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Null controllability of the Lotka-McKendrick system with spatial diffusion
Quadratic BSDEs with mean reflection
1. | Institut de Recherche Mathématique de Rennes, Université Rennes 1, 35042 Rennes Cedex, France |
2. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
3. | School of Mathematical Sciences, Shanghai Jiao Tong University, 200240 Shanghai, China |
4. | Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France |
5. | Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland |
6. | Zhongtai Securities Institute for Financial Studies and Institute for Advanced Research, Shandong University, Jinan 250100, China |
The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the $z$ argument. This work is the sequel of [
References:
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S. Ankirchner, P. Imkeller and G. dos Reis,
Classical and variational differentiability of BSDEs with quadratic growth, Electron. J. Probab, 12 (2007), 1418-1453.
doi: 10.1214/EJP.v12-462. |
[2] |
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. |
[3] |
B. Bouchard, R. Elie and A. Réveillac,
BSDEs with weak terminal condition, Ann. Probab., 43 (2015), 572-604.
doi: 10.1214/14-AOP913. |
[4] |
P. Briand, P. E. Chaudru de Raynal, A. Guillin and C. Labart, Particles systems and numerical schemes for mean reflected stochastic differential equations, preprint, arXiv: 1612.06886 |
[5] |
P. Briand and F. Confortola,
BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838.
doi: 10.1016/j.spa.2007.06.006. |
[6] |
P. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, Ann. Appl. Probab., 28 (2018), 482–510, arXiv: 1605.06301
doi: 10.1214/17-AAP1310. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
R. Buckdahn and Y. Hu,
Pricing of American contingent claims with jump stock price and constrained portfolios, Math. Oper. Res., 23 (1998), 177-203.
doi: 10.1287/moor.23.1.177. |
[11] |
R. Buckdahn and Y. Hu,
Hedging contingent claims for a large investor in an incomplete market, Adv. in Appl. Probab., 30 (1998), 239-255.
doi: 10.1239/aap/1035228002. |
[12] |
J. F. Chassagneux, R. Elie and I. Kharroubi,
A note on existence and uniqueness for solutions of multidimensional reflected BSDEs, Electron. Commun. Probab., 16 (2011), 120-128.
doi: 10.1214/ECP.v16-1614. |
[13] |
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. |
[14] |
P. Cheridito and K. Nam,
Multidimensional quadratic and subquadratic BSDEs with special structure, Stochastics, 87 (2015), 871-884.
doi: 10.1080/17442508.2015.1013959. |
[15] |
J. Cvitanić and I. Karatzas,
Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056.
doi: 10.1214/aop/1041903216. |
[16] |
J. Cvitanić, I. Karatzas and H. M. Soner,
Backward stochastic differential equations with constraints on the gains-process, Ann. Probab., 26 (1998), 1522-1551.
doi: 10.1214/aop/1022855872. |
[17] |
N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez,
Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737.
doi: 10.1214/aop/1024404416. |
[18] |
N. El Karoui, E. Pardoux and M. C. Quenez, Reflected backward SDEs and American options,
in Numerical Methods in Finance (eds. L. C. G. Rogers and D. Talay), Cambridge Univ. Press,
13 (1997), 215–231. |
[19] |
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. |
[20] |
C. Frei and G. dos Reis,
A financial market with interatcting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182.
doi: 10.1007/s11579-011-0039-0. |
[21] |
S. Hamadene and M. Jeanblanc,
On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192.
doi: 10.1287/moor.1060.0228. |
[22] |
S. Hamadene and J. Zhang,
Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl., 120 (2010), 403-426.
doi: 10.1016/j.spa.2010.01.003. |
[23] |
J. Harter and A. Richou, A stability approach for solving multidimensional quadratic BSDEs, preprint, arXiv: 1606.08627 |
[24] |
Y. Hu, P. Imkeller and M. Müller,
Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712.
doi: 10.1214/105051605000000188. |
[25] |
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. |
[26] |
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. |
[27] |
N. Kazamaki,
Continuous Exponential Martingales and BMO, Springer-Verlag, Berlin, 1994.
doi: 10.1007/BFb0073585. |
[28] |
M. Kobylanski,
Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.
doi: 10.1214/aop/1019160253. |
[29] |
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. |
[30] |
D. Nualart,
The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer, Berlin, 2006.
doi: 10.1007/3-540-28329-3. |
[31] |
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. |
[32] |
S. Peng and M. Xu,
Reflected BSDE with a constraint and its applications in an incomplete market, Bernoulli, 16 (2010), 614-640.
doi: 10.3150/09-BEJ227. |
[33] |
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. |
[34] |
H. Xing and G. Zitkovic, A class of globally solvable Markovian quadratic BSDE systems and
applications, Ann. Probab., 46 (2018), 491–550, arXiv: 1603.00217
doi: 10.1214/17-AOP1190. |
show all references
References:
[1] |
S. Ankirchner, P. Imkeller and G. dos Reis,
Classical and variational differentiability of BSDEs with quadratic growth, Electron. J. Probab, 12 (2007), 1418-1453.
doi: 10.1214/EJP.v12-462. |
[2] |
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. |
[3] |
B. Bouchard, R. Elie and A. Réveillac,
BSDEs with weak terminal condition, Ann. Probab., 43 (2015), 572-604.
doi: 10.1214/14-AOP913. |
[4] |
P. Briand, P. E. Chaudru de Raynal, A. Guillin and C. Labart, Particles systems and numerical schemes for mean reflected stochastic differential equations, preprint, arXiv: 1612.06886 |
[5] |
P. Briand and F. Confortola,
BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 118 (2008), 818-838.
doi: 10.1016/j.spa.2007.06.006. |
[6] |
P. Briand, R. Elie and Y. Hu, BSDEs with mean reflection, Ann. Appl. Probab., 28 (2018), 482–510, arXiv: 1605.06301
doi: 10.1214/17-AAP1310. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
R. Buckdahn and Y. Hu,
Pricing of American contingent claims with jump stock price and constrained portfolios, Math. Oper. Res., 23 (1998), 177-203.
doi: 10.1287/moor.23.1.177. |
[11] |
R. Buckdahn and Y. Hu,
Hedging contingent claims for a large investor in an incomplete market, Adv. in Appl. Probab., 30 (1998), 239-255.
doi: 10.1239/aap/1035228002. |
[12] |
J. F. Chassagneux, R. Elie and I. Kharroubi,
A note on existence and uniqueness for solutions of multidimensional reflected BSDEs, Electron. Commun. Probab., 16 (2011), 120-128.
doi: 10.1214/ECP.v16-1614. |
[13] |
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. |
[14] |
P. Cheridito and K. Nam,
Multidimensional quadratic and subquadratic BSDEs with special structure, Stochastics, 87 (2015), 871-884.
doi: 10.1080/17442508.2015.1013959. |
[15] |
J. Cvitanić and I. Karatzas,
Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056.
doi: 10.1214/aop/1041903216. |
[16] |
J. Cvitanić, I. Karatzas and H. M. Soner,
Backward stochastic differential equations with constraints on the gains-process, Ann. Probab., 26 (1998), 1522-1551.
doi: 10.1214/aop/1022855872. |
[17] |
N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez,
Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737.
doi: 10.1214/aop/1024404416. |
[18] |
N. El Karoui, E. Pardoux and M. C. Quenez, Reflected backward SDEs and American options,
in Numerical Methods in Finance (eds. L. C. G. Rogers and D. Talay), Cambridge Univ. Press,
13 (1997), 215–231. |
[19] |
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. |
[20] |
C. Frei and G. dos Reis,
A financial market with interatcting investors: Does an equilibrium exist?, Math. Financ. Econ., 4 (2011), 161-182.
doi: 10.1007/s11579-011-0039-0. |
[21] |
S. Hamadene and M. Jeanblanc,
On the starting and stopping problem: Application in reversible investments, Math. Oper. Res., 32 (2007), 182-192.
doi: 10.1287/moor.1060.0228. |
[22] |
S. Hamadene and J. Zhang,
Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl., 120 (2010), 403-426.
doi: 10.1016/j.spa.2010.01.003. |
[23] |
J. Harter and A. Richou, A stability approach for solving multidimensional quadratic BSDEs, preprint, arXiv: 1606.08627 |
[24] |
Y. Hu, P. Imkeller and M. Müller,
Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), 1691-1712.
doi: 10.1214/105051605000000188. |
[25] |
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. |
[26] |
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. |
[27] |
N. Kazamaki,
Continuous Exponential Martingales and BMO, Springer-Verlag, Berlin, 1994.
doi: 10.1007/BFb0073585. |
[28] |
M. Kobylanski,
Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28 (2000), 558-602.
doi: 10.1214/aop/1019160253. |
[29] |
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. |
[30] |
D. Nualart,
The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer, Berlin, 2006.
doi: 10.1007/3-540-28329-3. |
[31] |
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. |
[32] |
S. Peng and M. Xu,
Reflected BSDE with a constraint and its applications in an incomplete market, Bernoulli, 16 (2010), 614-640.
doi: 10.3150/09-BEJ227. |
[33] |
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. |
[34] |
H. Xing and G. Zitkovic, A class of globally solvable Markovian quadratic BSDE systems and
applications, Ann. Probab., 46 (2018), 491–550, arXiv: 1603.00217
doi: 10.1214/17-AOP1190. |
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