ann. Appl. Probab., 21 (2011), 1933-1964. 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), 3173-3208. 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), 259-276. 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), 53-75. doi: 10.1137/S0363012901387550.  Google Scholar

[29]

R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth, Stochastic Process. Appl., 118 (2008), 503-515. 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), 1831-1863. doi: 10.1214/12-AOP743.  Google Scholar

[2]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404. 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), 109-129. 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), 2921-2939. 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), 604-618. 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), 543-567. doi: 10.1007/s00440-007-0093-y.  Google Scholar

[7]

J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs. arXiv:1307.5741, 2013. Google Scholar

[8]

P. Cheridito and K. Nam, BSDEs with terminal conditions that have bounded Malliavin derivative, J. Funct. Anal., 266 (2014), 1257-1285. doi: 10.1016/j.jfa.2013.12.004.  Google Scholar

[9]

P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, arXiv:1309.6716, 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), 559-574. 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), 1-71. 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), 161-182. 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), 1691-1712. doi: 10.1214/105051605000000188.  Google Scholar

[14]

Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 123-137. doi: 10.1137/S0363012901391330.  Google Scholar

[15]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602. 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), 1696-1721. 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), 15pp.  Google Scholar

[18]

M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15 (2005), 2113-2143. 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), 121-150. doi: 10.1007/s00780-008-0079-3.  Google Scholar

[20]

M. A. Morlais, Utility maximization in a jump market model, Stochastics, 81 (2009), 1-27. 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), 1966-1995. 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), 55-61. 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, NC, 1991) (eds. B. L. Rozovskii and R. B. Sowers), Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992, 200-217. 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), 221-229. doi: 10.1137/120873777.  Google Scholar

[25]

A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, Ann. Appl. Probab., 21 (2011), 1933-1964. 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), 3173-3208. 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), 259-276. 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), 53-75. doi: 10.1137/S0363012901387550.  Google Scholar

[29]

R. Tevzadze, Solvability of backward stochastic differential equations with quadratic growth, Stochastic Process. Appl., 118 (2008), 503-515. doi: 10.1016/j.spa.2007.05.009.  Google Scholar

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