[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.

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

Upcoming events in which I am involved:

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.


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:

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.

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.

Unpublished papers:

Simulation of multivariate diffusion bridges

Co-authors: Mogens Bladt and Samuel Finch

With a view to likelihood inference for discretely observed multivariate stochastic differential equations, we propose simple methods of simulating multivariate diffusion bridges, approximately and exactly. Diffusion bridge simulation plays a fundamental role in simulation-based likelihood and Bayesian inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes the one-dimensional bridge-simulation method proposed by Bladt and Sørensen (2014) to the multi-variate setting. A method of simulating approximate, but often very accurate, diffusion bridges is proposed. These approximate bridges are used as proposal for easily implementable MCMC algorithms that produce exact diffusion bridges. The new method is more generally applicable than previous methods because it does not require the existence of a Lamperti transformation, which rarely exists for multivariate diffusions. Another advantage is that the new method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. In a simulation study the approximate method is shown to provide a good fit to the distribution of a diffusion bridge, except for bridges that are unlikely to occur in applications to statistical inference for discretely observed diffusions. The usefulness of the new method is illustrated by an application to Bayesian estimation for the multivariate hyperbolic diffusion model.


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.


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.


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.

Full list of publications and CV:

List of publications   


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) (2005)
  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)
  At Jonas Ströjby's Ph.D. defence in Lund (2010):   No. 1   No. 2   No. 3
  In Budapest (2013)

This homepage was last updated on March 3, 2015.