\(L^1\) solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under general assumptions.

*(English)*Zbl 07107395Summary: We establish several existence, uniqueness and comparison results for \(L^1\) solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumptions that the generator \(g\) satisfies a one-sided Osgood condition together with a very general growth condition in \(y\), a uniform continuity condition and/or a sub-linear growth condition in \(z\), and a generalized Mokobodzki condition for reflected BSDEs which relates the growth of \(g\) and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of \(L^1\) solutions of the reflected BSDEs. We also prove that the \(L^1\) solutions of reflected BSDEs can be approximated by a penalization method and by some sequences of \(L^1\) solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the \(L^1\) solutions of non-reflected BSDEs and reflected BSDEs.

##### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

##### Keywords:

reflected backward stochastic differential equation; existence and uniqueness; comparison theorem; stability theorem; \(L^1\) solution##### References:

[1] | Khaled Bahlali, Saïd Hamadène, and Brahim Mezerdi, Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stochastic Process. Appl. 115 (2005), no. 7, 1107-1129. |

[2] | Erhan Bayraktar and Song Yao, Doubly reflected BSDEs with integrable parameters and related Dynkin games, Stochastic Process. Appl. 125 (2015), no. 12, 4489-4542. |

[3] | Philippe Briand, Bernard Delyon, Ying Hu, Etienne Pardoux, and L. Stoica, \(L^p\) solutions of backward stochastic differential equations, Stochastic Process. Appl. 108 (2003), no. 1, 109-129. |

[4] | Philippe Briand and Ying Hu, BSDE with quadratic growth and unbounded terminal value, Probab. Theory Related Fields 136 (2006), no. 4, 604-618. |

[5] | Philippe Briand, Jean-Pierre Lepeltier, and Jaime San Martin, One-dimensional backward stochastic differential equations whose coefficient is monotonic in \(y\) and non-Lipschitz in \(z\), Bernoulli 13 (2007), no. 1, 80-91. |

[6] | Rainer Buckdahn, Ying Hu, and Shanjian Tang, Existence of solution to scalar BSDEs with \({L}\exp \left (\mu \sqrt{2\log (1+L)} \right )\)-integrable terminal values, Electron. Commun. Probab. 23 (2018), Paper No. 59, 8pp. |

[7] | Shaokuan Chen, \(L^p\) solutions of one-dimensional backward stochastic differential equations with continuous coefficients, Stoch. Anal. Appl. 28 (2010), no. 5, 820-841. |

[8] | Jakša Cvitanić and Ioannis Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Probab. 24 (1996), no. 4, 2024-2056. |

[9] | Brahim El Asri, Saïd Hamadène, and H. Wang, \( {L}^p\)-solutions for doubly reflected backward stochastic differential equations, Stoch. Anal. Appl. 29 (2011), no. 6, 907-932. |

[10] | Nicole El Karoui, C. Kapoudjian, Etienne Pardoux, Shige Peng, and Marie Claire Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, Ann. Probab. 25 (1997), no. 2, 702-737. |

[11] | Nicole El Karoui, Etienne Pardoux, and Marie Claire Quenez, Reflected backward SDEs and American options, Numerical methods in finance, Publ. Newton Inst., vol. 13, Cambridge Univ. Press, Cambridge, 1997, pp. 215-231. |

[12] | Nicole El Karoui, Shige Peng, and Marie Claire Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. |

[13] | Shengjun Fan, \({L}^p\) solutions of multidimensional BSDEs with weak monotonicity and general growth generators, J. Math. Anal. Appl. 432 (2015), 156-178. |

[14] | Shengjun Fan, Bounded solutions, \({L}^p\ (p>1)\) solutions and \({L}^1\) solutions for one-dimensional BSDEs under general assumptions, Stochastic Process. Appl. 126 (2016), 1511-1552. |

[15] | Shengjun Fan, Existence of solutions to one-dimensional BSDEs with semi-linear growth and general growth generators, Statist. Probab. Lett. 109 (2016), 7-15. |

[16] | Shengjun Fan, Existence, uniqueness and approximation for \({L}^p\) solutions of reflected BSDEs under weaker assumptions, Acta Mathematica Sinica, English Series 33 (2017), no. 6, 807-833. |

[17] | Shengjun Fan, \({L}^1\) solutions to one-dimensional BSDEs with sublinear growth generators in \(z\), arXiv:1701.04151v1[math.PR]16Jan2017 (2017). |

[18] | Shengjun Fan, \({L}^p\) solutions of doubly reflected BSDEs under general assumptions, arXiv:1701.04158v2[math.PR]30Apr2017 (2017). |

[19] | Shengjun Fan, Existence, uniqueness and stability of \({L}^1\) solutions for multidimensional BSDEs with generators of one-sided osgood type, J. Theoret. Probab. 31 (2018), 1860-1899. |

[20] | Shengjun Fan and Long Jiang, Uniqueness result for the BSDE whose generator is monotonic in \(y\) and uniformly continuous in \(z\), C. R. Math. Acad. Sci. Paris 348 (2010), no. 1-2, 89-92. |

[21] | Shengjun Fan and Long Jiang, One-dimensional BSDEs with left-continuous, lower semi-continuous and linear-growth generators, Statist. Probab. Lett. 82 (2012), 1792-1798. |

[22] | Shengjun Fan and Long Jiang, Multidimensional BSDEs with weak monotonicity and general growth generators, Acta Mathematica Sinica, English Series 29 (2013), no. 10, 1885-1906. |

[23] | Shengjun Fan, Long Jiang, and Matt Davison, Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators, C. R. Math. Acad. Sci. Paris 348 (2010), no. 11-12, 683-686. |

[24] | Shengjun Fan, Long Jiang, and Matt Davison, Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type, Front. Math. China 8 (2013), no. 4, 811-824. |

[25] | Shengjun Fan and Dequn Liu, A class of BSDEs with integrable parameters, Statist. Probab. Lett. 80 (2010), no. 23, 2024-2031. |

[26] | Saïd Hamadène and Mohammed Hassani, BSDEs with two reflecting barriers: the general result, Probab. Theory Related Fields 132 (2005), no. 2, 237-264. |

[27] | Saïd Hamadène, Mohammed Hassani, and Youssef Ouknine, Backward SDEs with two \(rcll\) reflecting barriers without Mokobodski’s hypothesis, Bull. Sci. Math. 134 (2010), no. 8, 874-899. |

[28] | Saïd Hamadène and Jean-Pierre Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Process. Appl. 85 (2000), no. 2, 177-188. |

[29] | Saïd Hamadène, Jean-Pierre Lepeltier, and Anis Matoussi, Double barrier backward SDEs with continuous coefficient, Backward stochastic differential equations (Paris, 1995-1996), Pitman Res. Notes Math. Ser., vol. 364, Longman, Harlow, 1997, pp. 161-175. |

[30] | Saïd Hamadène, Jean-Pierre Lepeltier, and Zhen Wu, Infinite horizon reflected backward stochastic differential equations and applications in mixed control and game problems, Probab. Math. Statist. 19 (1999), no. 2, Acta Univ. Wratislav. No. 2198, 211-234. |

[31] | Saïd Hamadène and Jianfeng Zhang, Switching problem and related system of reflected backward SDEs, Stochastic Process. Appl. 120 (2010), no. 4, 403-426. |

[32] | Ying Hu and Shanjian Tang, Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields 147 (2010), no. 1-2, 89-121. |

[33] | Ying Hu and Shanjian Tang, Multi-dimensional backward stochastic differential equations of diagonally quadratic generators, Stochastic Process. Appl. 126 (2015), no. 4, 1066-1086. |

[34] | Ying Hu and Shanjian Tang, Existence of solution to scalar BSDEs with \({L}\exp \sqrt{\frac{2}{\lambda }\log (1+L)} \)-integrable terminal values, Electron. Commun. Probab. 23 (2018), Paper No. 27, 11pp. |

[35] | Guangyan Jia, A uniqueness theorem for the solution of backward stochastic differential equations, C. R. Math. Acad. Sci. Paris 346 (2008), no. 7-8, 439-444. |

[36] | Guangyan Jia, Backward stochastic differential equations with a uniformly continuous generator and related \(g\)-expectation, Stochastic Process. Appl. 120 (2010), no. 11, 2241-2257. |

[37] | Guangyan Jia and Mingyu Xu, Construction and uniqueness for reflected BSDE under linear increasing condition, arXiv:0801.3718v1[math.SG] (2014). |

[38] | Tomasz Klimsiak, Reflected BSDEs with monotone generator, Electron. J. Probab. 17 (2012), Paper No. 107, 25pp. |

[39] | Tomasz Klimsiak, BSDEs with monotone generator and two irregular reflecting barriers, Bulletin des Sciences Mathématiques 137 (2013), no. 3, 268-321. |

[40] | Jean-Pierre Lepeltier, Anis Matoussi, and Mingyu Xu, Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions, Adv. in Appl. Probab. 37 (2005), no. 1, 134-159. |

[41] | Jean-Pierre Lepeltier and Jaime San Martin, Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), no. 4, 425-430. |

[42] | Min Li and Yufeng Shi, Solving the double barrier reflected BSDEs via penalization method, Statist. Probab. Lett. 110 (2016), 74-83. |

[43] | Jin Ma and Jianfeng Zhang, Representations and regularities for solutions to BSDEs with reflections, Stochastic Process. Appl. 115 (2005), no. 4, 539-569. |

[44] | Xuerong Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process. Appl. 58 (1995), no. 2, 281-292. |

[45] | Anis Matoussi, Reflected solutions of backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 34 (1997), no. 4, 347-354. |

[46] | Etienne Pardoux, BSDEs, weak convergence and homogenization of semilinear PDEs, Nonlinear Analysis, Differential Equations and Control (F.H. Clarke and R.J. Stern, eds.), Kluwer Academic, New York, 1999, pp. 503-549. |

[47] | Etienne Pardoux and Shige Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett. 14 (1990), no. 1, 55-61. |

[48] | Shige Peng, Backward SDE and related \(g\)-expectation, Backward stochastic differential equations (Paris,1995-1996) (Nicole El Karoui and L. Mazliak, eds.), Pitman Research Notes Mathematical Series, vol. 364, Longman, Harlow, 1997, pp. 141-159. |

[49] | Shige Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type, Probab. Theory Related Fields 113 (1999), no. 4, 473-499. |

[50] | Shige Peng, Nonlinear expectations, nonlinear evaluations and risk measures, Stochastic methods in finance, Lecture Notes in Math., vol. 1856, Springer, Berlin, 2004, pp. 165-253. |

[51] | Shige Peng and Mingyu Xu, The smallest \(g\)-supermartingale and reflected BSDE with single and double \(L^2\) obstacles, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 605-630. |

[52] | Shige Peng and Mingyu Xu, Reflected BSDE with a constraint and its applications in an incomplete market, Bernoulli 16 (2010), no. 3, 614-640. |

[53] | Yong Ren and Mohamed El Otmani, Doubly reflected bsdes driven by a lévy process, Nonlinear Analysis: Real World Applications 13 (2012), 1252-1267. |

[54] | Emanuela Rosazza Gianin, Risk measures via \(g\)-expectations, Insurance Math. Econom. 39 (2006), no. 1, 19-34. |

[55] | Andrzej Rozkosz and Leszek Słomiński, \({L}^p\) solutions of reflected BSDEs under monotonicity condition, Stochastic Process. Appl. 122 (2012), no. 12, 3875-3900. |

[56] | Mingyu Xu, Reflected backward SDEs with two barriers under monotonicity and general increasing conditions, J. Theoret. Probab. 20 (2007), no. 4, 1005-1039. |

[57] | Mingyu Xu, Backward stochastic differential equations with reflection and weak assumptions on the coefficients, Stochastic Process. Appl. 118 (2008), no. 6, 968-980. |

[58] | Song Yao, \({L}^p\) solutions of backward stochastic differential equations with jumps, Stochastic Process. Appl. 127 (2017), no. 11, 3465-3511. |

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