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Backward Stochastic Differential Equations with No Driving Martingale, Markov Processes and Associated Pseudo-Partial Differential Equations: Part II—Decoupled Mild Solutions and Examples

Author

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  • Adrien Barrasso

    (Université d’Évry Val d’Essonne)

  • Francesco Russo

    (Unité de Mathématiques appliquées)

Abstract

Let $$(\mathbb {P}^{s,x})_{(s,x)\in [0,T]\times E}$$ ( P s , x ) ( s , x ) ∈ [ 0 , T ] × E be a family of probability measures, where E is a Polish space, defined on the canonical probability space $${\mathbb D}([0,T],E)$$ D ( [ 0 , T ] , E ) of E-valued càdlàg functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator a is well-posed. We consider also an associated semilinear Pseudo-PDE for which we introduce a notion of so-called decoupled mild solution and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of backward stochastic differential equations (BSDEs) without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map a is not a PDE operator.

Suggested Citation

  • Adrien Barrasso & Francesco Russo, 2021. "Backward Stochastic Differential Equations with No Driving Martingale, Markov Processes and Associated Pseudo-Partial Differential Equations: Part II—Decoupled Mild Solutions and Examples," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1110-1148, September.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:3:d:10.1007_s10959-021-01092-7
    DOI: 10.1007/s10959-021-01092-7
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    References listed on IDEAS

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    1. Issoglio, Elena & Jing, Shuai, 2020. "Forward–backward SDEs with distributional coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 47-78.
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
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