[Paper Review] A note on Harnack and Transportation inequalities For Stochastic Differential Equations with reflections

Brahim Boufoussi, Soufiane Mouchtabih|arXiv (Cornell University)|May 3, 2019

Stochastic processes and financial applicationsEconomics, Econometrics and Finance13 references3 citations

TL;DR

This paper establishes transportation cost inequalities and Harnack inequalities for reflected stochastic differential equations (RSDEs) and stochastic differential equations with local times (SDELs). Using Girsanov transformation and coupling methods, it derives $T_1(C)$ and $T_2(C)$ inequalities with respect to uniform and $L^2$ metrics, leading to concentration of measure and exponential ergodicity. A key result is a log-Harnack inequality for the associated semigroups under mild regularity conditions on coefficients.

ABSTRACT

We establish transportation cost inequalities, with respect to the uniform and $L_2$-metric, on the path space of continuous functions, for laws of solutions of stochastic differential equations with reflections. We also consider the case of stochastic differential equations involving local times. Harnack inequalities for the associated semigroups are also established.

Motivation & Objective

• To derive $T_1(C)$ and $T_2(C)$ transportation inequalities for laws of solutions to reflected SDEs with respect to uniform and $L^2$ metrics.
• To establish log-Harnack and Harnack inequalities for the semigroups associated with reflected and local time SDEs.
• To analyze small and large time asymptotics of SDEs with local times using Wasserstein distance estimates.
• To prove exponential convergence in Wasserstein distance to the invariant measure under suitable conditions.
• To provide concentration inequalities for additive functionals of the solution using the derived transportation inequalities.

Proposed method

• Employing Girsanov transformation to derive transportation cost inequalities for the law of reflected SDEs on path space.
• Applying coupling techniques to establish log-Harnack and Harnack inequalities for the semigroup $P_t f(x) = \mathbb{E}[f(X^x(t))]$.
• Transforming SDEs with local times into ordinary SDEs via a change of measure and applying known results from Wang (2011) on Harnack inequalities.
• Using the Wasserstein distance $W_p^d(\mu, \nu)$ and relative entropy $H(\nu/\mu)$ to define and verify $T_p(C)$ inequalities.
• Applying Djellout et al. (2004) results to prove exponential convergence of the transition kernel to the invariant measure in $W_2$ distance.
• Deriving concentration inequalities via the $T_1(C)$ inequality, such as Hoeffding-type bounds for time averages and suprema of the process.

Experimental results

Research questions

• RQ1 Does the law of a reflected SDE satisfy a transportation inequality with respect to the uniform and $L^2$ metrics on path space?
• RQ2 Can log-Harnack and Harnack inequalities be established for the semigroup of reflected SDEs under mild regularity conditions on drift and diffusion coefficients?
• RQ3 What is the rate of convergence to the invariant measure for SDEs with local times in the Wasserstein $W_2$ distance?
• RQ4 How do transportation inequalities for SDEs with local times lead to concentration of measure for additive functionals?
• RQ5 What are the small and large time asymptotic behaviors of the solution to SDEs with local times, as captured by Wasserstein estimates?

Key findings

• The law of the solution to a reflected SDE satisfies $T_1(C)$ and $T_2(C)$ inequalities with respect to the uniform and $L^2$ metrics, implying concentration of measure.
• For any Lipschitz functional $F_V$ on path space, the concentration inequality $\mathbb{P}\big(\frac{1}{T}\int_0^T V(X(t)) dt - \mathbb{E}[V(X(t))] > r\big) \leq \exp\big(-\frac{r^2}{2C\alpha^2}\big)$ holds.
• For the supremum functional $F_\infty$, the concentration bound $\mathbb{P}\big(\sup_{t\in[0,T]}|X(t)-x| - \mathbb{E}[\sup_{t\in[0,T]}|X(t)-x|] > r\big) \leq \exp\big(-\frac{r^2}{2C}\big)$ is established.
• Under conditions $H(7)$ and $H(8)$, the semigroup satisfies the Harnack inequality: $(P_T f(y))^p \leq (P_T f^p(x)) \exp\big[\cdots\big]$ for $p > (1 + \gamma/\lambda)^2$, with explicit dependence on $|x-y|^2$.
• The Wasserstein distance between the transition kernel and the invariant measure decays exponentially: $W_2(P_t(x,\cdot), \mu) \leq \frac{M}{m} e^{-\delta t} \big(\int |x-y|^2 \mu(dy)\big)^{1/2}$.
• For a specific SDE with $\sigma(x) = I_{x<0} + \frac{1+\beta}{1-\beta}I_{x\geq0}$, $b(x) = -\delta x \sigma(x)$, and $\nu = \beta \delta_0$, the transformed process is an Ornstein-Uhlenbeck process, and all results apply.