[Michael Sørensen]

Michael Sørensen

Professor and Head of the Department of Mathematical Sciences, University of Copenhagen.

Member of the Dynamical Systems Interdisciplinary Network under the University of Copenhagen Excellence Programme for Interdisciplinary Research.

Here are some ways of contacting me:

E-mail: michael@math.ku.dk

Telephone (direct): +45 3533 0402
                    (dept.): +45 3532 0723/+45 3532 0724

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.

Full list of publications and CV:         List of publications       CV


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.

[cover] Jointly with Mathieu Kessler and Alexander Lindner, I have edited the book Statistical Methods for Stochastic Differential Equations, where you can read about my work (and that of others) in this area. I have written the chapter on "Estimating functions for diffusion type processes".

Textbook in Danish:

The most comprehensive version of my introduction to probability theory for first year students is the 9th edition: En Introduktion til Sandsynlighedsregning.

Extended abstract collections:

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

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

Selected recent publications:

Efficient estimation for diffusions sampled at high frequency over a fixed time interval. Co-author: Nina Munkholt Jakobsen. Bernoulli, 23, 2017, 1874 - 1910.

Simulation of multivariate diffusion bridges. Co-authors: Mogens Bladt and Samuel Finch. J. Roy. Statist. Soc. B, 78, 2016, 343 - 369.

Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Co-author: Mogens Bladt. Bernoulli, 20, 2014, 645 - 675.

A transformation approach to modelling multi-modal diffusions. Co-author: Julie Lyng Forman. Journal of Statistical Planning and Inference, 146, 2014, 56 - 69.

Statistical inference for discrete-time samples from affine stochastic delay differential equations. Co-author: Uwe Küchler. Bernoulli, 19, 2013, 409 - 425.

Estimating functions for diffusion-type processes. In Kessler, M., Lindner, A. and Sørensen, M. (eds.): Statistical Methods for Stochastic Differential Equations, CRC Press - Chapmann and Hall, 2012, 1 - 107.

Prediction-based estimating functions: review and new developments. Brazilian Journal of Probability and Statistics, 25, 2011, 362 - 391.

Maximum likelihood estimation for integrated diffusion processes. Co-author: Fernando Baltazar-Larios. In Chiarella, C. and Novikov, A. (eds.): Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer, Heidelberg, 2010, 407 - 423.

A simple estimator for discrete-time samples from affine stochastic delay differential equations. Co-author: Uwe Küchler. Statistical Inference for stochastic Processes, 13, 2010, 125 - 132.

Estimating functions for discretely sampled diffusion-type models. Co-authors: Bo Martin Bibby and Martin Jacobsen. In Ait-Sahalia, Y. and Hansen, L.P. (eds.): Handbook of Financial Econometrics, North Holland, Oxford, 2010, 203 - 268.

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.

Unpublished papers:

A review of asymptotic theory of estimating functions

Co-author: Jean Jacod

Asymptotic statistical theory for estimating functions is reviewed in a generality suitable for stochastic processes. Conditions concerning existence of a consistent estimator, uniqueness, rate of convergence, and the asymptotic distribution are treated separately. Our conditions are not minimal, but can be verified for many interesting stochastic process models. Several examples illustrate the wide applicability of the theory and why the generality is needed.

Estimating functions for jump-diffusions

Co-author: Nina Munkholt Jakobsen

The theory of approximate martingale estimating functions for continuous diffusions is well developed and encompasses many estimators proposed in the literature. This paper extends the asymptotic theory for approximate martingale estimating functions to diffusions with finite-activity jumps. The primary aim is to shed light on the question of rate optimality and efficiency of estimators when observations of a jump-diffusion process are made at increasing frequency, with terminal sampling time going to infinity. Under mild assumptions, it is shown that approximate martingale estimating functions yield consistent and asymptotically normal estimators in the presence of jumps, and a consistent estimator of the asymptotic variance is provided. The estimators are rate optimal for parameters of the drift and jump components of the process. Additional conditions are derived, under which estimators of a diffusion coefficient parameter are rate optimal and therefore converge at a faster rate. These are supplemented with conditions ensuring efficiency of the estimators. Interestingly, the efficiency conditions for jump parameters are much more restrictive than for parameters of the drift and diffusion coefficients. The conditions for both rate optimality and efficiency are, in the established framework of approximate martingale estimating functions, very restrictive. However, these conditions contribute valuable insight into the characteristics of asymptotically well-performing estimating functions, and thus indicate a potentially fruitful direction for further development of estimation for diffusions with jumps.

Langevin diffusions on the torus: estimation and applications

Co-authors: Eduardo García-Portugués, Kanti V. Mardia and Thomas Hamelryck

We introduce stochastic models for continuous-time evolution of angles and develop their estimation. We focus on studying Langevin diffusions with stationary distributions equal to well- known distributions from directional statistics, since such diffusions can be regarded as toroidal analogues of the Ornstein-Uhlenbeck process. Their likelihood function is a product of transition densities with no analytical expression, but that can be calculated by solving the Fokker-Planck equation numerically through adequate schemes. We propose three approximate likelihoods that are computationally tractable: (i) a likelihood based on the stationary distribution; (ii) toroidal adaptations of the Euler and Shoji-Ozaki pseudo-likelihoods; (iii) a likelihood based on a specific approximation to the transition density of the wrapped normal process. A simulation study compares, in dimensions one and two, the approximate transition densities to the exact ones, and investigates the empirical performance of the approximate likelihoods. Finally, two diffusions are used to model the evolution of the backbone angles of the 1GB1 protein during a molecular dynamics simulation. The software package sdetorus implements the estimation methods and applications presented in the paper.


Co-author: Bent Jesper Christensen

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.


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.


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.


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.


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.


J. Møller and M. Sørensen: Parametric models of spatial birth-and-death processes with a view to modelling linear dune fields
Research Report No. 184, Department of Theoretical Statistics, University of Aarhus, 1990.

This homepage was last updated on November 29, 2017.