[Paper Review] Stability of densities for perturbed Diffusions and Markov Chains

Valentin Konakov, Anna Kozhina+1 more|arXiv (Cornell University)|Jun 29, 2015

Advanced Mathematical Modeling in Engineering31 references3 citations

TL;DR

This paper establishes quantitative stability bounds for the transition densities of non-degenerate diffusion processes and Markov chains under perturbations of their coefficients. Using the parametrix method and Hölder continuity assumptions, it derives explicit error estimates in terms of coefficient differences, with convergence rates depending on the regularity of drift and diffusion coefficients, particularly under Hölder or polynomial decay conditions.

ABSTRACT

We are interested in studying the sensitivity of diffusion processes or their approximations by Markov Chains with respect to a perturbation of the coefficients.

Motivation & Objective

• To quantify the sensitivity of transition densities of non-degenerate diffusions and Markov chains to perturbations in drift and diffusion coefficients.
• To provide explicit error bounds for the difference between the true and perturbed densities under minimal regularity assumptions on coefficients.
• To extend stability results to discrete-time Markov chain approximations of SDEs, particularly in the context of Euler scheme weak error analysis.
• To analyze the impact of coefficient perturbations in applications such as model misspecification, parameter estimation, and numerical scheme convergence.
• To establish convergence rates for the density difference under Hölder continuity and polynomial decay assumptions, with explicit dependence on the perturbation size and regularity parameters.

Proposed method

• Employs the parametrix method to represent the transition densities of diffusions and Markov chains via series expansions involving Gaussian kernels and iterated integrals.
• Uses the McKean–Singer approach to the parametrix expansion, which is robust to lack of time smoothness and adaptable to discrete-time Markov chains.
• Derives bounds on the difference of densities by analyzing the perturbation of the parametrix series terms, particularly focusing on second- and third-order derivatives.
• Applies Hölder inequalities and Gaussian-type bounds to control the difference in drift and diffusion coefficients, especially under Hölder continuity assumptions.
• Introduces a convolution analysis for densities with polynomial decay (e.g., $ q_m(t,x) riangleq c^d (t)^{-d/2} Q_{M-(d+5+ u)}(x / t^{1/2}) $) to handle off-diagonal and diagonal regimes in the convolution integrals.
• Uses induction and Gamma function identities to control the growth of iterated convolutions in the parametrix expansion, leading to explicit convergence rates in terms of $ au^{rac{r heta}{2}} $.

Experimental results

Research questions

• RQ1 How does the transition density of a non-degenerate diffusion process change under small perturbations of its drift and diffusion coefficients?
• RQ2 What is the rate of convergence of the density of a Markov chain approximation to the true diffusion density when coefficients are perturbed?
• RQ3 How do Hölder continuity and polynomial decay conditions on the coefficients affect the stability of the density under perturbation?
• RQ4 Can the parametrix method be used to derive explicit error bounds for the weak error of the Euler scheme with irregular coefficients?
• RQ5 What is the role of higher-order derivatives in the density difference, and how is the concentration loss controlled in the parametrix expansion?

Key findings

• Under Hölder continuity of coefficients, the difference between the true and perturbed densities is bounded by $ rac{ ilde{ ho}_{ ext{sup}}}{(t_j - t_k)^{1 - rac{ u}{2}}} $, where $ ilde{ ho}_{ ext{sup}} $ captures the sup-norm of coefficient differences.
• For the Markov chain approximation, the density difference satisfies $ | ilde{p}^h oxtimes H^{h,(r)} - ilde{p}_ u^h oxtimes H_ u^{h,(r)}| riangleq ext{error} riangleq (r+1) ho_{ u, u, ext{sup}} rac{igracevert (1 igvee T^{(1- u)/2})c_1 igracevert^{r+1} [ ext{B}( u/2)]^r}{ ext{B}(1 + r u/2)} rac{c^d}{(t_j - t_i)^{d/2}} Q_{M-(d+5+ u)}ig( rac{y-x}{(t_j - t_i)^{1/2}/c} ig) (t_j - t_i)^{r u/2} $, showing explicit convergence in $ (t_j - t_i)^{r u/2} $.
• The analysis of convolutions of polynomially decaying densities shows that $ I_{t_k}^1(t_i,t_j,x,y) riangleq ig| ig( q_m(t_k - t_i, ullet - x) * q_m(t_j - t_k, ullet - y) ig)(y) ig| riangleq ext{convolution} riangleq ar{c} q_m(t_j - t_i, y - x) $, uniformly in $ i < k < j $, under $ m > d $.
• The method controls the concentration loss in the parametrix expansion by splitting the domain and integrating $ |w|^3 $ and $ |w|^4 $ terms, which leads to the need for $ m > d+5+ u $ to ensure integrability.
• The convergence rate of the density difference is $ O((t_j - t_i)^{r u/2}) $, which is optimal under the assumed Hölder regularity $ u $, and the bound is uniform in space and time.
• The results are robust to coefficient perturbations in the sense of $ ho_{ u, u, ext{sup}} $, which measures the sup-norm of the difference between $ (b, u) $ and $ (b_ u, u_ u) $, and the bound scales linearly with this perturbation.