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Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

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Abstract

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.

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Authors and Affiliations

  1. School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    QingFeng Zhu

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    QingFeng Zhu & YuFeng Shi

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  1. QingFeng Zhu
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  2. YuFeng Shi
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Correspondence to YuFeng Shi.

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Zhu, Q., Shi, Y. Forward-backward doubly stochastic differential equations and related stochastic partial differential equations. Sci. China Math. 55, 2517–2534 (2012). https://doi.org/10.1007/s11425-012-4411-1

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