Here are two ways of contacting me:
E-mail: michael@math.ku.dk
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
Books:
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 publications from the last 10
years:
Likelihood estimation for
stochastic differential equations with mixed effects.
Co-authors: Fernando Baltazar-Larios and Mogens Bladt.
Scand. J. Statist., 53, 2026, 1061
- 1080.
https://doi.org/10.1111/sjos.70068
Prediction-based inference
for integrated diffusions with high frequency data. Co-author:
Emil S. Jørgensen. Statistics
and Risk Modelling, 2026.
https://doi.org/10.1515/strm-2025-0033
A family of toroidal
diffusions with exact likelihood inference. Co-author: Eduardo
Garcéia-Portuguées. Biometrika,
113, 2026, asaf050.
https://doi.org/10.1093/biomet/asaf050.
Ole E. Barndorff-Nielsen:
Sand, wind and statistics. Bernoulli, 32, 2026, 49
- 67.
Prediction-based estimation
for diffusion models with high-frequency data. Co-author: Emil
S. Jørgensen Japanese Journal of Statistics and Data
Science, 4, 2021, 483
- 511.
Estimating functions for
jump-diffusions. Co-author: Nina Munkholt
Jakobsen. Stoch. Proc. Appl., 129, 2019, 3282
- 3318.
Langevin diffusions on the
torus: estimation and applications. Co-authors: Eduardo
Garcia-Portugues, Kanti V. Mardi and Thomas Hamelryck. Statistics
and Computing, 29, 2019, 1 - 22.
A review of asymptotic
theory of estimating functions. Co-author: Jean Jacod
Statistical Inference for Stochastic Processes,
21, 2018, 415 - 434.
A generative angular model
of protein structure evolution Co-authors: Michael Golden, Eduardo
Garcia-Portugues, Kanti V. Mardi, Thomas Hamelryck and Jotun
Hein. Molecular Biology and Evolution,
34, 2017, 2085 - 2100.
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.
See also
Corrigendum to "Simulation of
multivariate diffusion bridges". Co-authors: Mogens Bladt and
Samuel Finch. J. Roy. Statist. Soc. B,
84, 2022, 1581 - 1585.
Unpublished papers:
PREDICTION-BASED INFERENCE FOR
INTEGRATED DIFFUCIONS WITH HIGH FREQUENCY DATA
Co-author: Emil S. Jørgensen
We consider parametric inference for an ergodic and stationary
diffusion process, when the data are high-frequency observations of
the integral of the diffusion process. Such data are obtained via
certain measurement devices, or if positions are recorded and speed is
modelled by a diffusion. In finance, realized volatility or
variations thereof can be used to construct observations of the latent
integrated volatility process. Specifically, we assume that the
integrated process is observed at equidistant, deterministic time
points and consider the high-frequency/infinite horizon asymptotic
scenario, where the number of observations, the sampling frequency and
the time of the last observation all go to infinity. Subject to mild
standard regularity conditions on the diffusion model, we prove the asymptotic
existence and uniqueness of a consistent estimator for useful and
tractable classes of prediction-based estimating functions. Asymptotic
normality of the estimator is obtained under an additional assumption
on the rates. The proofs are based on the useful Euler-Ito expansions
of transformations of diffusions and integrated diffusions, which we
study in some detail.
EFFICIENT ESTIMATION FOR ERGODIC DIFFUSION
PROCESSES SAMPLED AT HIGH FREQUENCY
A general theory of efficient estimation for ergodic diffusion
processes sampled at high frequency with an infinite time horizon is
presented. High frequency sampling is common in many applications,
with finance as a prominent example. 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. Easily checked conditions ensuring
that an estimating function is an approximate martingale are derived,
and general conditions ensuring consistency and asymptotic normality
of estimators are given. Most importantly, simple conditions are given
that ensure rate optimality and efficiency. Rate optimal estimators of
parameters in the diffusion coefficient converge faster than
estimators of drift coefficient parameters because they take advantage
of the information in the quadratic variation. The conditions
facilitate the choice among the multitude of estimators that have been
proposed for diffusion models. Optimal martingale estimating functions
in the sense of Godambe and Heyde and their high frequency
approximations are, under weak conditions, shown to satisfy the
conditions for rate optimality and efficiency. This provides a natural
feasible method of constructing explicit rate optimal and efficient
estimating functions by solving a linear equation.
OPTIMAL INFERENCE IN DYNAMIC MODELS WITH
CONDITIONAL MOMENT RESTRICTIONS
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.
SMALL DISPERSION ASYMPTOTICS FOR DIFFUSION
MARTINGALE ESTIMATING FUNCTIONS
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
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
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.
A preprint with details that did not appear in the version that was
published in the Scandinavian Journal of Statistics:
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
September 8, 2026.
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