BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients. (English) Zbl 1524.60117
Summary: This paper deals with a class of backward stochastic differential equation driven by two mutually independent fractional Brownian motions. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H05 | Stochastic integrals |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60G22 | Fractional processes, including fractional Brownian motion |
Keywords:
backward stochastic differential equation; stochastic Lipschitz coefficients; Malliavin derivative and fractional Itô’s formulaReferences:
| [1] | S. Aidara and A. B. Sow, Generalized fractional backward stochastic differential equation with non Lipschitz coefficients. Afr. Mat. 27(2016), no. 3-4, 443-455. · Zbl 1386.60189 |
| [2] | S. Aidara Backward stochastic differential equations driven by two mutually independent fractional Brownian motions. Applied Mathematics and Nonlinear Science-D-19-00001R1, (2019). |
| [3] | L. Decreusefond, A.S. Ustunel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1998), 177-214,. · Zbl 0924.60034 |
| [4] | Fei, W. & Xia, D. & Zhang, S. . Solutions to BSDEs driven by Both Standard and Fractional Brownian Motions. Acta Mathematicae Applicatae Sinica, English Series, 329-354, (2013). · Zbl 1329.60180 |
| [5] | Y. Hu, Integral transformation and anticipative calculus for fractional Brownian motion. Mem. Amer. Math. Soc. 175 (2005). · Zbl 1072.60044 |
| [6] | Y. Hu, S. Peng, Backward stochastic differential equation driven by fractional Brownian motion. SIAM J. Control Optim. 48 (2009). 1675-1700, · Zbl 1284.60117 |
| [7] | E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation. Systems Control Lett, 114 (1990), 55-61. · Zbl 0692.93064 |
| [8] | S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab. 37 (2009), 877-902 · Zbl 1186.60053 |
| [9] | Y. Wang, Z. Huang, Backward stochastic differential equations with non Lipschitz coefficients equations. Statistics and Probability Letters. 79 (2009), 1438-1443. · Zbl 1172.60322 |
| [10] | L.C. Young, An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 (1936), 251-282. · JFM 62.0250.02 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
