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Backward doubly SDEs with continuous and stochastic linear growth coefficients

  • Jean Marc Owo ORCID logo EMAIL logo
Published/Copyright: August 7, 2018

Abstract

We study backward doubly stochastic differential equations when the coefficients are continuous with stochastic linear growth. Via an approximation and comparison theorem, the existence of minimal and maximal solutions are obtained.

MSC 2010: 60H10; 60H20

Communicated by Mykola Portenko


References

[1] A. Aman and J. M. Owo, Generalized backward doubly stochastic differential equations driven by Lévy processes with continuous coefficients, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 10, 2011–2020. 10.1007/s10114-012-0506-4Search in Google Scholar

[2] A. Aman and J. M. Owo, Reflected backward doubly stochastic differential equations with discontinuous generator, Random Oper. Stoch. Equ. 20 (2012), no. 2, 119–134. 10.1515/rose-2012-0005Search in Google Scholar

[3] K. Bahlali, A. Elouaflin and M. N’zi, Backward stochastic differential equations with stochastic monotone coefficients, J. Appl. Math. Stoch. Anal. (2004), no. 4, 317–335. 10.1155/S1048953304310038Search in Google Scholar

[4] K. Bahlali, A. Elouaflin and M. N’zi, RBSDEs with stochastic monotone and polynomial growth condition, Afr. Diaspora J. Math. 6 (2008), no. 1, 55–73. Search in Google Scholar

[5] K. Bahlali, S. Hamadène and B. Mezerdi, Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stochastic Process. Appl. 115 (2005), no. 7, 1107–1129. 10.1016/j.spa.2005.02.005Search in Google Scholar

[6] N. El Karoui and S.-J. Huang, A general result of existence and uniqueness of backward stochastic differential equations, Backward Stochastic Differential Equations (Paris 1995–1996), Pitman Res. Notes Math. Ser. 364, Longman, Harlow (1997), 27–36. Search in Google Scholar

[7] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1–71. 10.1111/1467-9965.00022Search in Google Scholar

[8] A. Elouaflin and M. N’zi, Reflected bacward stochastic differential equation with stochastic Lipschitz coefficient and random terminal time, submitted. Search in Google Scholar

[9] S. Hamadène and J. P. Lepeltier, Backward equations, stochastic control and zero-sum stochastic differential games, Stochastics Stochastics Rep. 54 (1995), no. 3–4, 221–231. 10.1080/17442509508834006Search in Google Scholar

[10] S. Hamadene and J.-P. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett. 24 (1995), no. 4, 259–263. 10.1016/0167-6911(94)00011-JSearch in Google Scholar

[11] B. Han, Y. Shi and Z. Bo, Backward doubly SDE with non-Lipschitz coefficients, preprint (2004), http://www.paper.edu.cn. Search in Google Scholar

[12] J. P. Lepeltier and J. San Martin, Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), no. 4, 425–430. 10.1016/S0167-7152(96)00103-4Search in Google Scholar

[13] M. N’zi and J.-M. Owo, Backward doubly stochastic differential equations with non-Lipschitz coefficients, Random Oper. Stoch. Equ. 16 (2008), no. 4, 307–324. 10.1515/ROSE.2008.018Search in Google Scholar

[14] M. N’zi and J.-M. Owo, Backward doubly stochastic differential equations with discontinuous coefficients, Statist. Probab. Lett. 79 (2009), no. 7, 920–926. 10.1016/j.spl.2008.11.011Search in Google Scholar

[15] J.-M. Owo, Backward doubly stochastic differential equations with stochastic Lipschitz condition, Statist. Probab. Lett. 96 (2015), 75–84. 10.1016/j.spl.2014.09.012Search in Google Scholar

[16] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. 10.1016/0167-6911(90)90082-6Search in Google Scholar

[17] E. Pardoux and S. G. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Stochastic Partial Differential Equations and Their Applications (Charlotte 1991), Lect. Notes Control Inf. Sci. 176, Springer, Berlin (1992), 200–217. 10.1007/BFb0007334Search in Google Scholar

[18] E. Pardoux and S. G. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields 98 (1994), no. 2, 209–227. 10.1007/BF01192514Search in Google Scholar

[19] S. G. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics Stochastics Rep. 37 (1991), no. 1–2, 61–74. 10.1080/17442509108833727Search in Google Scholar

[20] S. G. Peng and Y. Shi, Infinite horizon forward-backward stochastic differential equations, Stochastic Process. Appl. 85 (2000), no. 1, 75–92. 10.1016/S0304-4149(99)00066-6Search in Google Scholar

[21] Y. Shi, Y. Gu and K. Liu, Comparison theorems of backward doubly stochastic differential equations and applications, Stoch. Anal. Appl. 23 (2005), no. 1, 97–110. 10.1081/SAP-200044444Search in Google Scholar

[22] S. F. Zhou, X. Y. Cao and X. Guo, Backward doubly stochastic differential equations, Math. Appl. (Wuhan) 17 (2004), no. 1, 95–103. Search in Google Scholar

Received: 2017-07-14
Accepted: 2018-06-12
Published Online: 2018-08-07
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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