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Convergence rates for Chernoff-type approximations of convex monotone semigroups

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  • Blessing, Jonas
  • Jiang, Lianzi
  • Kupper, Michael
  • Liang, Gechun

Abstract

We provide explicit convergence rates for Chernoff-type approximations of convex monotone semigroups which have the form S(t)f=limn→∞I(tn)nf for bounded continuous functions f. Under suitable conditions on the one-step operators I(t) regarding the time regularity and consistency of the approximation scheme, we obtain ‖S(t)f−I(tn)nf‖∞≤cn−γ for bounded Lipschitz continuous functions f, where c≥0 and γ>0 are determined explicitly. Moreover, the mapping t↦S(t)f is Hölder continuous. These results are closely related to monotone approximation schemes for viscosity solutions but are obtained independently by following a recently developed semigroup approach to Hamilton–Jacobi–Bellman equations which uniquely characterizes semigroups via their Γ-generators. The different approach allows to consider convex rather than sublinear equations and the results can be extended to unbounded functions by modifying the norm with a suitable weight function. Furthermore, up to possibly different consistency errors for the operators I(t), the upper and lower bound for the error between the semigroup and the iterated operators are symmetric. The abstract results are applied to Nisio semigroups and limit theorems for convex expectations.

Suggested Citation

  • Blessing, Jonas & Jiang, Lianzi & Kupper, Michael & Liang, Gechun, 2025. "Convergence rates for Chernoff-type approximations of convex monotone semigroups," Stochastic Processes and their Applications, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:spapps:v:189:y:2025:i:c:s0304414925001413
    DOI: 10.1016/j.spa.2025.104700
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    References listed on IDEAS

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    1. repec:dau:papers:123456789/5524 is not listed on IDEAS
    2. Denk, Robert & Kupper, Michael & Nendel, Max, 2020. "A semigroup approach to nonlinear Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1616-1642.
    3. Blessing, Jonas & Kupper, Michael & Nendel, Max, 2023. "Convergence of Infintesimal Generators and Stability of Convex Montone Semigroups," Center for Mathematical Economics Working Papers 680, Center for Mathematical Economics, Bielefeld University.
    4. Max Nendel, 2025. "Lower semicontinuity of monotone functionals in the mixed topology on C b $C_{b}$," Finance and Stochastics, Springer, vol. 29(1), pages 261-287, January.
    5. Jonas Blessing & Michael Kupper & Alessandro Sgarabottolo, 2024. "Discrete approximation of risk-based prices under volatility uncertainty," Papers 2411.00713, arXiv.org.
    6. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    7. Nendel, Max, 2025. "Lower semicontinuity of monotone functionals in the mixed topology on C b," Center for Mathematical Economics Working Papers 723, Center for Mathematical Economics, Bielefeld University.
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