Drift parameter estimation in fractional diffusions driven by perturbed random walks

We estimate the drift parameter in a simple linear model driven by fractional Brownian motion. We propose maximum likelihood estimators (MLE) for the drift parameter construct by using a random walk approximation of the fractional Brownian motion.     

Asymptotic normality and efficiency of the maximum likelihood estimator for the parameter of a ballistic random walk in a random environment

We consider a one-dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove asymptotic normality for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cramér-Rao bound. We also explore in a simulation setting the numerical behavior of asymptotic confidence regions for the parameter value.     

Parametric estimation for cusp-type signal driven by fractional Brownian motion

Abstract

We investigate the asymptotic properties of the maximum likelihood estimator of the drift parameter in a cusp-type signal driven by a fractional Brownian motion.     

Parametric estimation for linear stochastic differential equations driven by mixed fractional Brownian motion

ABSTRACT

We investigate the asymptotic properties of the maximum likelihood estimator and Bayes estimator of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by a mixed fractional Brownian motion. We obtain a Bernstein–von Mises-type theorem also for such a class of processes.     

Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process. This revised version was published online in August 2006 with corrections to the Cover Date.     

Second Order Expansions for the Moments of Minimum Point of an Unbalanced Two-sided Normal Random Walk

In this paper, the second order expansions for the first two moments of the minimum point of an unbalanced two-sided normal random walk are obtained when the drift parameters approach zero. The basic technique is the uniform strong renewal theorem in the exponential family. The comparison with numerical values shows that the approximations are very accurate. It is shown, particularly, that the first moment is significantly different from its continuous Brownian motion analog while the second moments are the same in the first order. The results can be used to study properties of the maximum likelihood estimator for the change point.     

Asymmetric skew Bessel processes and their applications to finance

In this paper, we extend the Harrison and Shepp's construction of the skew Brownian motion (1981) and we obtain a diffusion similar to the two-dimensional Bessel process with speed and scale densities discontinuous at one point. Natural generalizations to multi-dimensional and fractional order Bessel processes are then discussed as well as invariance properties. We call this family of diffusions asymmetric skew Bessel processes in opposition to skew Bessel processes as defined in Barlow et al. [On Walsh's Brownian motions, Séminaire de Probabilitiés XXIII, Lecture Notes in Mathematics, vol. 1372, Springer, Berlin, New York, 1989, pp. 275–293]. We present factorizations involving (asymmetric skew) Bessel processes with random time. Finally, applications to the valuation of perpetuities and Asian options are proposed.     

Maximum likelihood estimator for the drift of a Brownian flow

The maximum likelihood estimator for the drift of a Brownian flow on ℝ^d, d ⩾ 2, is found with the assumption that the covariance is known. By approximation of the drift with known functions, the statistical model is reduced to a parametric one that is a curved exponential family. The data is the n‐point motion of the Brownian flow throughout the time interval [0, T]. The asymptotic properties of the MLE are also investigated. Copyright © 2000 John Wiley & Sons, Ltd.     

Arbitrage in skew Brownian motion models

Empirical skewness of asset returns can be reproduced by stochastic processes other than the Brownian motion with drift. Some authors have proposed the skew Brownian motion for pricing as well as interest rate modelling. Although the asymmetric feature of random return involved in the stock price process is driven by a parsimonious one-dimensional model, we will show how this is intrinsically incompatible with a modern theory of arbitrage in continuous time. Application to investment performance and to the Black-Scholes pricing model clearly emphasize how this process can provide some kind of arbitrage.     

Optimal estimation of a signal perturbed by a sub-fractional Brownian motion

We consider the problem of optimal estimation of the vector parameter θ of the drift term in a sub-fractional Brownian motion. We obtain the maximum likelihood estimator as well as Bayesian estimator when the prior distribution is Gaussian.     

Some Limit Theorems for Heights of Random Walks on a Spider

A simple random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker and the Brownian motion can go on the legs in n steps. The heights on the legs are also investigated when the number of legs goes to infinity.     

Exact inequalities for the maximum of a skew Brownian motion

The maximal inequality for the skew Brownian motion being a generalization of the well-known inequalities for the standard Brownian motion and its module is obtained in the paper. The proof is based on the solution to an optimal stopping problem for which we find the cost function and optimal stopping time.     

The problem of the most visited site in random environment

We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remains open. Received: 17 April 1998     

Parameter estimation for stochastic equations with additive fractional Brownian sheet

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.      

Maximum likelihood estimation for multivariate skew normal mixture models

This paper provides a flexible mixture modeling framework using the multivariate skew normal distribution. A feasible EM algorithm is developed for finding the maximum likelihood estimates of parameters in this context. A general information-based method for obtaining the asymptotic covariance matrix of the maximum likelihood estimators is also presented. The proposed methodology is illustrated with a real example and results are also compared with those obtained from fitting normal mixtures.     

Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment

We consider a one-dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. For that purpose, we adapt the method developed in the ballistic case by Comets et al. (2014) and Falconnet et al. (2014). Using a supplementary assumption due to the special nature of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphasize the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.     

On the character of convergence to Brownian local time. I

Summary Many results are known about the convergence of some processes to Brownian local time. Among such processes are the process of occupation times of Brownian motion, the number of downcrossings of Brownian motion over smaller and smaller intervals before timet, the number of visits of the recurrent integer-valued random walk to some point duringn steps and others. In this paper we consider the asymptotic behaviour of the differences between Brownian local time and some of the processes which converge to it.     

Schur-Weyl duality and the heat kernel measure on the unitary group

We investigate a relation between the Brownian motion on the unitary group and the most natural random walk on the symmetric group, based on Schur-Weyl duality. We use this relation to establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. This expectation turns out to be the generating series of certain paths in the Cayley graph of the symmetric group. Using our expansion, we recover asymptotic results of Xu, Biane and Voiculescu. We give an interpretation of our main expansion in terms of random ramified coverings of a disk.     

Strong approximation for the general Resten-Spitzer random walk in independent random scenery

This paper is to prove that, if a one-dimensional random walk can be approximated by a Brownian motion, then the related random walk in a general independent scenery can be approximated by a Brownian motion in Brownian scenery.