[Paper Review] Random Perturbation of Some Multi-dimensional Non-Lipschitz Ordinary Differential Equations

Liangquan Zhang|arXiv (Cornell University)|Feb 19, 2012

Stochastic processes and financial applications13 references3 citations

TL;DR

This paper studies the effect of small random perturbations on multi-dimensional ordinary differential equations (ODEs) with non-Lipschitz drift, particularly when the drift has an isolated zero. It shows that adding a small Brownian noise leads to a unique strong solution in the SDE formulation, and as noise intensity vanishes, only certain solutions of the original ODE emerge as limits—specifically, those that are selected by the stochastic perturbation.

ABSTRACT

For multi-dimensional ODE,there is a general local existence theory if the right hand side is only supposed to be continuous (Peano's Theorem), even though uniqueness may lost in this case. However, when adding a small random noise (standard Brownian motion), one can obtain a perturbed SDE which admits a unique strong solution even when the drift is assumed to be continuous and bounded. Moreover, when the noise intensity tends to zero, the solutions to the perturbed SDEs converge, in a suitable sense, to the solutions of the ODE. If the drift has an isolated zero and is non Lipschitz continuous at zero, the ODE may have infinitely many solutions. Our main result is to characterize which solutions of the ODE can be the limits of the solutions of the perturbed SDEs.

Motivation & Objective

• To understand the selection mechanism of solutions in multi-dimensional ODEs with non-Lipschitz drift, especially when uniqueness fails.
• To investigate the behavior of solutions to stochastic differential equations (SDEs) with small noise when the drift is continuous but not Lipschitz.
• To characterize which solutions of the original ODE can arise as limits of solutions to the perturbed SDEs as noise intensity tends to zero.
• To resolve the ambiguity in solution selection for ODEs with isolated zeros in the drift by introducing stochastic perturbations.

Proposed method

• Use of Peano's Theorem to establish local existence of solutions for the ODE with continuous drift.
• Introduction of a small random noise (standard Brownian motion) to the ODE, transforming it into a stochastic differential equation (SDE).
• Application of strong solution theory for SDEs to guarantee uniqueness of solutions under the perturbed dynamics.
• Analysis of the limit behavior of SDE solutions as the noise intensity approaches zero, using convergence in distribution or pathwise convergence.
• Identification of the limiting ODE solutions via the stochastic perturbation mechanism, focusing on the role of the isolated zero in the drift.
• Use of probabilistic techniques to characterize the set of ODE solutions that can be selected by the noise-induced dynamics.

Experimental results

Research questions

• RQ1 Which solutions of a non-Lipschitz ODE with an isolated zero in the drift can arise as limits of solutions to perturbed SDEs as noise intensity vanishes?
• RQ2 How does the addition of small random noise restore uniqueness in ODEs where uniqueness fails due to non-Lipschitz continuity?
• RQ3 What is the relationship between the structure of the drift near its isolated zero and the selection of limiting solutions?
• RQ4 Can the stochastic perturbation mechanism preferentially select specific solutions over others in the presence of infinitely many ODE solutions?
• RQ5 Under what conditions does the solution of the perturbed SDE converge to a solution of the original ODE in the zero-noise limit?

Key findings

• The perturbed SDE admits a unique strong solution even when the drift is only continuous and bounded, due to the noise-induced regularization.
• As the noise intensity tends to zero, the solutions of the SDE converge to a specific subset of solutions of the original ODE.
• Only solutions that are 'stochastically selected' by the noise mechanism can emerge as limits; not all ODE solutions are accessible this way.
• The selection mechanism depends critically on the behavior of the drift near its isolated zero, particularly its non-Lipschitz nature.
• The limiting solutions are characterized by the stability properties induced by the stochastic perturbation, favoring certain trajectories over others.