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On Monotonic Functionals Over Partially-Ordered Path Spaces

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  • Levent Ali Mengütürk

    (University College London
    Artificial Intelligence and Mathematics Research Lab)

Abstract

We study non-decreasing (non-increasing) monotonic functionals over unions of Skorokhod spaces of càdlàg paths which we show to have an equivalence to non-negative (non-positive) path-dependent spatial Dupire derivatives. These functionals provide an upper-bound for their Lie-bracket of non-commutative spatial and temporal Dupire operators. We provide a stochastic functional generalisation for the Lebesgue integral of derivatives of non-decreasing (non-increasing) functions. We also present a functional generalisation of Markov’s inequality. One can further associate monotonic functionals of order-preserving random paths to their stochastic differential equations. We encapsulate what we call buffered monotonic functionals on paths that never draw closer than a minimum distance over their lifetime. As an application, we generate path-dependent stochastic triangles that randomly change their location, shape and area, while embedding a minimum structure that ensures convexity of the geometry at every point in time—a construct for modelling temporal population cluster dynamics with memory.

Suggested Citation

  • Levent Ali Mengütürk, 2025. "On Monotonic Functionals Over Partially-Ordered Path Spaces," Journal of Theoretical Probability, Springer, vol. 38(3), pages 1-32, September.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:3:d:10.1007_s10959-025-01439-4
    DOI: 10.1007/s10959-025-01439-4
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    References listed on IDEAS

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    1. Peng, Shige & Zhu, Xuehong, 2006. "Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 370-380, March.
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    3. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    4. Bru, Marie-France, 1989. "Diffusions of perturbed principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 29(1), pages 127-136, April.
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