We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Edgeworth type expansions of third order for transition densities are proved. This is done for time horizons that converge to 0. For this purpose we represent the transition density as a functional of densities of sums of i.i.d. variables. This will be done by application of the parametrix method. Then we apply Edgeworth expansions to the densities. The resulting series gives our Edgeworth-type expansion for the transition density of Markov chains. The research is motivated by applications to high frequency data that are available on a very fine grid but are approximated by a diffusion model on a more rough grid.
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition densities are proved. The observation time [0,T] may be fixed or lim n T = 0, where nh = T and h is a mesh between two neighboring observation points.
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points.
We are interested in studying the sensitivity of diffusion processes or their approximations by Markov Chains with respect to a perturbation of the coefficients.
The Markov chain approximation of a one-dimensional symmetric diffusion is investigated in this paper. Given an irreducible reflecting diffusion on a closed interval with scale function $s$ and speed measure $m$, the approximating Markov chains are constructed explicitly through the trace of the Dirichlet form corresponding to the diffusion. One feature of our approach is that it does not require uniform ellipticity on diffusion coefficient of the limit object or uniform regularity on conductances of the approximative Markov chains, as imposed usually in the previous related works.
In this paper we present a novel method for estimating the parameters of a parametric diffusion processes. Our approach is based on a closed-form Maximum Likelihood estimator for an approximating Continuous Time Markov Chain (CTMC) of the diffusion process. Unlike typical time discretization approaches, such as psuedo-likelihood approximations with Shoji-Ozaki or Kessler's method, the CTMC approximation introduces no time-discretization error during parameter estimation, and is thus well-suited for typical econometric situations with infrequently sampled data. Due to the structure of the CTMC,…
Is studied asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. The view of regular and singular parts of solution is found.
In this paper, we study small noise asymptotics of Markov-modulated diffusion processes in the regime that the modulating Markov chain is rapidly switching. We prove the joint sample-path large deviations principle for the Markov-modulated diffusion process and the occupation measure of the Markov chain (which evidently also yields the large deviations principle for each of them separately by applying the contraction principle). The structure of the proof is such that we first prove exponential tightness, and then establish a local large deviations principle (where the latter part is split int…
In this article we consider a family of real-valued diffusion processes on the time interval $[0,1]$ indexed by their prescribed initial value $x \in \mathbb{R}$ and another point in space, $y \in \mathbb{R}$. We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in $y$ at time $t=1$. Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an Itô diffusion? We eventually illustrate our precise answer with several examples.
The goal of the paper is to describe the large time behaviour of a Markov process associated with a symmetric diffusion in a high-contrast random environment and to characterize the limit semigroup and the limit process under the diffusive scaling.
We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions. We discuss examples and formulate open problems.
Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been proposed to approximate the underlying Markov chain or diffusion. A fundamental question is how the computational savings trade off against the statistical error incurred due to approximations. This paper develops general results that address this question. We bound the Wasserstei…
In this paper we study limit behavior for a Markov-modulated (MM) binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obta…
We study the rate of weak convergence of Markov chains to diffusion processes under suitable but quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the speed of convergence of this hybrid approach and we provide an example in finance, applying our results to a tree-finite difference approximation in the Heston or Bates m…
In this paper, we study the asymptotic of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in …
In a companion paper, the authors have characterized all deterministic semigroups, and all Markov semigroups, whose trajectories are Carathe'odory solutions to a given ODE x'=f(x), with f possibly discontinuous. The present paper establishes two approximation results. Namely, every deterministic semigroup can be obtained as the pointwise limit of the flows generated by a sequence of ODEs $x'=f_n(x) with smooth right hand sides. Moreover, every Markov semigroup can be obtained as limit of a sequence of diffusion processes with smooth drifts and with diffusion coefficients approaching zero.
In this note, we propose two different approaches to rigorously justify a pseudo-Markov property for controlled diffusion processes which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. The first approach develops a sketch of proof proposed by Fleming and Souganidis~\cite{fleming-souganidis}. The second approach is based on an enlargement of the original state space and a controlled martingale problem. We clarify some measurability and topological issues raised by these two approaches.
We propose a discrete time discrete space Markov chain approximation with a Brownian bridge correction for computing curvilinear boundary crossing probabilities of a general diffusion process on a finite time interval. For broad classes of curvilinear boundaries and diffusion processes, we prove the convergence of the constructed approximations in the form of products of the respective substochastic matrices to the boundary crossing probabilities for the process as the time grid used to construct the Markov chains is getting finer. Numerical results indicate that the convergence rate for the p…
This paper demonstrates a lower and upper solution method to investigate the asymptotic behaviour of the conservative reaction-diffusion systems associated with Markovian process algebra models. In particular, we have proved the uniform convergence of the solution to its constant equilibrium for a case study as time tends to infinity, together with experimental results illustrations.
For a wide class of continuous-time Markov processes, including all irreducible hypoelliptic diffusions evolving on an open, connected subset of $\RL^d$, the following are shown to be equivalent: (i) The process satisfies (a slightly weaker version of) the classical Donsker-Varadhan conditions; (ii) The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in a strong sense in terms of an associated operator norm; (iii) The resolvent kernel of the process is `$v$-separable', that is, it can be approximated arbitrarily well in operator norm by finite-ran…
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. In Part I of this work, by applying a…
We consider a diffusion given by a small noise perturbation of a dynamical system driven by a potential function with a finite number of local minima. The classical results of Freidlin and Wentzell show that the time this diffusion spends in the domain of attraction of one of these local minima is approximately exponentially distributed and hence the diffusion should behave approximately like a Markov chain on the local minima. By the work of Bovier and collaborators, the local minima can be associated with the small eigenvalues of the diffusion generator. Applying a Markov mapping theorem, we…
Let $Y$ be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite state Markov process $X$: $dY_t=-λ(X_t)Y_tdt+σ(X_t)dB_t$, $Y_0$ given. Under ergodicity condition, we get quantitative estimates for the long time behavior of $Y$. We also establish a trichotomy for the tail of the stationary distribution of $Y$: it can be heavy (only some moments are finite), exponential-like (only some exponential moments are finite) or Gaussian-like (its Laplace transform is bounded below and above by Gaussian ones). The critical moments are characterized by the parameters of the model.
This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [32], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [45], which considered the same problem for metastable reversible diffusion processes. Our proof …
"Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schrödinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit o…
Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.
This paper provides a general and abstract approach to approximate ergodic regimes of Markov and Feller processes. More precisely, we show that the recursive algorithm presented in Lamberton & Pages (2002) and based on simulation algorithms of stochastic schemes with decreasing step can be used to build invariant measures for general Markov and Feller processes. We also propose applications in three different configurations: Approximation of Markov switching Brownian diffusion ergodic regimes using Euler scheme, approximation of Markov Brownian diffusion ergodic regimes with Milstein scheme an…
Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.
Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such "limitless" approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter $n$ is taken large,…
Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dY_t=a(X_t)Y_t dt+σ(X_t) dW_t, Y_0=y_0. Ergodicity conditions for Y have been obtained. Here we investigate the tail propriety of the stationary distribution of this model. A characterization of either heavy or light tail case is established. The method is based on a renewal theorem for systems of equations with distributions on R.
Motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics, we study existence, uniqueness and the strong Markov property of weak solutions to a class of degenerate stochastic differential equations. The stochastic differential equations considered in our article admit solutions supported in the set $[0,\infty)^n\times\mathbb{R}^m$, and they are degenerate in the sense that the diffusion matrix is not strictly elliptic, as the smallest eigenvalue converges to zero proportional to the distance to the boundary of the domain, and …
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schrödinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, weaker than irreducibility of the quantum Markov semigroup, neve…
We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretised approximations of diffusions, such as the Euler--Maruyama scheme. Our approach is based on particle marginal Metropolis--Hastings, a particle filter, randomised multilevel Monte Carlo, and importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a b…
Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a combination of the two. Motivated by models with multiple time-scales arising in systems biology, we present a general approach to proving a central limit theorem capturing the fluctuations of the original model around the deterministic limit. The central limit theorem provides a method for deriving an appropriate diffusion (Langevin) approximation.
In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under some "curvature" assumptions. Our notion of curvature coincides with the usual $Γ_2$ curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results …
Consider a non-symmetric generalized diffusion $X(\cdot)$ in ${\bbR}^d$ determined by the differential operator $A(\msx)=-\sum_{ij} \partial_ia_{ij}(\msx)\partial_j +\sum_i b_i(\msx)\partial_i$. In this paper the diffusion process is approximated by Markov jump processes $X_n(\cdot)$, in homogeneous and isotropic grids $G_n \subset {\bbR}^d$, which converge in distribution to the diffusion $X(\cdot)$. The generators of $X_n(\cdot)$ are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for $d\geq3$ can be applied to processes for which the diffusion tenso…
We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simul…
Filtering theory gives an explicit models for the flow of information and thereby quantifies the rates of change of information supplied to and dissipated from the filter's memory. Here we extend the analysis of Mitter and Newton from linear Gaussian models to general nonlinear filters involving Markov diffusions.The rates of entropy production are now generally the average squared-field (co-metric) of various logarithmic probability densities, which may be interpreted as Fisher information associate with Gaussian perturbations (via de Bruijn's identity). We show that the central connection is…
In this article, we continue to explore Probability Bracket Notation (PBN), proposed in our previous article. Using both Dirac vector bracket notation (VBN) and PBN, we define induced Hilbert space and induced sample space, and propose that there exists an equivalence relation between a Hilbert space and a sample space constructed from the same base observable(s). Then we investigate Markov transition matrices and their eigenvectors to make diffusion maps with two examples: a simple graph theory example, to serve as a prototype of bidirectional transition operator; a famous text document examp…
One of the key ingredients to successfully apply Stein's method for distributional approximation are solutions to the Stein equations and their derivatives. Using Barbour's generator approach, one can solve for the solutions to the Stein equation in terms of the semi-group of a Markov process, which is typically a diffusion process if it is a continuous distribution. For an arbitrary diffusion it can a difficult task to evaluate the semi-group and its derivatives. In this paper, for polynomial test functions, instead of calculating the semi-group of a diffusion, via a duality argument, we inst…
We consider the optimal stopping problem consisting in, given a strong Markov process, a reward function and a discount rate, finding the stopping time such that the expected reward at the stopping time is maximum. The approach we follow, has two main components: the Dynkin's characterization of the value function as the smallest excessive function dominating the reward; and the Riesz representation of excessive functions in terms of the Green kernel, the main reference being Salminen 85. In the context of one-dimensional diffusions we give a complete characterization of the solution under som…
We develop a new simulation method for multidimensional diffusions with sticky boundaries. The challenge comes from simulating the sticky boundary behavior, for which standard methods like the Euler scheme fail. We approximate the sticky diffusion process by a multidimensional continuous time Markov chain (CTMC), for which we can simulate easily. We develop two ways of constructing the CTMC: approximating the infinitesimal generator of the sticky diffusion by finite difference using standard coordinate directions, and matching the local moments using the drift and the eigenvectors of the covar…
The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. …
This paper uses the generator approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. Until now, the standard way to invoke Stein's method for this problem was to use the Poisson equation for the diffusion as a starting point. The main technical difficulty with this approach is obtaining bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the discrete Poisson equation of the Markov chain. An important step in our approach is extending the d…
The paper proposes a class of financial market models which are based on inhomogeneous telegraph processes and jump diffusions with alternating volatilities. It is assumed that the jumps occur when the tendencies and volatilities are switching. We argue that such a model captures well the stock price dynamics under periodic financial cycles. The distribution of this process is described in detail. For this model we obtain the structure of the set of martingale measures. This incomplete model can be completed by adding another asset based on the same sources of randomness. Explicit closed-form …
It is well known that a regular diffusion on an interval $I$ without killing inside is uniquely determined by a canonical scale function $s$ and a canonical speed measure $m$. Note that $s$ is a strictly increasing and continuous function and $m$ is a fully supported Radon measure on $I$. In this paper we will associate a general triple $(I,s,m)$, where $s$ is only assumed to be increasing and $m$ is not necessarily fully supported, to certain Markov processes by way of Dirichlet forms. Using two transformations, called scale completion and darning respectively, to rebuild the topology of $I$,…
For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the law of the Markov process itself. In the last section we want to give an application to the Wasserstein diffusion.
We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the 'averaged' fluid limit exists. Here, we establish conv…
This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily single-node) queues with Markov modulation, two variants were distinguished: one in which the server speed is modulated, and one in which the service requirement is modulated (i.e., depends on the state of the background process upon arrival). The setup of the present paper, however, is more general, as we allow both the server speed and the service requirem…
We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition prob…
