[Paper Review] Stochastic differential equations driven by G-Brownian motion and ordinary differential equations

Peng Luo, Falei Wang|arXiv (Cornell University)|Sep 20, 2013

Stochastic processes and financial applications10 references4 citations

TL;DR

This paper establishes a novel method to reduce stochastic differential equations (SDEs) driven by G-Brownian motion to parametrized ordinary differential equations (ODEs) via sample path analysis. By leveraging G-stochastic calculus, the authors prove that the solution of a G-SDE can be represented as a function of G-Brownian motion and a finite variation process, enabling the application of classical ODE theory and leading to a new comparison theorem for G-SDEs with necessary and sufficient conditions.

ABSTRACT

In this paper, we show that the integration of a stochastic differential equations driven by G-Brownian motion in R can be reduced to the integration of an ordinary differential equations parametrized by a variable in (Ω,F). We study the sample solutions of G-SDEs by an extention of G-Itô formula. And then we also get a comparison theorem for G-SDEs and its applications.

Motivation & Objective

• To establish a connection between G-SDEs and ODEs by analyzing sample solutions under the G-framework.
• To overcome the challenge of proving sample solutions belong to the space $ M^{2}_{G}(0,T) $ using G-stochastic calculus techniques.
• To derive a comparison theorem for G-SDEs and identify necessary and sufficient conditions for its validity.
• To provide a theoretical and computational framework that enables the use of classical ODE results in the analysis of G-SDEs.

Proposed method

• Represent the solution of a G-SDE as a function of G-Brownian motion and a finite variation process via sample path decomposition.
• Use G-stochastic calculus to prove that sample solutions belong to $ M^{2}_{G}(0,T) $, ensuring integrability and well-posedness.
• Apply the $ G $-Itô formula to derive pathwise dynamics of the solution, reducing the SDE to a family of ODEs indexed by $ \omega \in \Omega $.
• Utilize the $ G $-normal distribution and sublinear expectation structure to handle model uncertainty in the SDE dynamics.
• Establish comparison results by analyzing the ODE parametrization and deriving sufficient conditions on drift and diffusion coefficients.
• Leverage existing ODE theory (e.g., existence, uniqueness, comparison) to infer properties of the G-SDE solution.

Experimental results

Research questions

• RQ1 Can the solution of a G-SDE be represented as a solution to a family of ODEs indexed by sample paths in $ \Omega $?
• RQ2 Under what conditions does the sample solution of a G-SDE belong to the space $ M^{2}_{G}(0,T) $?
• RQ3 What is the necessary and sufficient condition for a comparison theorem to hold for G-SDEs?
• RQ4 How can classical ODE theory be applied to analyze G-SDEs under model uncertainty?
• RQ5 Can the dynamics of a G-SDE be decomposed into a G-Brownian motion and a finite variation process to simplify analysis?

Key findings

• The solution of a G-SDE can be represented as a function of G-Brownian motion and a finite variation process, enabling reduction to ODEs on each sample path.
• The sample solution of a G-SDE belongs to $ M^{2}_{G}(0,T) $, which is established via advanced G-stochastic calculus techniques.
• A new comparison theorem for G-SDEs is derived, with a necessary and sufficient condition involving drift and diffusion coefficients.
• The comparison condition $ b^1(x) - b^2(x) + 2G(h^1(x) - h^2(x)) \leq 0 $ is shown to be sufficient but not necessary for pathwise dominance.
• An asymptotic pathwise bound is obtained: $ |X_t| \leq C|B_t| + C\int_0^t e^{C|B_s|}ds + |X_0| $ for q.s. $ \omega $, under boundedness assumptions on coefficients.
• The method allows direct application of classical ODE results to G-SDEs, significantly simplifying theoretical and computational analysis.