[Michael Sørensen]

Michael Sørensen

Professor and Associate Chair for Research at the Department of Mathematical Sciences, University of Copenhagen.

Principal investigator of the Statistics Network under the University of Copenhagen Programme of Excellence.


Here are some ways of contacting me:

E-mail: michael@math.ku.dk

Telephone (direct): +45 3532 0680
                    (dept.): +45 3532 0899

Fax: +45 3532 0772

Address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.



My main research interests are:

Statistical inference for stochastic processes, in particular discrete time sampling of continuous time processes such as models given by stochastic differential equations and jump processes. The interaction between statistics and finance. Stochastic models and related statistical problems, particularly in the physics of blown sand, turbulence and biology.


Books:

[cover]
You can read about my work on exponential families of stochastic processes in my book Exponential Families of Stochastic Processes, co-authored with Uwe Küchler (Humboldt- University of Berlin).



[cover]
You may also be interested in the book Empirical Process Techniques for Dependent Data that I have edited jointly with Thomas Mikosch and Herold Dehling.


Extended abstract collections:

Workshop on Stochastic Partial Differential Equations: Statistical Issues and Applications, co-edited with Marianne Huebner     postscript     pdf

Workshop on Dynamic Stochastic Modeling in Biology, co-edited with Marianne Huebner     postscript     pdf


Selected recent publications:

Parametric inference for discretely sampled stochastic differential equations. In Andersen, T.G. Davis, R.A., Kreiss, J.-P. and Mikosch, T. (eds.): Handbook of Financial Time Series, Springer, Heidelberg, 2009, 531 - 553.

Estimation for stochastic differential equations with a small diffusion coefficient. Co-author: Arnaud Gloter. Stoch. Proc. Appl., 119, 2009, 679 - 699.

Efficient estimation of transition rates between credit ratings from observations at discrete time points. Co-author: Mogens Bladt. Quantitative Finance, 9, 2009, 147 - 160.

 The Pearson diffusions: A class of statistically tractable diffusion processes. Co-author: Julie Lyng Forman. Scand. J. Statist., 35, 2008, 438 - 465.

The vertical variation of particle speed and flux density in aeolian saltation: measurement and modeling. Co-author: Keld Rømer Rasmussen. J. Geophys. Res., 113, 2008, F02S12, doi:10.1029/2007JF000774.

Diffusion models for exchange rates in a target zone. Co-author: Kristian Stegenborg Larsen. Mathematical Finance, 17, 2007, 285 - 306.

Dynamics of particles in aeolian saltation. Co-author: Keld Rømer Rasmussen. In Garcia-Rojo, R., Herrmann, H.J. and McNamara, S. (eds.): Powders and Grains 2005, Vol. 2, Balkema, Rotterdam, 2005, 967 - 972.

Statistical inference for discretely observed Markov jump processes. Co-author: Mogens Bladt. J. Roy. Statist. Soc., ser. B, 67, 2005, 395 - 410.

Diffusion-type models with given marginal and autocorrelation function. Co-authors: Bo Martin Bibby and Ib Michael Skovgaard. Bernoulli, 11, 2005, 191 - 220.

On time-reversibility and estimating functions for Markov processes. Co-author: Mathieu Kessler. Statistical Inference for Stochastic Processes, 8, 2005, 95 - 107.

Martingale estimating functions for discretely observed stochastic differential equation models. In Romanelli, S., Mininni, R.M. and Lucente, S. (eds.): Interplay between (C_0)-semigroups and PDEs: Theory and applications. Aracne Editrice, Rome, 2004, 213 - 236.

Inference for observations of integrated diffusion processes. Co-author: S. Ditlevsen. Scand. J. Statist., 31, 2004, 417 - 429.

Estimation for discretely observed diffusions using transform functions. Co-authors: Leah Kelly and Eckhard Platen. J. Appl. Prob., 41A, 2004, 99 - 118.

On the rate of aeolian sand transport. Geomorphology, 59, 2004, 53 - 62.

Small-diffusion asymptotics for discretely sampled stochastic differential equations. Co-author: Masayuki Uchida. Bernoulli, 9, 2003, 1051 - 1069.

Hyperbolic processes in finance. Co-author: B.M. Bibby. In S. Rachev (ed.): Handbook of Heavy Tailed Distributions in Finance, Elsevier Science, 2003, 211 - 248.

Prediction-based estimating functions. Econometrics Journal, 3, 2000, 123 - 147.



Post script and pdf files of my recent unpublished papers:


Estimating functions for diffusion-type processes

pdf

A broad review is given of the theory of estimating functions for discretely sampled diffusion-type processes. Martingale and non-martingale estimating functions are presented for directly observed solutions to stochastic differential equations. Particular attention is given to explicit estimating functions. Prediction-based estimating functions are presented as a useful generalization for non-Markovian observations like diffusions observed with measurement errors and stochastic volatility models. A number of examples of non-Markovian data of the diffusion-type are discussed. The asymptotic theory is presented in detail including several asymptotic scenarios relevant to diffusion models. Also optimality and efficiency is considered.


Maximum likelihood estimation for integrated diffusion processes

Co-author: Fernando Baltazar-Larios

pdf

We propose a method for obtaining maximum likelihood estimates of parameters in diffusion models when the data is a discrete time sample of the integral of the process, while no direct observations of the process itself are available. The data are, moreover, assumed to be contaminated by measurement errors. Integrated volatility is an example of this type of observations. Another example is ice-core data on oxygen isotopes used to investigate paleo-temperatures. The data can be viewed as incomplete observations of a model with a tractable likelihood function. Therefore we propose a simulated EM-algorithm to obtain maximum likelihood estimates of the parameters in the diffusion model. As part of the algorithm, we use a recent simple method for approximate simulation of diffusion bridges. In a simulation study for the Ornstein-Uhlenbeck process the methods works well. The method is applied to a set of integrated paleo-temperature data obtained from an ice-core.


Simple simulation of diffusion bridges with application to likelihood inference for diffusions

Co-author: Mogens Bladt

pdf

With a view to likelihood inference for discretely observed diffusion type models, we propose a simple method of simulating approximations to diffusion bridges. The method is applicable to all one-dimensional diffusion processes and has the advantage that simple simulation methods like the Euler scheme can be applied to bridge simulation. Another advantage over other bridge simulation methods is that the proposed method works well when the diffusion bridge is defined in a long interval because the computational complexity of the method is linear in the length of the interval. In a simulation study we investigate the accuracy and efficiency of the new method and compare it to exact simulation methods. In the study the method provides a very good approximation to the distribution of a diffusion bridge for bridges that are likely to occur in applications to likelihood inference. To illustrate the usefulness of the new method, we present an EM-algorithm for a discretely observed diffusion process. We demonstrate how this estimation method simplifies for exponential families of diffusions and very briefly consider Bayesian inference.


A SIMPLE ESTIMATOR FOR DISCRETE-TIME SAMPLES FROM AFFINE STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Co-author: Uwe Küchler

pdf

Estimation for discrete time observations of an affine stochastic delay differential equation is considered. The delay measure is assumed to be concentrated on a finite set. A simple estimator is obtained by discretization of the continuous-time likelihood function, and its asymptotic properties are investigated. The estimator is very easy to calculate works well at high sampling frequencies, but it is shown to have a significant bias when the sampling frequency is low.


OPTIMAL INFERENCE IN DYNAMIC MODELS WITH CONDITIONAL MOMENT RESTRICTIONS

Co-author: Bent Jesper Christensen

postscript pdf

By an application of the theory of optimal estimating function, optimal instruments for dynamic models with conditional moment restrictions are derived. The general efficiency bound is provided, along with estimators attaining the bound. It is demonstrated that the optimal estimators are always at least as efficient as the traditional optimal generalized method of moments estimator, and usually more efficient. The form of our optimal instruments resembles that from Newey (1990), but involves conditioning on the history of the stochastic process. In the special case of i.i.d.\ observations, our optimal estimator reduces to Newey's. Specification and hypothesis testing in our framework are introduced. We derive the theory of optimal instruments and the associated asymptotic distribution theory for general cases including non-martingale estimating functions and general history dependence. Examples involving time-varying conditional volatility and stochastic volatility are offered.


EFFICIENT ESTIMATION FOR ERGODIC DIFFUSIONS SAMPLED AT HIGH FREQUENCY

postscript pdf

A general theory of efficient estimation for ergodic diffusions sampled at high frequency is presented. High frequency sampling is now possible in many applications, in particular in finance. The theory is formulated in term of approximate martingale estimating functions and covers a large class of estimators including most of the previously proposed estimators for diffusion processes, for instance GMM-estimators and the maximum likelihood estimator. Simple conditions are given that ensure rate optimality, where estimators of parameters in the diffusion coefficient converge faster than estimators of parameters in the drift coefficient, and for efficiency. The conditions turn out to be equal to those implying small $\Delta$-optimality in the sense of Jacobsen and thus gives an interpretation of this concept in terms of classical statistical concepts. Optimal martingale estimating functions in the sense of Godambe and Heyde are shown to be give rate optimal and efficient estimators under weak conditions.


STATISTICAL INFERENCE FOR DISCRETE-TIME SAMPLES FROM AFFINE STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

Co-author: Uwe Küchler

postscript pdf

Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. Also a more general class of prediction-based estimating functions is investigated. In particular, the optimal prediction-based estimating function and the asymptotic properties of the estimators are derived. The maximum pseudo-likelihood estimator is a particular case, and an expression is found for the efficiency loss when using the maximum pseudo-likelihood estimator rather than the computationally more involved optimal prediction-based estimator. The distribution of the pseudo-likelihood estimator is investigated in a simulation study. For models where the delay measure is concentrated on a finite set, an estimator obtained by discretization of the continuous-time likelihood function is presented, and its asymptotic properties are investigated. The estimator is very easy to calculate, but it is shown to have a significant bias when the sampling frequency is low. Two examples of affine stochastic delay equation are considered in detail.


ON THE SIZE DISTRIBUTION OF SAND

postscript pdf

A model is presented of the development of the size distribution of sand while it is transported from a source to a deposit. The model provides a possible explanation of the log-hyperbolic shape that is frequently found in unimodal grain size distributions in natural sand deposits, as pointed out by Bagnold. It implies that the size distribution of a sand deposit is a logarithmic normal-inverse Gaussian (NIG) distribution, which is one of the generalized hyperbolic distributions. The model extends previous models by taking into account that individual grains do not have the same travel time from the source to the deposit. The travel time is assumed to be random so that the wear on the individual grains vary randomly. The model provides an interpretation of the parameters the NIG-distribution, and relates the mean, variance and skewness of the log-size distribution to the physical parameters of the model. This might be useful when comparing empirical size-distributions from different deposits.



ESTIMATING FUNCTIONS FOR DISCRETELY SAMPLED DIFFUSION-TYPE MODELS

Co-authors: B.M. Bibby and M. Jacobsen

postscript pdf

The theory of estimating functions for diffusion-type models is reviewed with a view to applications in finance. Several types of estimating functions are presented, including some explicit estimating functions. Special attention is given to martingale estimating functions. Also prediction-based estimating functions are considered. Large sample asymptotics is discussed in detail. The theory of optimal estimating functions is presented and applied to diffusion models. The classical theory as well as the new theory of small Delta-optimality are considered. The focus is on diffusion models and stochastic volatility models of the diffusion-type, but a simple diffusion with jumps is treated too. Several examples are given.


SMALL DISPERSION ASYMPTOTICS FOR DIFFUSION MARTINGALE ESTIMATING FUNCTIONS   (Preprint No. 2000-2)

postscript pdf

Martingale estimating functions provide a flexible and powerful framework for statistical inference about diffusion models based on discrete time observations. We supplement the standard results on large sample asymptotics by results on small dispersion asymptotics, which can be applied in situations where the noise term is sufficiently small, compared to the drift term, that a Gaussian approximation to the diffusion can be used. The theory, which is based on the stochastic Taylor expansion, covers proper likelihood inference too. It is remarkable that the martingale property of an estimating function also for small dispersion asymptotics ensures that estimators are consistent. A model from mathematical finance is considered in detail. For this example the range of applicability of the small dispersion asymptotics is investigated in a simulation study of the distribution of estimators.


EXPONENTIAL FAMILY INFERENCE FOR DIFFUSION MODELS   (RR No. 383)

postscript pdf

We consider ergodic diffusion processes for which the class of invariant measures is an exponential family, and study inference based on the class of invariant probability measures when the diffusion has been observed at discrete time points. When the drift depends linearly on the parameters, the invariant measures form an exponential family. It is investigated how the usual exponential family inference, which can be done by means of standard statistical computer packages, works when the observations are from a diffusion process. In particular, the limit distributions of estimators and test statistics are derived. As an example, we consider classes of diffusions with generalized inverse Gaussian marginals. A particular instance is the well-known Cox-Ingersoll-Ross model from mathematical finance.


ON THE MOMENTS OF SOME FIRST PASSAGE TIMES FOR EXPONENTIAL FAMILIES OF PROCESSES   (RR No. 302)

postscript pdf

For curved exponential families of stochastic processes a natural and often studied sequential procedure is to stop observation when a linear combination of the coordinates of the canonical process crosses a prescribed level. Conditions are given which ensure that such first passage times, or a function of them, have finite moments. Also results about L_p convergence as the prescribed level tends to infinity are given.


Full list of publications and CV:

List of publications:    postscript   pdf.

CV:    postscript   pdf.

Some photos with friends and colleagues around the world:
  In Brasil (1998)
  In Tokyo (2001)
  In Bari (Italy) (2004)
  At Oberwolfach (2004)
  At Daydream Island (Australia)
  In Amsterdam (2005):   No. 1   No. 2   No. 3   No. 4
  In Guanajuato (Mexico) (2006): No. 1 No. 2 No. 3
  In Princeton (2007)
  In Le Mans (2009)
  In Copenhagen (2009)


Documentation for a computer program for prediction based estimation for stochastic volatility models


This homepage was last updated on October 20, 2009.