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Generator Estimation of Markov Jump Processes Based on Incomplete Observations Nonequidistant in Time
Markov jump processes can be used to model the effective dynamics of observables in applications ranging from molecular dynamics to finance. In this paper we present a different method which allows the inverse modeling of Markov jump processes based on incomplete observations in time: We consider the case of a given time series of the discretely observed jump process. We show how to compute efficiently the maximum likelihood estimator of its infinitesimal generator and demonstrate in detail that the method allows us to handle observations nonequidistant in time. The method is based on the work of and Bladt and Sørensen [J. R. Stat. Soc. Ser. B (Stat. Methodol.) 67, 395 (2005)] but scales much more favorably than it with the length of the time series and the dimension and size of the state space of the jump process. We illustrate its performance on a toy problem as well as on data arising from simulations of biochemical kinetics of a genetic toggle switch
Approximation of epidemic models by diffusion processes and their statistical inference
Multidimensional continuous-time Markov jump processes on
form a usual set-up for modeling -like epidemics. However,
when facing incomplete epidemic data, inference based on is not easy
to be achieved. Here, we start building a new framework for the estimation of
key parameters of epidemic models based on statistics of diffusion processes
approximating . First, \previous results on the approximation of
density-dependent -like models by diffusion processes with small diffusion
coefficient , where is the population size, are
generalized to non-autonomous systems. Second, our previous inference results
on discretely observed diffusion processes with small diffusion coefficient are
extended to time-dependent diffusions. Consistent and asymptotically Gaussian
estimates are obtained for a fixed number of observations, which
corresponds to the epidemic context, and for . A
correction term, which yields better estimates non asymptotically, is also
included. Finally, performances and robustness of our estimators with respect
to various parameters such as (the basic reproduction number), ,
are investigated on simulations. Two models, and , corresponding to
single and recurrent outbreaks, respectively, are used to simulate data. The
findings indicate that our estimators have good asymptotic properties and
behave noticeably well for realistic numbers of observations and population
sizes. This study lays the foundations of a generic inference method currently
under extension to incompletely observed epidemic data. Indeed, contrary to the
majority of current inference techniques for partially observed processes,
which necessitates computer intensive simulations, our method being mostly an
analytical approach requires only the classical optimization steps.Comment: 30 pages, 10 figure
Modelling FX smile : from stochastic volatility to skewness
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HMM based scenario generation for an investment optimisation problem
This is the post-print version of the article. The official published version can be accessed from the link below - Copyright @ 2012 Springer-Verlag.The Geometric Brownian motion (GBM) is a standard method for modelling financial time series. An important criticism of this method is that the parameters of the GBM are assumed to be constants; due to this fact, important features of the time series, like extreme behaviour or volatility clustering cannot be captured. We propose an approach by which the parameters of the GBM are able to switch between regimes, more precisely they are governed by a hidden Markov chain. Thus, we model the financial time series via a hidden Markov model (HMM) with a GBM in each state. Using this approach, we generate scenarios for a financial portfolio optimisation problem in which the portfolio CVaR is minimised. Numerical results are presented.This study was funded by NET ACE at OptiRisk Systems
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