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Solutions to BSDEs driven by both standard and fractional Brownian motions

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Abstract

The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.

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Authors and Affiliations

  1. Department of Applied Mathematics, Anhui Polytechnic University, Wuhu, 241000, China

    Wei-yin Fei

  2. Department of Applied Mathematics, Anhui Polytechnic University, Wuhu, 241000, China

    Deng-Feng Xia

  3. Department of Statistics and Finance, University of Science and Technology of China, Hefei, 230026, China

    Shu-guang Zhang

Authors
  1. Wei-yin Fei
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  2. Deng-Feng Xia
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  3. Shu-guang Zhang
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Correspondence to Wei-yin Fei.

Additional information

Supported by National Basic Research Program of China (973 Program, No. 2007CB814901), National Natural Science Foundation of China (No. 71171003), Anhui Natural Science Foundation (No. 090416225) and Anhui Natural Science Foundation of Universities (No. KJ2010A037).

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Fei, Wy., Xia, DF. & Zhang, Sg. Solutions to BSDEs driven by both standard and fractional Brownian motions. Acta Math. Appl. Sin. Engl. Ser. 29, 329–354 (2013). https://doi.org/10.1007/s10255-013-0219-1

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  • DOI: https://doi.org/10.1007/s10255-013-0219-1

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