October  2019, 39(10): 5571-5601. doi: 10.3934/dcds.2019245

Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure

Higher Institute of Applied Sciences and Technologies of Gabès & LR17ES11, Tunisia

Received  March 2018 Revised  April 2019 Published  July 2019

We consider backward doubly stochastic differential equations (BDSDEs in short) driven by a Brownian motion and an independent Poisson random measure. We give sufficient conditions for the existence and the uniqueness of solutions of equations with Lipschitz generator which is, first, standard and then depends on the values of a solution in the past. We also prove comparison theorem for reflected BDSDEs.

Citation: Monia Karouf. Reflected solutions of backward doubly SDEs driven by Brownian motion and Poisson random measure. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5571-5601. doi: 10.3934/dcds.2019245
References:
[1]

K. BahlaliS. Hamadène and B. Mezerdi, BSDEs with two reflecting barriers and continuous with quadratic growth coefficient, Stoch. Proc. Appl., 115 (2005), 1107-1129.  doi: 10.1016/j.spa.2005.02.005.

[2]

K. BahlaliM. HassaniB. Mansouri and N. Mrhardy, One barrier reflected backward doubly stochastic differential equations with continuous generator, C. R. Math., 347 (2009), 1201-1206.  doi: 10.1016/j.crma.2009.08.001.

[3]

V. Bally and A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, Theor. Probab., 14 (2001), 125-164.  doi: 10.1023/A:1007825232513.

[4]

G. BarlesR. Buchdahn and E. Pardoux, BSDE's and integral-partial differential equations, Stoch. Stoch. Rep., 60 (1997), 57-83.  doi: 10.1080/17442509708834099.

[5]

E. Bayraktar and S. Yao, Doubly reflected BSDEs with integrable parameters and related Dynkin games, Stoch. Proc. Appl., 125 (2015), 4489-4542.  doi: 10.1016/j.spa.2015.07.007.

[6]

R. Buckdahn and J. Li, Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 381-420.  doi: 10.1007/s00030-009-0022-0.

[7]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations Ⅰ, Stoch. Proc. Appl., 93 (2001), 181-204.  doi: 10.1016/S0304-4149(00)00093-4.

[8]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations Ⅱ, Stoch. Proc. Appl., 93 (2001), 205-228.  doi: 10.1016/S0304-4149(00)00092-2.

[9]

J. Cvitanic and I. Karatzas, Backward SDEs with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056.  doi: 10.1214/aop/1041903216.

[10]

J. Cvitanic and J. Ma, Reflected backward-forward SDEs and obstacle problems with boundary conditions, Appl. Math. Stoch. Anal., 14 (2001), 113-138.  doi: 10.1155/S1048953301000090.

[11]

C. Dellacherie, Capacités et Processus Stochastiques, Springer-Verlag, Berlin-New York, 1972.

[12]

C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Chap. Ⅰ à Ⅳ, Hermann, Paris, 1975.

[13]

C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Chap.Ⅴ-Ⅷ, Hermann, Paris, 1980.

[14]

L. Delong, Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management, Appl. Math., 39 (2012), 463-488.  doi: 10.4064/am39-4-5.

[15]

L. Delong and P. Imkeller, Backward stochastic differential equations with time delayed generator - Results and counterexamples, Ann. Appl. Probab., 20 (2010), 1512-1536.  doi: 10.1214/09-AAP663.

[16]

L. Delong and P. Imkeller, On Malliavin's differentiability of time delayed BSDEs driven by Brownian motions and Poisson random measures, Stoch. Proc. Appl., 120 (2010), 1748-1775.  doi: 10.1016/j.spa.2010.05.001.

[17]

G. Dos ReisA. Reveillac and J. Zhang, FBSDE with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity, Stoch. Proc. Appl., 121 (2011), 2114-2150.  doi: 10.1016/j.spa.2011.05.002.

[18]

B. El AsriS. Hamadène and H. Wang, Lp-solutions for doubly reflected backward stochastic differential equations, Stoch. Anal. Appl., 29 (2011), 907-932.  doi: 10.1080/07362994.2011.564442.

[19]

N. El-KarouiC. KapoudjianE. PardouxS. 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.

[20]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[21]

H. Essaky, Reflected backward stochastic differential equation with jumps and rcll obstacle, Bull. Sci. Math., 132 (2008), 690-710.  doi: 10.1016/j.bulsci.2008.03.005.

[22]

P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.

[23]

S. Hamadène, BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stoch. Proc. Appl., 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0.

[24]

S. Hamadène and M. Hassani, BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game, EJP, 11 (2006), 121-145.  doi: 10.1214/EJP.v11-303.

[25]

S. HamadèneM. Hassani and Y. Ouknine, BSDEs with two general discontinuous reflecting barriers without Mokobodski's condition, Bull. Sci. Math., 134 (2010), 874-899.  doi: 10.1016/j.bulsci.2010.03.001.

[26]

S. Hamadène and I. Hdhiri, Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator, Appl. Math. Stoch. Anal., 2006 (2006). Article ID 95818, 28 pages. doi: 10.1155/JAMSA/2006/95818.

[27]

S. Hamadène and J.-P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stoc. Stoc. Reports, 54 (1995), 221-231.  doi: 10.1080/17442509508834006.

[28]

S. Hamadène and J.-P. Lepeltier, Reflected BSDEs and mixed game problem, Stoch. Proc. Appl., 85 (2000), 177-188.  doi: 10.1016/S0304-4149(99)00072-1.

[29]

S. Hamadène, J.-P. Lepeltier and A. Matoussi, Double barrier reflected BSDEs with continuous coefficient, in N. El Karoui, L. Mazliak (Eds), Pitman Research Notes Math. series, 364 (1997), 161–175.

[30]

S. Hamadène and Y. Ouknine, Backward stochastic differential equations with jumps and random obstacle, EJP, 8 (2003), 1-20.  doi: 10.1214/EJP.v8-124.

[31]

S. Hamadène and Y. Ouknine, Reflected backward SDEs with general jumps, Teor. Veroyatnost. i Primenen., 60 (2016), 263-280.  doi: 10.1137/S0040585X97T987648.

[32]

S. Hamadène, E. Rotenstein and A. Zặlinescu, A generalized mixed zero-sum stochastic differential game and double barrier reflected BSDEs with quadratic growth coefficient, Analelle Stiintifice ale Universitatii Alexandru Ioan Cuza din Iasi, Seria nova Mathematica (ISI), Tomul LV, f.2, 55 (2009), 419–444.

[33]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Math. Library 24, North-Holland, Amsterdam; Kodansha, Tokyo, 1981.

[34]

I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculs, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.

[35]

M. Karouf, Reflected BSDE's with discontinuous barrier and time delayed generators, ESAIM: PS, 19 (2015), 194-203.  doi: 10.1051/ps/2014021.

[36]

M. Karouf, Reflected and doubly reflected backward stochastic differential equations with time-delayed generators, J. Theor. Probab., 32 (2019), 216-248.  doi: 10.1007/s10959-018-0829-x.

[37]

W. Lu, Y. Ren and L. Hu, Multivalued backward doubly stochastic differential equations with time delayed generators, J. Mathematics, (2013), 14 pages.

[38]

J. Luo, Y. Zhang and Zhi. Li, Backward doubly stochastic differential equations with time delayed generators, Beijing: Sciencepaper, 2013, Available from: http://www.paper.edu.cn/releasepaper/content/201301-26.

[39]

B. Mansouri, I. Salhi and L. Tamer, Reflected backward doubly stochastic differential equations with time delayed generators, preprint, arXiv: 1703.10532.

[40]

X. Mao, Adapted solution of backward stochastic differential equations with non-Lipschitz coefficients, Stoch. Proc. Appl., 58 (1995), 281-292.  doi: 10.1016/0304-4149(95)00024-2.

[41]

E. Pardoux, Stochastic partial differential equations, Fudan Lecture Notes, (2007), 87 pages.

[42]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[43]

E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theor. Related Fields, 98 (1994), 209-227.  doi: 10.1007/BF01192514.

[44]

S. Peng and Y. Shi, A type of time-symmetric forward-backward stochastic differential equations, C. R. Math. Acad. Sci., Paris 336 (2003), 773–778. doi: 10.1016/S1631-073X(03)00183-3.

[45]

J. T. Shi, Optimal control of backward stochastic differential equations with time delayed generators, in Pro. 30th Chinese Control Conference, Yantai, P. R. China, (2011), 1285–1289.

[46]

J. T. Shi, Optimal control of BSDEs with time delayed generators driven by Brownian motion and Poisson random measures, in Pro. 32th Chinese Control Conference, Xi'an, P. R. China, (2013), 1575–1580.

[47]

J. T. Shi and G. Wang, A non-zero sum differential game of BSDE with time delayed generator and applications, in IEEE Transactions on Automatic Control, (2015), 1959–1964. doi: 10.1109/TAC.2015.2480335.

[48]

A. B. Sow, Backward doubly stochastic differential equations driven by Lévy process: The case of non-Lipschitz coefficients, J. Numer. Math. Stoch., 3 (2011), 71-79. 

[49]

A. B. Sow, BSDE with jumps and non-Lipschitz coefficients: Application to large deviations, Braz. J. Probab. Stat., 28 (2014), 96-108.  doi: 10.1214/12-BJPS197.

[50]

X. Sun and Y. Lu, The property for solutions of the multi-dimensional BDSDEs, Chinese J. Appl. Probab. Stat., 24 (2008), 73-82. 

[51]

S. Tang and X. Li, Necessary condition for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.

[52]

Y. Wang and Z. Huang, Backward stochastic differential equations with non Lipschitz coefficients equations, Stat. Probab. Lett., 79 (2009), 1438-1443.  doi: 10.1016/j.spl.2009.03.003.

[53]

F. Xi-liang and R. Yong, Reflected backward doubly stochastic differential equation with jumps, Mathematica Applicata, 22 (2009), 778-784. 

[54]

Q. Zhou and Y. Ren, Reflected backward stochastic differential equations with time delayed generators, Stat. Probab. Lett., 82 (2012), 979-990.  doi: 10.1016/j.spl.2012.02.012.

[55]

B. Zhu and B. Han, Comparison theorems for the multidimensional BDSDEs and applications, J. Appl. Math., 2012 (2012), Art. ID 304781, 14 pp. doi: 10.1155/2012/304781.

show all references

References:
[1]

K. BahlaliS. Hamadène and B. Mezerdi, BSDEs with two reflecting barriers and continuous with quadratic growth coefficient, Stoch. Proc. Appl., 115 (2005), 1107-1129.  doi: 10.1016/j.spa.2005.02.005.

[2]

K. BahlaliM. HassaniB. Mansouri and N. Mrhardy, One barrier reflected backward doubly stochastic differential equations with continuous generator, C. R. Math., 347 (2009), 1201-1206.  doi: 10.1016/j.crma.2009.08.001.

[3]

V. Bally and A. Matoussi, Weak solutions for SPDEs and backward doubly stochastic differential equations, Theor. Probab., 14 (2001), 125-164.  doi: 10.1023/A:1007825232513.

[4]

G. BarlesR. Buchdahn and E. Pardoux, BSDE's and integral-partial differential equations, Stoch. Stoch. Rep., 60 (1997), 57-83.  doi: 10.1080/17442509708834099.

[5]

E. Bayraktar and S. Yao, Doubly reflected BSDEs with integrable parameters and related Dynkin games, Stoch. Proc. Appl., 125 (2015), 4489-4542.  doi: 10.1016/j.spa.2015.07.007.

[6]

R. Buckdahn and J. Li, Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers, NoDEA Nonlinear Differential Equations Appl., 16 (2009), 381-420.  doi: 10.1007/s00030-009-0022-0.

[7]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations Ⅰ, Stoch. Proc. Appl., 93 (2001), 181-204.  doi: 10.1016/S0304-4149(00)00093-4.

[8]

R. Buckdahn and J. Ma, Stochastic viscosity solutions for nonlinear stochastic partial differential equations Ⅱ, Stoch. Proc. Appl., 93 (2001), 205-228.  doi: 10.1016/S0304-4149(00)00092-2.

[9]

J. Cvitanic and I. Karatzas, Backward SDEs with reflection and Dynkin games, Ann. Probab., 24 (1996), 2024-2056.  doi: 10.1214/aop/1041903216.

[10]

J. Cvitanic and J. Ma, Reflected backward-forward SDEs and obstacle problems with boundary conditions, Appl. Math. Stoch. Anal., 14 (2001), 113-138.  doi: 10.1155/S1048953301000090.

[11]

C. Dellacherie, Capacités et Processus Stochastiques, Springer-Verlag, Berlin-New York, 1972.

[12]

C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Chap. Ⅰ à Ⅳ, Hermann, Paris, 1975.

[13]

C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Chap.Ⅴ-Ⅷ, Hermann, Paris, 1980.

[14]

L. Delong, Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management, Appl. Math., 39 (2012), 463-488.  doi: 10.4064/am39-4-5.

[15]

L. Delong and P. Imkeller, Backward stochastic differential equations with time delayed generator - Results and counterexamples, Ann. Appl. Probab., 20 (2010), 1512-1536.  doi: 10.1214/09-AAP663.

[16]

L. Delong and P. Imkeller, On Malliavin's differentiability of time delayed BSDEs driven by Brownian motions and Poisson random measures, Stoch. Proc. Appl., 120 (2010), 1748-1775.  doi: 10.1016/j.spa.2010.05.001.

[17]

G. Dos ReisA. Reveillac and J. Zhang, FBSDE with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity, Stoch. Proc. Appl., 121 (2011), 2114-2150.  doi: 10.1016/j.spa.2011.05.002.

[18]

B. El AsriS. Hamadène and H. Wang, Lp-solutions for doubly reflected backward stochastic differential equations, Stoch. Anal. Appl., 29 (2011), 907-932.  doi: 10.1080/07362994.2011.564442.

[19]

N. El-KarouiC. KapoudjianE. PardouxS. 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.

[20]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[21]

H. Essaky, Reflected backward stochastic differential equation with jumps and rcll obstacle, Bull. Sci. Math., 132 (2008), 690-710.  doi: 10.1016/j.bulsci.2008.03.005.

[22]

P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.

[23]

S. Hamadène, BSDEs and risk sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations, Stoch. Proc. Appl., 107 (2003), 145-169.  doi: 10.1016/S0304-4149(03)00059-0.

[24]

S. Hamadène and M. Hassani, BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game, EJP, 11 (2006), 121-145.  doi: 10.1214/EJP.v11-303.

[25]

S. HamadèneM. Hassani and Y. Ouknine, BSDEs with two general discontinuous reflecting barriers without Mokobodski's condition, Bull. Sci. Math., 134 (2010), 874-899.  doi: 10.1016/j.bulsci.2010.03.001.

[26]

S. Hamadène and I. Hdhiri, Backward stochastic differential equations with two distinct reflecting barriers and quadratic growth generator, Appl. Math. Stoch. Anal., 2006 (2006). Article ID 95818, 28 pages. doi: 10.1155/JAMSA/2006/95818.

[27]

S. Hamadène and J.-P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stoc. Stoc. Reports, 54 (1995), 221-231.  doi: 10.1080/17442509508834006.

[28]

S. Hamadène and J.-P. Lepeltier, Reflected BSDEs and mixed game problem, Stoch. Proc. Appl., 85 (2000), 177-188.  doi: 10.1016/S0304-4149(99)00072-1.

[29]

S. Hamadène, J.-P. Lepeltier and A. Matoussi, Double barrier reflected BSDEs with continuous coefficient, in N. El Karoui, L. Mazliak (Eds), Pitman Research Notes Math. series, 364 (1997), 161–175.

[30]

S. Hamadène and Y. Ouknine, Backward stochastic differential equations with jumps and random obstacle, EJP, 8 (2003), 1-20.  doi: 10.1214/EJP.v8-124.

[31]

S. Hamadène and Y. Ouknine, Reflected backward SDEs with general jumps, Teor. Veroyatnost. i Primenen., 60 (2016), 263-280.  doi: 10.1137/S0040585X97T987648.

[32]

S. Hamadène, E. Rotenstein and A. Zặlinescu, A generalized mixed zero-sum stochastic differential game and double barrier reflected BSDEs with quadratic growth coefficient, Analelle Stiintifice ale Universitatii Alexandru Ioan Cuza din Iasi, Seria nova Mathematica (ISI), Tomul LV, f.2, 55 (2009), 419–444.

[33]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Math. Library 24, North-Holland, Amsterdam; Kodansha, Tokyo, 1981.

[34]

I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculs, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0302-2_2.

[35]

M. Karouf, Reflected BSDE's with discontinuous barrier and time delayed generators, ESAIM: PS, 19 (2015), 194-203.  doi: 10.1051/ps/2014021.

[36]

M. Karouf, Reflected and doubly reflected backward stochastic differential equations with time-delayed generators, J. Theor. Probab., 32 (2019), 216-248.  doi: 10.1007/s10959-018-0829-x.

[37]

W. Lu, Y. Ren and L. Hu, Multivalued backward doubly stochastic differential equations with time delayed generators, J. Mathematics, (2013), 14 pages.

[38]

J. Luo, Y. Zhang and Zhi. Li, Backward doubly stochastic differential equations with time delayed generators, Beijing: Sciencepaper, 2013, Available from: http://www.paper.edu.cn/releasepaper/content/201301-26.

[39]

B. Mansouri, I. Salhi and L. Tamer, Reflected backward doubly stochastic differential equations with time delayed generators, preprint, arXiv: 1703.10532.

[40]

X. Mao, Adapted solution of backward stochastic differential equations with non-Lipschitz coefficients, Stoch. Proc. Appl., 58 (1995), 281-292.  doi: 10.1016/0304-4149(95)00024-2.

[41]

E. Pardoux, Stochastic partial differential equations, Fudan Lecture Notes, (2007), 87 pages.

[42]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61.  doi: 10.1016/0167-6911(90)90082-6.

[43]

E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theor. Related Fields, 98 (1994), 209-227.  doi: 10.1007/BF01192514.

[44]

S. Peng and Y. Shi, A type of time-symmetric forward-backward stochastic differential equations, C. R. Math. Acad. Sci., Paris 336 (2003), 773–778. doi: 10.1016/S1631-073X(03)00183-3.

[45]

J. T. Shi, Optimal control of backward stochastic differential equations with time delayed generators, in Pro. 30th Chinese Control Conference, Yantai, P. R. China, (2011), 1285–1289.

[46]

J. T. Shi, Optimal control of BSDEs with time delayed generators driven by Brownian motion and Poisson random measures, in Pro. 32th Chinese Control Conference, Xi'an, P. R. China, (2013), 1575–1580.

[47]

J. T. Shi and G. Wang, A non-zero sum differential game of BSDE with time delayed generator and applications, in IEEE Transactions on Automatic Control, (2015), 1959–1964. doi: 10.1109/TAC.2015.2480335.

[48]

A. B. Sow, Backward doubly stochastic differential equations driven by Lévy process: The case of non-Lipschitz coefficients, J. Numer. Math. Stoch., 3 (2011), 71-79. 

[49]

A. B. Sow, BSDE with jumps and non-Lipschitz coefficients: Application to large deviations, Braz. J. Probab. Stat., 28 (2014), 96-108.  doi: 10.1214/12-BJPS197.

[50]

X. Sun and Y. Lu, The property for solutions of the multi-dimensional BDSDEs, Chinese J. Appl. Probab. Stat., 24 (2008), 73-82. 

[51]

S. Tang and X. Li, Necessary condition for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.

[52]

Y. Wang and Z. Huang, Backward stochastic differential equations with non Lipschitz coefficients equations, Stat. Probab. Lett., 79 (2009), 1438-1443.  doi: 10.1016/j.spl.2009.03.003.

[53]

F. Xi-liang and R. Yong, Reflected backward doubly stochastic differential equation with jumps, Mathematica Applicata, 22 (2009), 778-784. 

[54]

Q. Zhou and Y. Ren, Reflected backward stochastic differential equations with time delayed generators, Stat. Probab. Lett., 82 (2012), 979-990.  doi: 10.1016/j.spl.2012.02.012.

[55]

B. Zhu and B. Han, Comparison theorems for the multidimensional BDSDEs and applications, J. Appl. Math., 2012 (2012), Art. ID 304781, 14 pp. doi: 10.1155/2012/304781.

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