Goodness of fit test for small diffusions by discrete time observations

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional small diffusions. Our test is based on discrete time observation of the processes, and the diffusion coefficient is a nuisance function which is “estimated” in some sense in our testing procedure. We prove that the limit distribution of our test is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. We also show that our test is consistent under any fixed alternative.     

Goodness of fit test for ergodic diffusion processes

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypothesis and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact rejection regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.     

Edgeworth expansion for ergodic diffusions

The Edgeworth expansion for an additive functional of an ergodic diffusion is validated under fairly weak conditions. The validation procedure does not depend on the stationarity or the geometric mixing property, but exploits the strong Markov property of the process. In particular for an Itô-diffusion of dimension one, verifiable conditions for the validity of the expansion are given in terms of the coefficients of the corresponding stochastic differential equation. The maximum likelihood estimator for the CIR process is treated as example.     

Empirical $$L^2$$ L 2 -distance test statistics for ergodic diffusions

Statistical Inference for Stochastic Processes - The aim of this paper is to introduce a new type of test statistic for simple null hypothesis on one-dimensional ergodic diffusion processes sampled...     

Asymptotic statistical equivalence for scalar ergodic diffusions

For scalar diffusion models with unknown drift function asymptotic equivalence in the sense of Le Cam's deficiency between statistical experiments is considered under long-time asymptotics. A local asymptotic equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed. The authors acknowledge the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH     

Estimating functions for noisy observations of ergodic diffusions

In this article, general estimating functions for ergodic diffusions sampled at high frequency with noisy observations are presented. The theory is formulated in terms of approximate martingale estimating functions based on local means of the observations, and simple conditions are given for rate optimality. The estimation of the diffusion parameter is faster than the estimation of the drift parameter, and the rate of convergence is classical for the drift parameter but not classical for the diffusion parameter. The link with specific minimum contrast estimators is established, as an example.     

On ergodic control of degenerate diffusions

For the ergodic control problem with degenerate diffusions, the existence of an optimal solution is established for various interesting classes of solutions.This research was supported by Grant No. 26/01/92-G from the Department of Atomic Energy, Government of India, Delhi, India.     

Stochastic quantization and ergodic theorem for density of diffusions

For a given probability density function ρ(x) on Rd,we construct a(non-stationary) diffusion process xt,starting at any point x in Rd,such that 1/T∫T0 δ(xt-x)dt converges to ρ(x) almost surely.The rate of this convergence is also investigated.To find this rate,we mainly use the Clark-Ocone formula from Malliavin calculus and the Girsanov transformation technique.     

Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions

The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.     

Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case

Asymptotic local equivalence in the sense of Le Cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of Gaussian shift experiments. The nonparametric local neighbourhoods can be attained for any dimension, provided the regularity of the drift is sufficiently large. In addition, a heteroskedastic Gaussian regression experiment is given, which is also locally asymptotically equivalent and which does not depend on the centre of localisation. For one direction of the equivalence an explicit Markov kernel is constructed.     

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Goodness of fit assessment for a fractal model of stock markets

An assessment of the goodness of fit of a new stochastic model of stock dynamics is investigated. The model is the multifractional Brownian motion (mBm), introduced independently by Péltier and Lévy Véhel (1995) [2] and Benassi (1997) [3]. The analysis concerns the (un)conditional distributions of log-variations of the Dow Jones Industrial Average (DJIA). By comparing the performance of mBm with respect to a Garch (1,1), we argue that the former captures the distributional features as well as the pathwise empirical ones displayed by the U.S. Dow Jones index, while the Garch (1,1) works better in global terms.     

Comparison of interacting diffusions and an application to their ergodic theory

Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.     

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Goodness of wit test Goodness of wit test #14: Outcome analysis     

The Lagrange approach to ergodic control of diffusions with cost constraints

This article concerns n-dimensional controlled diffusion processes. The main problem is to maximize a certain long-run average reward (also known as an ergodic reward) in such a way that a given long-run average cost is bounded above by a constant. Under suitable assumptions, the existence of optimal controls for such constrained control problems is a well-known fact. In this article we go a bit further and our goal is to introduce a technique to compute optimal controls. To this end, we follow the Lagrange multipliers approach. An example on a linear-quadratic system illustrates our results.     

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Goodness of wit test Goodness of wit test #12: Straightforward!