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ARS CONJECTANDI
A celebration of 300 years of stochastics
Freiburg and Basel, May 21 - 24, 2013
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| Jean Jacod, Paris Basel, May 23, 2013, 15:00–16:00 |
| Lévy, Itô, Doob, Meyer and Beyond: Birth and Development of Semimartingales |
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Abstract:
The theory of semimartingales took more than 40 years to reach its full
extension, and the main aim of the talk is to give an account of how this
happened, and of the main steps on the road. Although in many respects the
theory is now achieved, it still undergoes interesting developments on some specific points,
and our secondary aim is to sketch some of these recent extensions.
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| Tom Kurtz, Madison Freiburg, May 24, 2013, 15:30–16:30 |
| Diffusion Approximations for Markov Processes with Multiple Time Scales |
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Abstract: Motivated by models from systems biology, techniques for deriving diffusion
approximations will be discussed. Fluctuations in the process may arise
both from random sampling and from rapid variation in the natural drift of
the process. Classical martingale methods are exploited to deal with both
sources simultaneously. Examples include Michaelis-Menten type enzyme
reactions as well as models of more complicated reaction networks. The talk
is based on joint work with Hye-Won Kang and Lea Popovic.
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| Michael Sørensen, Copenhagen Freiburg, May 22, 2013, 16:30–17:30 |
| Statistics for Stochastic Differential Equations – Past and Present |
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Abstract:
The first statistical methods for stochastic differential equations were
developed for continuous-time data, where an entire trajectory is
assumed to be observed in an interval. Such data cannot be obtained in
practice, but an explicit likelihood function is given by the Girsanov
formula, and a beautiful theory was developed, in particular for
exponential families of diffusion processes. Although this theory is not
directly applicable in practice, it has in several ways informed the
development of statistical methods for discretely sampled stochastic
differential equations, and the continuous-time likelihood plays an
important role in some modern approaches to likelihood inference for
discrete-time observations. For discrete-time data, there is usually no
explicit expression for the likelihood function, which is a product of
transition densities. Therefore, the likelihood function must be
approximated, which can be done in several ways. The talk will review
these developments, including recent methods for simulation-based
likelihood inference.
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