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Set-valued stochastic integrals for convoluted L\'{e}vy processes

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  • Weixuan Xia

Abstract

In this paper we study set-valued Volterra-type stochastic integrals driven by L\'{e}vy processes. Upon extending the classical definitions of set-valued stochastic integral functionals to convoluted integrals with square-integrable kernels, set-valued convoluted stochastic integrals are defined by taking the closed decomposable hull of the integral functionals for generic time. We show that, aside from well-established results for set-valued It\^{o} integrals, while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are guaranteed to be integrably bounded for bounded integrands, this is not true when the random measure is of infinite variation. For indefinite integrals, we prove that it is a mutual effect of kernel singularity and jumps that the set-valued convoluted integrals are possibly explosive and take extended vector values. These results have some important implications on how set-valued fractional dynamical systems are to be constructed in general. Two classes of set-monotone processes are studied for practical interests in economic and financial modeling.

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  • Weixuan Xia, 2023. "Set-valued stochastic integrals for convoluted L\'{e}vy processes," Papers 2312.01730, arXiv.org, revised Nov 2024.
    • Handle: RePEc:arx:papers:2312.01730
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    1. Liang Wang & Weixuan Xia, 2022. "Power‐type derivatives for rough volatility with jumps," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(7), pages 1369-1406, July.
    2. Lyasoff, Andrew, 2017. "Stochastic Methods in Asset Pricing," MIT Press Books, The MIT Press, edition 1, volume 1, number 026203655x, December.
    3. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
    4. Michał Kisielewicz, 1993. "Properties of solution set of stochastic inclusions," International Journal of Stochastic Analysis, Hindawi, vol. 6, pages 1-19, January.
    5. Jost, Céline, 2006. "Transformation formulas for fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1341-1357, October.
    6. Michał Kisielewicz, 2013. "Stochastic Differential Inclusions," Springer Optimization and Its Applications, in: Stochastic Differential Inclusions and Applications, edition 127, chapter 0, pages 147-179, Springer.
    7. Zongxia Liang & Ming Ma, 2020. "Robust consumption‐investment problem under CRRA and CARA utilities with time‐varying confidence sets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1035-1072, July.
    8. Michał Kisielewicz, 2020. "Set-Valued Stochastic Integrals and Applications," Springer Optimization and Its Applications, Springer, number 978-3-030-40329-4, December.
    9. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
    10. Michał Kisielewicz, 2013. "Stochastic Differential Inclusions and Applications," Springer Optimization and Its Applications, Springer, edition 127, number 978-1-4614-6756-4, December.
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