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Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities

Author

Listed:
  • Lucian Maticiuc

    (“Alexandru Ioan Cuza” University
    “Gheorghe Asachi” Technical University)

  • Tianyang Nie

    (Shandong University
    CNR-UMR 6205 Université de Bretagne Occidentale
    University of Sydney)

Abstract

In the framework of fractional stochastic calculus, we study the existence and the uniqueness of the solution for a backward stochastic differential equation, formally written as: $$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)= f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array} \right. \end{aligned}$$ { − d Y ( t ) = f ( t , η ( t ) , Y ( t ) , Z ( t ) ) d t − Z ( t ) δ B H ( t ) , t ∈ [ 0 , T ] , Y ( T ) = ξ , where $$\eta $$ η is a stochastic process given by $$\eta (t)=\eta (0) +\int _{0}^{t}\sigma (s) \delta B^{H}(s)$$ η ( t ) = η ( 0 ) + ∫ 0 t σ ( s ) δ B H ( s ) , $$t\in [0,T]$$ t ∈ [ 0 , T ] , and $$B^{H}$$ B H is a fractional Brownian motion with Hurst parameter greater than $$1/2$$ 1 / 2 . The stochastic integral used in above equation is the divergence-type integral. Based on Hu and Peng’s paper, Backward stochastic differential equation driven by fractional Brownian motion, SIAM J Control Optim (2009), we develop a rigorous approach for this equation. Moreover, we study the existence of the solution for the multivalued backward stochastic differential equation $$\begin{aligned} \left\{ \begin{array}{l} -\hbox {d}Y(t)+\partial \varphi (Y(t))\hbox {d}t\ni f(t,\eta (t),Y(t),Z(t))\hbox {d}t-Z(t)\delta B^{H}\left( t\right) ,\quad t\in [0,T],\\ Y(T)=\xi , \end{array}\right. \end{aligned}$$ { − d Y ( t ) + ∂ φ ( Y ( t ) ) d t ∋ f ( t , η ( t ) , Y ( t ) , Z ( t ) ) d t − Z ( t ) δ B H ( t ) , t ∈ [ 0 , T ] , Y ( T ) = ξ , where $$\partial \varphi $$ ∂ φ is a multivalued operator of subdifferential type associated with the convex function $$\varphi $$ φ .

Suggested Citation

  • Lucian Maticiuc & Tianyang Nie, 2015. "Fractional Backward Stochastic Differential Equations and Fractional Backward Variational Inequalities," Journal of Theoretical Probability, Springer, vol. 28(1), pages 337-395, March.
  • Handle: RePEc:spr:jotpro:v:28:y:2015:i:1:d:10.1007_s10959-013-0509-9
    DOI: 10.1007/s10959-013-0509-9
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    References listed on IDEAS

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    1. Mémin, Jean & Mishura, Yulia & Valkeila, Esko, 2001. "Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 51(2), pages 197-206, January.
    2. Alòs, Elisa & Mazet, Olivier & Nualart, David, 2000. "Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 121-139, March.
    3. Pardoux, Etienne & Rascanu, Aurel, 1998. "Backward stochastic differential equations with subdifferential operator and related variational inequalities," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 191-215, August.
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    Cited by:

    1. Sin, Myong-Guk & Ri, Kyong-Il & Kim, Kyong-Hui, 2022. "Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 190(C).
    2. Pei Zhang & Adriana Irawati Nur Ibrahim & Nur Anisah Mohamed, 2023. "Anticipated BSDEs Driven by Fractional Brownian Motion with a Time-Delayed Generator," Mathematics, MDPI, vol. 11(23), pages 1-13, December.
    3. Kyong-Il, Ri & Myong-Guk, Sin, 2024. "Existence and uniqueness of solution for fully coupled fractional forward–backward stochastic differential equations with delay and anticipated term," Statistics & Probability Letters, Elsevier, vol. 206(C).

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