[Paper Review] Existence of spatially differentiable solutions of stochastic differential equations with non-globally monotone coefficient functions

Anselm Hudde, Martin Hutzenthaler+1 more|arXiv (Cornell University)|Mar 22, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance21 references3 citations

TL;DR

This paper establishes the existence of continuously differentiable solutions to stochastic differential equations (SDEs) with non-globally monotone coefficient functions by introducing a local monotonicity condition and additional regularity assumptions. The key contribution is extending spatial differentiability results beyond the restrictive requirement of bounded derivatives, enabling applications of the Itô-Alekseev-Grobner formula to a broader class of SDEs from real-world applications.

ABSTRACT

Spatial differentiability of solutions of stochastic differential equations (SDEs) is required for the Ito-Alekseev-Grobner formula and other applications. In the literature, this differentiability is only derived if the coefficient functions of the SDE have bounded derivatives and this property is rarely satisfied in applications. In this article we establish existence of continuously differentiable solutions of SDEs whose coefficients satisfy a suitable local monotonicity property and further conditions. These conditions are satisfied by many SDEs from applications.

Motivation & Objective

• To address the limitation in existing literature that requires bounded derivatives for spatial differentiability of SDE solutions.
• To identify weaker conditions than global monotonicity and bounded derivatives that still ensure spatial differentiability.
• To establish a framework applicable to SDEs arising in practical applications where coefficient functions often lack global monotonicity or bounded derivatives.
• To provide a theoretical foundation for using the Itô-Alekseev-Grobner formula in broader classes of SDEs.

Proposed method

• Introduces a local monotonicity condition on the drift and diffusion coefficients of the SDE as a key structural assumption.
• Imposes additional regularity conditions on the coefficients to ensure sufficient smoothness for differentiability.
• Applies stochastic calculus and a priori estimates to control the behavior of solutions in local neighborhoods.
• Uses a localization technique to extend local differentiability to global continuous differentiability under the proposed conditions.
• Relies on fixed-point arguments and comparison principles in the context of SDEs with irregular coefficients.
• Establishes differentiability through a bootstrap argument based on the regularity of the coefficient functions and the local monotonicity property.

Experimental results

Research questions

• RQ1 Under what conditions can SDEs with non-globally monotone coefficients still admit spatially differentiable solutions?
• RQ2 Can the requirement of bounded derivatives in existing differentiability results be relaxed for SDEs with irregular coefficients?
• RQ3 What structural properties of the coefficient functions ensure the existence of continuously differentiable solutions?
• RQ4 How can local monotonicity be leveraged to prove global differentiability of SDE solutions?
• RQ5 To what extent do the proposed conditions cover SDEs arising in practical applications?

Key findings

• The paper proves the existence of continuously differentiable solutions to SDEs under a local monotonicity condition on the coefficients, even when global monotonicity or bounded derivatives do not hold.
• The proposed conditions are satisfied by many SDEs from applications, extending the range of applicability of the Itô-Alekseev-Grobner formula.
• The differentiability result is established without requiring global Lipschitz or bounded derivative conditions, relaxing standard assumptions in the literature.
• The method relies on a novel combination of localization, a priori estimates, and fixed-point arguments tailored to irregular coefficients.
• The framework enables the analysis of SDEs with coefficients that are locally monotone but not globally, broadening the scope of differentiability theory.
• The result provides a theoretical basis for sensitivity analysis and control applications involving SDEs with non-smooth or non-globally monotone coefficients.