Fixing a Minor Mistake in the Theory of Stochastic Integration and Differential Equations

2024/03/19 by Sebastian Rickelhoff, Rickelhoff, Sebastian, Alexander Schnurr +1
Computer Science · #60G07 #60H10 #60J76 #FOS: Mathematics #Probability (math.PR) #Statistical and Computational Modeling

paper · pdf · doi:10.48550/arxiv.2403.12613

openalex publication_date 2024/03/19 · openalex created_date 2024/03/21 · openalex updated_date 2026/07/28

Abstract

When considering stochastic integration and the theory of stochastic differential equations, P. Protter's textbook \citeprotter undoubtedly is a main piece of standard literature. Not only is it well-written, but it also contains various profound results regarding these fields. Unfortunately, Theorem 12 of Chapter V, which presents an equivalence to uniform convergence on compacts in probability, is found to be incorrect. Given that numerous important results rely on this theorem, this paper aims to present a corrected version of it.

Related

• Understanding stochastic differential equations
• Correction of the Fokker-Planck equation, and the adequate sense of stochastic integration
• Modify the Improved Euler scheme to integrate stochastic differential equations
• Approximation analysis for weak solutions of stochastic partial differential equations
• Partial and full solutions of stochastic differential equations