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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

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

[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.Google Scholar

[39]

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

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

[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. Google Scholar

[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. Google Scholar

[53]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[33]

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

[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. Google Scholar

[35]

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

[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. Google Scholar

[37]

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

[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.Google Scholar

[39]

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

[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. Google Scholar

[41]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

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

[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. Google Scholar

[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. Google Scholar

[53]

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

[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. Google Scholar

[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. Google Scholar

[1]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401

[2]

Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565

[3]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[4]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[5]

Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296

[6]

Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

[7]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775

[8]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017

[9]

Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297

[10]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[11]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065

[12]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075

[13]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803

[14]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002

[15]

Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011

[16]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[17]

Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619

[18]

Oliver Knill. A deterministic displacement theorem for Poisson processes. Electronic Research Announcements, 1997, 3: 110-113.

[19]

J.-M. Deshouillers, G. Effinger, H. te Riele and D. Zinoviev. A complete Vinogradov 3-primes theorem under the Riemann hypothesis. Electronic Research Announcements, 1997, 3: 99-104.

[20]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (45)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]