We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a ψ-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224-247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings-Metropolis algorithms.
This paper studies the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of discrete-time block-monotone Markov chains under subgeometric drift conditions. The main result of this paper is to present an upper bound for the total variation distance between the stationary probability vectors of a block-monotone Markov chain and its LC-block-augmented truncation. The main result is extended to Markov chains that themselves may not be block monotone but are block-wise dominated by block-monotone Markov chains satisfying modified drift conditions. Fi…
We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed by Connor & Kendall (2007) concerning the existence of so-called 'tame' Markov chains. Furthermore, we show that our 'subsampled drift condition' implies the existence of finite moments for the return time to a small set.
In this note, we are concerned with the subgeometric rate of convergence of a Markov chain with discrete time parameter to its invariant measure in the $f$-norm. We clarify how three typical subgeometric rates of convergence are inherited under a discrete time version of Bochner's subordination. The crucial point is to establish the corresponding moment estimates for discrete time subordinators under some reasonable conditions on the underlying Bernstein function.
This paper deals with the ergodicity and the existence of a strong law of large numbers for adaptive Markov Chain Monte Carlo. We show that a diminishing adaptation assumption together with a drift condition for positive recurrence is enough to imply ergodicity. Strengthening the drift condition to a polynomial drift condition yields a strong law of large numbers for possibly unbounded functions. These results broaden considerably the class of adaptive MCMC algorithms for which rigorous analysis is now possible. As an example, we give a detailed analysis of the Adaptive Metropolis Algorithm of…
We establish subgeometric bounds on convergence rate of general Markov processes in the Wasserstein metric. In the discrete time setting we prove that the Lyapunov drift condition and the existence of a "good" $d$-small set imply subgeometric convergence to the invariant measure. In the continuous time setting we obtain the same convergence rate provided that there exists a "good" $d$-small set and the Douc-Fort-Guillin supermartingale condition holds. As an application of our results, we prove that the Veretennikov-Khasminskii condition is sufficient for subexponential convergence of strong s…
In this paper, we give quantitative bounds on the $f$-total variation distance from convergence of an Harris recurrent Markov chain on an arbitrary under drift and minorisation conditions implying ergodicity at a sub-geometric rate. These bounds are then specialized to the stochastically monotone case, covering the case where there is no minimal reachable element. The results are illustrated on two examples from queueing theory and Markov Chain Monte Carlo.
This paper studies limit theorems for Markov Chains with general state space under conditions which imply subgeometric ergodicity. We obtain a central limit theorem and moderate deviation principles for additive not necessarily bounded functional of the Markov chains under drift and minorization conditions which are weaker than the Foster-Lyapunov conditions. The regeneration-split chain method and a precise control of the modulated moment of the hitting time to small sets are employed in the proof.
This paper studies the subgeometric convergence of the stationary distribution in taking the infinite-level limit of a finite-level M/G/1-type Markov chain, that is, in letting the upper boundary level go to infinity. This study is performed through the fundamental deviation matrix, which is a block-decomposition-friendly solution for the Poisson equation of the deviation matrix. The fundamental deviation matrix yields a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial rol…
We derive sufficient conditions for subgeometric f-ergodicity of strongly Markovian processes. We first propose a criterion based on modulated moment of some delayed return-time to a petite set. We then formulate a criterion for polynomial f-ergodicity in terms of a drift condition on the generator. Applications to specific processes are considered, including Langevin tempered diffusions on R^n and storage models.
This paper considers the level-increment (LI) truncation approximation of M/G/1-type Markov chains. The LI truncation approximation is useful for implementing the M/G/1 paradigm, which is the framework for computing the stationary distribution of M/G/1-type Markov chains. The main result of this paper is a subgeometric convergence formula for the total variation distance between the original stationary distribution and its LI truncation approximation. Suppose that the equilibrium level-increment distribution is subexponential, and that the downward transition matrix is rank one. We then show t…
We are interested in the rate of convergence of a subordinate Markov process to its invariant measure. Given a subordinator and the corresponding Bernstein function (Laplace exponent) we characterize the convergence rate of the subordinate Markov process; the key ingredients are the rate of convergence of the original process and the (inverse of the) Bernstein function. At a technical level, the crucial point is to bound three types of moments (sub-exponential, algebraic and logarithmic) for subordinators as time $t$ tends to infinity. At the end we discuss some concrete models and we show tha…
We provide simple and constructive proofs of Harris-type theorems on the existence and uniqueness of an equilibrium and the speed of equilibration of discrete-time and continuous-time stochastic semigroups. Our results apply both to cases where the relaxation speed is exponential (also called geometric) and to those with no spectral gap, with non-exponential speeds (also called subgeometric). We give constructive estimates in the subgeometric case and discrete-time statements which seem both to be new. The method of proof also differs from previous works, based on semigroup and interpolation a…
In this paper, we provide sufficient conditions for the existence of the invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible. Compared to previous work, our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation. Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditio…
We extend the hypocoercivity framework for piecewise-deterministic Markov process (PDMP) Monte Carlo established in [Andrieu et. al. (2018)] to heavy-tailed target distributions, which exhibit subgeometric rates of convergence to equilibrium. We make use of weak Poincaré inequalities, as developed in the work of [Grothaus and Wang (2019)], the ideas of which we adapt to the PDMPs of interest. On the way we report largely potential-independent approaches to bounding explicitly solutions of the Poisson equation of the Langevin diffusion and its first and second derivatives, required here to cont…
We prove existence and uniqueness of a stationary distribution and absolute regularity for nonlinear GARCH and INGARCH models of order (p,q). In contrast to previous work we impose, besides a geometric drift condition, only a semi-contractive condition which allows us to include models which would be ruled out by a fully contractive condition. This results in a subgeometric rather than the more usual geometric decay rate of the mixing coefficients. The proofs are heavily based on a coupling of two versions of the processes.We prove existence and uniqueness of a stationary distribution and abso…
Many applications in networked control require intermittent access of a controller to a system, as in event-triggered systems or information constrained control applications. Motivated by such applications and extending previous work on Lyapunov-theoretic drift criteria, we establish both subgeometric and geometric rates of convergence for Markov chains under state dependent random time drift criteria. We quantify how the rate of ergodicity, nature of Lyapunov functions, their drift properties, and the distributions of stopping times are related. We finally study an application in networked co…
Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Ma…
Bounds on convergence rate to the invariant distribution for a class of stochastic differential equations (SDEs) with a gradient-type drift are obtained.
In this paper, we investigate the weak convergence rate of Euler-Maruyama's approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.
Applying Zvonkin's transform, the exponential convergence in Wasserstein distance for a class of functional SDEs with Hölder continuous drift is obtained. This combining with log-Harnack inequality implies the same convergence in the sense of entropy, which also yields the convergence in total variation norm by Pinsker's inequality.
We study the convergence of a drift implicit scheme for one-dimensional SDEs that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under general conditions, we obtain a strong convergence of order 1. In the CIR case, Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of order 1/2 for this scheme. Here, we obtain a strong convergence of order 1 under more restrictive assumptions on the CIR parameters.
This paper is concerned with strong convergence of a tamed $θ$-Euler-Maruyama scheme for neutral stochastic differential delay equations with superlinearly growing coefficients. We not only prove the strong convergence of implicit schemes, but also reveal the convergence rate for these equations driven by Brownian motion and pure jumps, respectively.
Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.
We consider SDEs with bounded and $α$-Hölder continuous drift, with $α\in (0,1)$, driven by multiplicative noise. We show that under sufficient conditions on the diffusion matrix, which guarantee the existence of a unique strong solution, the weak rate of convergence for the Euler-Maruyama scheme is almost $(1+α)/2$. The present paper forms part of the author's master's thesis.
The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.
This paper deals with the rate of convergence in 1-Wasserstein distance of the marginal law of a Brownian motion with drift conditioned not to have reached 0 towards the Yaglom limit of the process. In particular it is shown that, for a wide class of initial measures including Dirac measures and probability measures with compact support, the Wasserstein distance decays asymptotically as 1/t. Likewise, this speed of convergence is recovered for the convergence of marginal laws conditioned not to be absorbed up to a horizon time towards the Bessel-$3$ process, when the horizon time tends to infi…
In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale $\eps$ goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on $p$ respectively.
We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $ρ$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $ρ\ge d$, the order of convergence at…
Let $(X_n)_{n=0}^\infty$ denote a Markov chain on a Polish space that has a stationary distribution $\varpi$. This article concerns upper bounds on the Wasserstein distance between the distribution of $X_n$ and $\varpi$. In particular, an explicit geometric bound on the distance to stationarity is derived using generalized drift and contraction conditions whose parameters vary across the state space. These new types of drift and contraction allow for sharper convergence bounds than the standard versions, whose parameters are constant. Application of the result is illustrated in the context of …
In this paper, exploiting the regularities of the corresponding Kolmogorov equations involved we investigate strong convergence of exponential integrator scheme for a range of stochastic partial differential equations, in which the drift term is Hölder continuous, and reveal the rate of convergence.
Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.
In this paper, we consider a fixed delay Cox-Ingersoll-Ross process (CIR process) on the regime where it does not hit zero, the aim is to determine a positive preserving implicit Euler Scheme. On a time grid with constant stepsize our scheme extends the scheme proposed by Alfonsi in 2005 for the classical CIR model. Furthermore, we consider its piecewise linear interpolation, and, under suitable conditions, we establish the order of strong convergence in the uniform norm, thus extending the results of Dereich et al. in 2011.
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q -- L $ρ$ drift coefficient when the condition d $ρ$ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 -- (d $ρ$ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove …
In this paper, we consider a numerical approximation of the stochastic differential equation (SDE) $$X_{t}=x_{0}+ \int_{0}^{t} b(s, X_{s}) \mathrm{d}s + L_{t},~x_{0} \in \mathbb{R}^{d},~t \in [0,T],$$ where the drift coefficient $b:[0,T] \times \mathbb{R}^d \to \mathbb{R}^d$ is Hölder continuous in both time and space variables and the noise $L=(L_t)_{0 \leq t \leq T}$ is a $d$-dimensional Lévy process. We provide the rate of convergence for the Euler-Maruyama approximation when $L$ is a Wiener process or a truncated symmetric $α$-stable process with $α\in (1,2)$. Our technique is based on the…
We consider the stability problems of one dimensional SDEs when the diffusion coefficients satisfy the so called Nakao-Le Gall condition. The explicit rate of convergence of the stability problems are given by the Yamada-Watanabe method without the drifts. We also discuss the convergence rate for the SDEs driven by the symmetric $α$ stable process. These stability rate problems are extended to the case where the drift coefficients are bounded and in $L^1$. It is shown that the convergence rate is invariant under the removal of drift method for the SDEs driven by the Wiener process.
In this paper we study the existence and uniqueness of the random periodic solution for a stochastic differential equation with a one-sided Lipschitz condition (also known as monotonicity condition) and the convergence of its numerical approximation via the backward Euler-Maruyama method. The existence of the random periodic solution is shown as the limits of the pull-back flows of the SDE and discretized SDE respectively. We establish a convergence rate of the strong error for the backward Euler-Maruyama method and obtain the weak convergence result for the approximation of the periodic measu…
This work establishes the weak convergence of Euler-Maruyama's approximation for stochastic differential equations (SDEs) with singular drifts under the integrability condition in lieu of the widely used growth condition. This method is based on a skillful application of the dimension-free Harnack inequality. Moreover, when the drifts satisfy certain regularity conditions, the convergence rate is estimated. This method is also applicable when the diffusion coefficients are degenerate. A stochastic damping Hamiltonian system is studied as an illustrative example.
By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in[1,\infty)$. In particular, we show that for the Itô stochastic differential equation $$\d X_t=\d B_t+b(X_t)\,\d t,$$ if the drift term $b$ satisfies that for any $x,y\in\R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} K_1|x-y|^2,& |x-y|\le L; -K_2|x-y|^2,& |x-y|> L \end{cases}$$ holds with some positive constants $K_1$, $K_2$ and $L>0$,…
In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of $0$. We show the existence, uniqueness and positivity of the solutions. The estimations of moments, including the negative power moments, are given. Based on these estimations, strong convergence of the positivity preserving drift-implicit Euler-type scheme is proved, and optimal convergence rate is obtained. By using Lamperti transformation, we show th…
Quantitative QSD convergence in 1-Wasserstein distance via the Föllmer drift
Centro de Modelamiento Matemático, Universidad de Chile（智利大学数学建模中心）；IRL 2807 - CNRS（法国国家科学研究中心第2807国际研究实验室）；Department of Mathematics, UCLA（加州大学洛杉矶分校数学系）
In this paper, we are concerned with convergence rate of Euler-Maruyama scheme for stochastic differential equations with rough coefficients. The key contributions lie in (i), by means of regularity of non-degenerate Kolmogrov equation, we investigate convergence rate of Euler-Maruyama scheme for a class of stochastic differential equations, which allow the drifts to be Dini-continuous and unbounded; (ii) by the aid of regularization properties of degenerate Kolmogrov equation, we discuss convergence rate of Euler-Maruyama scheme for a range of degenerate stochastic differential equations, whe…
In this paper we develop via Girsanov's transformation a perturbation argument to investigate weak convergence of Euler-Maruyama (EM) scheme for path-dependent SDEs with Hölder continuous drifts. This approach is available to other scenarios, e.g., truncated EM schemes for non-degenerate SDEs with finite memory or infinite memory. Also, such trick can be applied to study weak convergence of truncated EM scheme for a range of stochastic Hamiltonian systems with irregular coefficients and with memory, which are typical degenerate dynamical systems. Moreover, the weak convergence of path-dependen…
In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler-Maruyama scheme for such a class of SDEs. We prove convergence of order $1/2$. Finally, we present numerical examples.
In this note we propose an exact simulation algorithm for the solution of dX_t=dW_t+b(X_t)dt (1) where b is a smooth real function except at point 0 where b(0+)\neq b(0-). The main idea is to sample an exact skeleton of X using an algorithm deduced from the convergence of the solutions of the skew perturbed equation dX^β_t=dW_t+b(X^β_t)dt + βdL^0_t {X^β} (2) towards X solution of (1) as βtends to 0. In this note, we show that this convergence induces the convergence of exact simulation algorithms proposed by the authors in \cite{etoremartinez1} for the solutions of (2) towards a limit algorith…
This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is Hölder continuous.
This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform …
We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence …
