# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [7] J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs. arXiv:1307.5741, 2013. [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. [9] P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, arXiv:1309.6716, 2015. doi: 10.1080/17442508.2015.1013959. [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. [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. [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. [13] Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [14] Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 123-137. doi: 10.1137/S0363012901391330. [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. [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. [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. [18] M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15 (2005), 2113-2143. doi: 10.1214/105051605000000395. [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. [20] M. A. Morlais, Utility maximization in a jump market model, Stochastics, 81 (2009), 1-27. doi: 10.1080/17442500802201425. [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. [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. [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. [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. [25] A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, Ann. Appl. Probab., 21 (2011), 1933-1964. doi: 10.1214/10-AAP744. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] J. F. Chassagneux and A. Richou, Numerical simulation of quadratic BSDEs. arXiv:1307.5741, 2013. [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. [9] P. Cheridito and K. Nam, Multidimensional quadratic and subquadratic BSDEs with special structure, arXiv:1309.6716, 2015. doi: 10.1080/17442508.2015.1013959. [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. [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. [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. [13] Y. Hu, P. Imkeller and M. Muller, Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712. doi: 10.1214/105051605000000188. [14] Y. Hu and X. Y. Zhou, Indefinite stochastic Riccati equations, SIAM J. Control Optim., 42 (2003), 123-137. doi: 10.1137/S0363012901391330. [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. [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. [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. [18] M. Mania and M. Schweizer, Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15 (2005), 2113-2143. doi: 10.1214/105051605000000395. [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. [20] M. A. Morlais, Utility maximization in a jump market model, Stochastics, 81 (2009), 1-27. doi: 10.1080/17442500802201425. [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. [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. [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. [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. [25] A. Richou, Numerical simulation of BSDEs with drivers of quadratic growth, Ann. Appl. Probab., 21 (2011), 1933-1964. doi: 10.1214/10-AAP744. [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. [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. [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. [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.
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