# American Institute of Mathematical Sciences

December  2021, 6(4): 301-318. doi: 10.3934/puqr.2021015

## General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

 1 School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China

f_s_j@126.com (Shengjun Fan)

Received  July 04, 2021 Accepted  October 27, 2021 Published  December 2021

Fund Project: The authors are greatly grateful to the referee for his/her careful reading and valuable suggestions.Fan’s  research  is  partially  supported  by  the  Fundamental  Research  Funds  for  the  CentralUniversities (Grant No. 2017XKZD11) and the National Natural Science Foundation of China(Grant No. 12171471).

This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.

Citation: Tingting Li, Ziheng Xu, Shengjun Fan. General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition. Probability, Uncertainty and Quantitative Risk, 2021, 6 (4) : 301-318. doi: 10.3934/puqr.2021015
##### References:
 [1] Bender, C. and Kohlmann, M., BSDES with stochastic lipschitz condition, In: CoFE-Diskussionspapiere/Zentrum für Finanzen und Ökonometrie, 2000, http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-4241. Google Scholar [2] Bismut, J., Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 1973, 44(2): 384−404. Google Scholar [3] Briand, P. and Confortola, F., BSDEs with stochastic lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 2008, 118(5): 818−838. Google Scholar [4] Briand, P., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L., $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 2003, 108(1): 109−129. doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar [5] Chen, Z. and Wang, B., Infinite time interval BSDEs and the convergence of $g\text{-}{\rm{martingales}}$, Journal of the Australian Mathematical Society (Series A), 2000, 69(2): 187−211. doi: 10.1017/S1446788700002172.  Google Scholar [6] Delaen, F. and Tang, S., Harmonic analysis of stochastic equations and backward stochastics differential equations, Probab. Theory Relat. Fields, 2010, 146(1−2): 291−336. doi: 10.1007/s00440-008-0191-5.  Google Scholar [7] Ding, X. and Wu, R., A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales, Stochastic Process. Appl., 1998, 78(2): 155−171. doi: 10.1016/S0304-4149(98)00051-9.  Google Scholar [8] El Karoui, N. and Huang, S., A general result of existence and uniqueness of backward stochastic differential equations, In: Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, Longman, London, 1997, 364: 27−36. Google Scholar [9] El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.  Google Scholar [10] Fan, S., $L^p$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators, J. Math. Anal. Appl., 2015, 432(1): 156−178. doi: 10.1016/j.jmaa.2015.06.049.  Google Scholar [11] Fan, S., Bounded solutions, $L^p\ (p>1)$ solutions and $L^1$ solutions for one-dimensional BSDEs under general assumptions, Stochastic Process. Appl., 2016, 126(5): 1511−1552. doi: 10.1016/j.spa.2015.11.012.  Google Scholar [12] Fan, S. and Jiang, L., Multidimensional BSDEs with weak monotonicity and general growth generators, Acta Mathematica Sinica, English Series, 2013, 29(10): 1885−1906. doi: 10.1007/s10114-013-2128-x.  Google Scholar [13] Fan, S., Jiang, L. and Davison, M., Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type, Front. Math. China, 2013, 8(4): 811−824. doi: 10.1007/s11464-013-0298-6.  Google Scholar [14] Kazamaki, N., Continuous exponential martingals and BMO, In: Lecture Notes in Math., Springer, Berlin, 1994. Google Scholar [15] Liu, Y., Li, D. and Fan, S., $L^p\ (p > 1)$ solutions of BSDEs with generators satisfying some non-uniform conditions in $t$ and $\omega$, Chinese Ann. Math. B, 2020, 41(3): 479−494. doi: 10.1007/s11401-020-0212-y.  Google Scholar [16] Luo, H. and Fan, S., Bounded solutions for general time interval BSDEs with quadratic growth coefficients and stochastic conditions, Stoch. Dynam., 2018, 18(5): 1850034. Google Scholar [17] Mao, X., Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process. Appl., 1995, 58(2): 281−292. doi: 10.1016/0304-4149(95)00024-2.  Google Scholar [18] Morlais, M. A., Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance Stoch., 2009, 13: 121−150. doi: 10.1007/s00780-008-0079-3.  Google Scholar [19] Pardoux, E., BSDEs, weak convergence and homogenization of semilinear PDEs, In: Clarke, F. and Stern, R. (eds.), Nonlinear Analysis, Differential Equations and Control, Kluwer Academic, New York, 1999. Google Scholar [20] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 1990, 14(1): 55−61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar [21] Pardoux, E. and Ră ${\underset{\raise0.4em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{{\rm{s}}} }$ scanu, A., Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer, Cham, 2014. Google Scholar [22] Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures, In: Stochastic Methods in Finance, Lecture Notes in Math, Springer, Berlin, 2004, 1856: 165−253. Google Scholar [23] Wang, J., Ran, Q. and Chen, Q., $L^p$ solutions of BSDEs with stochastic lipschitz condition, J. Appl. Math. Stoch. Anal., 2007, 2007: 78196. Google Scholar [24] Wang, X. and Fan, S., A class of stochastic Gronwall’s inequality and its application, Journal of Inequalities and Applications, 2018, 2018(1): 336. Google Scholar [25] Xiao, L. and Fan, S., General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general g-supermartingales, Stochastics, 2017, 89(5): 786−816. doi: 10.1080/17442508.2017.1282956.  Google Scholar

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##### References:
 [1] Bender, C. and Kohlmann, M., BSDES with stochastic lipschitz condition, In: CoFE-Diskussionspapiere/Zentrum für Finanzen und Ökonometrie, 2000, http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-4241. Google Scholar [2] Bismut, J., Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 1973, 44(2): 384−404. Google Scholar [3] Briand, P. and Confortola, F., BSDEs with stochastic lipschitz condition and quadratic PDEs in Hilbert spaces, Stochastic Process. Appl., 2008, 118(5): 818−838. Google Scholar [4] Briand, P., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L., $L^p$ solutions of backward stochastic differential equations, Stochastic Process. Appl., 2003, 108(1): 109−129. doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar [5] Chen, Z. and Wang, B., Infinite time interval BSDEs and the convergence of $g\text{-}{\rm{martingales}}$, Journal of the Australian Mathematical Society (Series A), 2000, 69(2): 187−211. doi: 10.1017/S1446788700002172.  Google Scholar [6] Delaen, F. and Tang, S., Harmonic analysis of stochastic equations and backward stochastics differential equations, Probab. Theory Relat. Fields, 2010, 146(1−2): 291−336. doi: 10.1007/s00440-008-0191-5.  Google Scholar [7] Ding, X. and Wu, R., A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales, Stochastic Process. Appl., 1998, 78(2): 155−171. doi: 10.1016/S0304-4149(98)00051-9.  Google Scholar [8] El Karoui, N. and Huang, S., A general result of existence and uniqueness of backward stochastic differential equations, In: Backward Stochastic Differential Equations, Pitman Research Notes in Mathematics Series, Longman, London, 1997, 364: 27−36. Google Scholar [9] El Karoui, N., Peng, S. and Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 1997, 7(1): 1−71. doi: 10.1111/1467-9965.00022.  Google Scholar [10] Fan, S., $L^p$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators, J. Math. Anal. Appl., 2015, 432(1): 156−178. doi: 10.1016/j.jmaa.2015.06.049.  Google Scholar [11] Fan, S., Bounded solutions, $L^p\ (p>1)$ solutions and $L^1$ solutions for one-dimensional BSDEs under general assumptions, Stochastic Process. Appl., 2016, 126(5): 1511−1552. doi: 10.1016/j.spa.2015.11.012.  Google Scholar [12] Fan, S. and Jiang, L., Multidimensional BSDEs with weak monotonicity and general growth generators, Acta Mathematica Sinica, English Series, 2013, 29(10): 1885−1906. doi: 10.1007/s10114-013-2128-x.  Google Scholar [13] Fan, S., Jiang, L. and Davison, M., Existence and uniqueness result for multidimensional BSDEs with generators of Osgood type, Front. Math. China, 2013, 8(4): 811−824. doi: 10.1007/s11464-013-0298-6.  Google Scholar [14] Kazamaki, N., Continuous exponential martingals and BMO, In: Lecture Notes in Math., Springer, Berlin, 1994. Google Scholar [15] Liu, Y., Li, D. and Fan, S., $L^p\ (p > 1)$ solutions of BSDEs with generators satisfying some non-uniform conditions in $t$ and $\omega$, Chinese Ann. Math. B, 2020, 41(3): 479−494. doi: 10.1007/s11401-020-0212-y.  Google Scholar [16] Luo, H. and Fan, S., Bounded solutions for general time interval BSDEs with quadratic growth coefficients and stochastic conditions, Stoch. Dynam., 2018, 18(5): 1850034. Google Scholar [17] Mao, X., Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process. Appl., 1995, 58(2): 281−292. doi: 10.1016/0304-4149(95)00024-2.  Google Scholar [18] Morlais, M. A., Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem, Finance Stoch., 2009, 13: 121−150. doi: 10.1007/s00780-008-0079-3.  Google Scholar [19] Pardoux, E., BSDEs, weak convergence and homogenization of semilinear PDEs, In: Clarke, F. and Stern, R. (eds.), Nonlinear Analysis, Differential Equations and Control, Kluwer Academic, New York, 1999. Google Scholar [20] Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 1990, 14(1): 55−61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar [21] Pardoux, E. and Ră ${\underset{\raise0.4em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{{\rm{s}}} }$ scanu, A., Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer, Cham, 2014. Google Scholar [22] Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures, In: Stochastic Methods in Finance, Lecture Notes in Math, Springer, Berlin, 2004, 1856: 165−253. Google Scholar [23] Wang, J., Ran, Q. and Chen, Q., $L^p$ solutions of BSDEs with stochastic lipschitz condition, J. Appl. Math. Stoch. Anal., 2007, 2007: 78196. Google Scholar [24] Wang, X. and Fan, S., A class of stochastic Gronwall’s inequality and its application, Journal of Inequalities and Applications, 2018, 2018(1): 336. Google Scholar [25] Xiao, L. and Fan, S., General time interval BSDEs under the weak monotonicity condition and nonlinear decomposition for general g-supermartingales, Stochastics, 2017, 89(5): 786−816. doi: 10.1080/17442508.2017.1282956.  Google Scholar
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