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Representation theorems for generators of backward stochastic differential equations and their applications

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  • Jiang, Long

Abstract

We prove that the generator g of a backward stochastic differential equation (BSDE) can be represented by the solutions of the corresponding BSDEs at point (t,y,z) if and only if t is a conditional Lebesgue point of generator g with parameters (y,z). By this conclusion, we prove that, if g is a Lebesgue generator and g is independent of y, then, Jensen's inequality for g-expectation holds if and only if g is super homogeneous; we also obtain a converse comparison theorem for deterministic generators of BSDEs.

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  • Jiang, Long, 2005. "Representation theorems for generators of backward stochastic differential equations and their applications," Stochastic Processes and their Applications, Elsevier, vol. 115(12), pages 1883-1903, December.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:12:p:1883-1903
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    References listed on IDEAS

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    1. 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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    Cited by:

    1. Sheng Jun Fan, 2018. "Existence, Uniqueness and Stability of $$L^1$$ L 1 Solutions for Multidimensional Backward Stochastic Differential Equations with Generators of One-Sided Osgood Type," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1860-1899, September.
    2. Zhou, Guangshuo & Du, Fengjiao & Fan, Shengjun, 2024. "Invariant representation for generators of general time interval quadratic BSDEs under stochastic growth conditions," Statistics & Probability Letters, Elsevier, vol. 205(C).
    3. Hu, Feng & Chen, Zengjing, 2010. "Generalized Peng's g-expectations and related properties," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 191-195, February.
    4. Zheng, Shiqiu & Li, Shoumei, 2015. "Representation theorems for generators of BSDEs with monotonic and convex growth generators," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 197-205.
    5. Liu, Haodong & Yang, Shuzhen, 2017. "Representation and converse comparison theorems for multidimensional BSDEs," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 67-74.
    6. Jiang, Long, 2006. "A note on g-expectation with comonotonic additivity," Statistics & Probability Letters, Elsevier, vol. 76(17), pages 1895-1903, November.
    7. Xiao, Lishun & Fan, Shengjun, 2017. "A representation theorem for generators of BSDEs with general growth generators in y and its applications," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 297-305.
    8. Jiang, Long, 2009. "A necessary and sufficient condition for probability measures dominated by g-expectation," Statistics & Probability Letters, Elsevier, vol. 79(2), pages 196-201, January.
    9. Dejian Tian & Xunlian Wang, 2023. "Dynamic star-shaped risk measures and $g$-expectations," Papers 2305.02481, arXiv.org.
    10. Jia, Guangyan, 2010. "Backward stochastic differential equations with a uniformly continuous generator and related g-expectation," Stochastic Processes and their Applications, Elsevier, vol. 120(11), pages 2241-2257, November.
    11. Fan, Sheng-Jun & Hu, Jian-Hua, 2008. "A limit theorem for solutions to BSDEs in the space of processes," Statistics & Probability Letters, Elsevier, vol. 78(8), pages 1024-1033, June.

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