On Monotonic Functionals Over Partially-Ordered Path Spaces

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.