[Paper Review] Large deviations for interacting Bessel-like processes and applications to systemic risk

Tomoyuki Ichiba, Mykhaylo Shkolnikov|arXiv (Cornell University)|Mar 12, 2013

Stochastic processes and financial applications6 references3 citations

TL;DR

This paper establishes a large deviation principle for systems of interacting Bessel-like diffusion processes with non-Lipschitz and degenerate diffusion coefficients, proving a hydrodynamic limit and propagation of chaos via a non-local McKean-Vlasov SDE. The key contribution is the first such result under explicit coefficient conditions, with applications to systemic risk in interbank lending systems.

ABSTRACT

We establish a process level large deviation principle for systems of interacting Bessel-like diffusion processes. By establishing weak uniqueness for the limiting non-local SDE of McKean-Vlasov type, we conclude that the latter describes the process level hydrodynamic limit of such systems and obtain a propagation of chaos result. This is the first instance of results of this type in the context of interacting diffusion processes under explicit assumptions on the coefficients, where the diffusion coefficients are allowed to be both non-Lipschitz and degenerate. In the second part of the paper, we explain how systems of this type naturally arise in the study of stability of the interbank lending system and describe some financial implications of our results.

Motivation & Objective

• To establish a process-level large deviation principle for interacting diffusion processes with degenerate, non-Lipschitz diffusion coefficients.
• To prove weak uniqueness and hydrodynamic limit for a non-local McKean-Vlasov SDE arising as the limit of finite systems.
• To demonstrate propagation of chaos in the mean-field limit under explicit assumptions on drift and diffusion coefficients.
• To apply the results to model systemic risk in interbank lending systems using Bessel-type dynamics.
• To analyze the long-time behavior and stationary distributions of the limiting SDEs, including Gamma-type stationary laws.

Proposed method

• Use of a process-level large deviation principle with scale $ n $, where the rate function is defined via a variational formula over control processes and Brownian motion.
• Application of weak uniqueness for the limiting non-local SDE of McKean-Vlasov type, ensuring the hydrodynamic limit is well-defined.
• Employment of Itô’s formula and moment estimates to derive ODEs for the first and second moments of the process.
• Derivation of a linear first-order PDE for the Laplace transform of the law of the process, enabling analysis of the time evolution of distributions.
• Use of the Fokker-Planck equation and Laplace transform techniques to characterize the stationary distribution as a Gamma law under specific coupling conditions.
• Analysis of the variance dynamics via an ODE driven by the interaction function $ \varphi $, showing initial variance growth under weak interaction.

Experimental results

Research questions

• RQ1 Under what conditions does a system of interacting Bessel-like diffusions satisfy a large deviation principle?
• RQ2 How does the hydrodynamic limit of such systems behave, and what is the nature of the limiting non-local SDE?
• RQ3 What conditions ensure weak uniqueness and propagation of chaos in systems with degenerate and non-Lipschitz diffusion coefficients?
• RQ4 How do the interaction function $ \varphi $ and initial conditions affect the long-time behavior and stationary distribution?
• RQ5 What are the financial implications of this framework for modeling systemic risk in interbank lending systems?

Key findings

• The sequence of empirical measures $ \rho^n $ satisfies a large deviation principle on the space $ \mathcal{X} $ with rate function $ J(\gamma) = \frac{1}{2} \inf_{(u,W)} \mathbb{E}\left[\int_0^T u(t)^2 dt\right] $, where the infimum is over controls that yield $ \gamma $ as law of a controlled SDE.
• The hydrodynamic limit is characterized by the unique strong solution of the non-local SDE $ dX(t) = b(X(t), \mathcal{L}(X(t))) dt + \sigma(X(t)) dW(t) $, with $ \sigma(x) = \sqrt{x} g(x) $, $ g $ continuous, bounded, and strictly positive.
• Propagation of chaos holds: the joint law of any fixed number of particles converges to the product of independent solutions of the mean-field SDE.
• The variance $ V(t) $ of the process increases initially when interaction is weak, governed by the ODE $ \frac{dV}{dt} = 4m_\lambda - 2\varphi(\mathcal{L}(X(t))) V(t) $, with $ V(0) = 0 $.
• Under recurrence, the stationary distribution is a Gamma distribution with shape $ \mathfrak{a} = 2/\varphi^* $ and rate $ \mathfrak{b} = \varphi^* m_\lambda / 2 $, where $ \varphi^* = \varphi(\alpha) $.
• The stationary law satisfies the Laplace transform equation $ (\varphi^* + 2x) \frac{d\mathfrak{u}}{dx} + \varphi^* m_\lambda \mathfrak{u} = 0 $, yielding $ \mathfrak{u}(x) = (1 + (2/\varphi^*)x)^{-\varphi^* m_\lambda / 2} $.