A Skorohod measurable universal functional representation of solutions to semimartingale SDEs

TL;DR

The paper addresses the problem of obtaining a universal Skorohod-measurable functional representation for solutions to general semimartingale-driven SDEs. It constructs a Skorohod-measurable universal functional $\Psi$ such that, for any admissible $(H,G,Y)$, the solution $X$ to $X_t = H_t + \int_0^t g(s,G,X) \mathrm{d}Y_s$ satisfies $X = \Psi(H,G,Y)$, with $\Psi$ built through a carefully designed iterative scheme and shown to be the limit of Skorohod-measurable approximants. This universality goes beyond the factorization lemma and yields a universal expression for Malliavin derivatives in the pure-jump Lévy setting, enabling uniform derivative formulas and hedging-type applications. The results have implications for computational stochastics and complexity analysis in jump-diffusion models, providing a cohesive framework for analyzing SDEs across a broad class of driving processes.

Abstract

In this paper we show the existence of a universal Skorohod measurable functional representation for a large class of semimartingale-driven stochastic differential equations. For this we prove that paths of the strong solutions of stochastic differential equations can be written as measurable functions of the paths of their driving processes into the space of all càdlàg functions equipped with the Borel sigma-field generated by all open sets with respect to the Skorohod metric. This result can be applied to calculate Malliavin derivatives for SDEs driven by pure-jump Lévy processes with drift.

Theorems & Definitions (17)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• Remark 3.2
• proof : Proof of Theorem \ref{['Main']}, part 1
• Definition 3.3: cf. Karandikar2019
• Remark 3.4
• Theorem 3.5: cf. Karandikar2019
• proof : Proof of Theorem \ref{['Main']}, part 2