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On Riemann–Liouville type operators, bounded mean oscillation, gradient estimates and approximation on the Wiener space

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  • Geiss, Stefan
  • Thuan, Nguyen Tran

Abstract

We discuss in a stochastic framework the interplay between Riemann–Liouville type operators applied to stochastic processes, bounded mean oscillation, real interpolation, and approximation. In particular, we investigate the singularity of gradient processes on the Wiener space arising from parabolic PDEs via the Feynman–Kac theory. The singularity is measured in terms of bmo-conditions on the fractional integrated gradient. As an application we treat an approximation problem for stochastic integrals on the Wiener space. In particular, we provide a discrete time hedging strategy for the binary option with a uniform local control of the hedging error under a shortfall constraint.

Suggested Citation

  • Geiss, Stefan & Thuan, Nguyen Tran, 2025. "On Riemann–Liouville type operators, bounded mean oscillation, gradient estimates and approximation on the Wiener space," Stochastic Processes and their Applications, Elsevier, vol. 187(C).
    • Handle: RePEc:eee:spapps:v:187:y:2025:i:c:s0304414925000924
    DOI: 10.1016/j.spa.2025.104651
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    References listed on IDEAS

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    1. Martin Schweizer & Christophe Stricker & Freddy Delbaen & Pascale Monat & Walter Schachermayer, 1997. "Weighted norm inequalities and hedging in incomplete markets," Finance and Stochastics, Springer, vol. 1(3), pages 181-227.
    2. Geiss, Stefan & Ylinen, Juha, 2020. "Weighted bounded mean oscillation applied to backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3711-3752.
    3. Laukkarinen, Eija, 2020. "Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4766-4792.
    4. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
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