Backward doubly stochastic differential equations driven by fractional Brownian motion with stochastic integral-Lipschitz coefficients. (English) Zbl 07812407
Summary: This paper deals with a class of backward doubly stochastic differential equations driven by fractional Brownian motion with Hurst parameter \(H\) greater than \(\frac{1}{2}\). We essentially establish the existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and stochastic integral-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 doubly stochastic differential equation; stochastic Lipschitz coefficients; Malliavin derivative and fractional Itô formulaReferences:
[1] | S. Aidara, Anticipated backward doubly stochastic differential equations with non-Liphschitz coefficients, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 9-19. · Zbl 1524.60109 |
[2] | S. Aidara and Y. Sagna, BSDEs driven by two mutually independent fractional Brownian motions with stochastic Lipschitz coefficients, Appl. Math. Nonlinear Sci. 4 (2019), no. 1, 151-162. · Zbl 1524.60117 |
[3] | S. Aidara and A. B. Sow, Generalized fractional BSDE with non Lipschitz coefficients, Afr. Mat. 27 (2016), no. 3-4, 443-455. · Zbl 1386.60189 |
[4] | L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal. 10 (1999), no. 2, 177-214. · Zbl 0924.60034 |
[5] | W. Fei, D. Xia and S. Zhang, Solutions to BSDEs driven by both standard and fractional Brownian motions, Acta Math. Appl. Sin. Engl. Ser. 29 (2013), no. 2, 329-354. · Zbl 1329.60180 |
[6] | Y. Hu, Integral transformations and anticipative calculus for fractional Brownian motions, Mem. Amer. Math. Soc. 175 (2005), 1-127. · Zbl 1072.60044 |
[7] | E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55-61. · Zbl 0692.93064 |
[8] | Y. Wang and Z. Huang, Backward stochastic differential equations with non-Lipschitz coefficients, Statist. Probab. Lett. 79 (2009), no. 12, 1438-1443. · Zbl 1172.60322 |
[9] | L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251-282. · JFM 62.0250.02 |
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