8,018 research outputs found
Fast MCMC sampling for Markov jump processes and extensions
Markov jump processes (or continuous-time Markov chains) are a simple and
important class of continuous-time dynamical systems. In this paper, we tackle
the problem of simulating from the posterior distribution over paths in these
models, given partial and noisy observations. Our approach is an auxiliary
variable Gibbs sampler, and is based on the idea of uniformization. This sets
up a Markov chain over paths by alternately sampling a finite set of virtual
jump times given the current path and then sampling a new path given the set of
extant and virtual jump times using a standard hidden Markov model forward
filtering-backward sampling algorithm. Our method is exact and does not involve
approximations like time-discretization. We demonstrate how our sampler extends
naturally to MJP-based models like Markov-modulated Poisson processes and
continuous-time Bayesian networks and show significant computational benefits
over state-of-the-art MCMC samplers for these models.Comment: Accepted at the Journal of Machine Learning Research (JMLR
A BSDE-based approach for the optimal reinsurance problem under partial information
We investigate the optimal reinsurance problem under the criterion of
maximizing the expected utility of terminal wealth when the insurance company
has restricted information on the loss process. We propose a risk model with
claim arrival intensity and claim sizes distribution affected by an
unobservable environmental stochastic factor. By filtering techniques (with
marked point process observations), we reduce the original problem to an
equivalent stochastic control problem under full information. Since the
classical Hamilton-Jacobi-Bellman approach does not apply, due to the infinite
dimensionality of the filter, we choose an alternative approach based on
Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize
the value process and the optimal reinsurance strategy in terms of the unique
solution to a BSDE driven by a marked point process.Comment: 30 pages, 3 figure
Uncoupled Analysis of Stochastic Reaction Networks in Fluctuating Environments
The dynamics of stochastic reaction networks within cells are inevitably
modulated by factors considered extrinsic to the network such as for instance
the fluctuations in ribsome copy numbers for a gene regulatory network. While
several recent studies demonstrate the importance of accounting for such
extrinsic components, the resulting models are typically hard to analyze. In
this work we develop a general mathematical framework that allows to uncouple
the network from its dynamic environment by incorporating only the
environment's effect onto the network into a new model. More technically, we
show how such fluctuating extrinsic components (e.g., chemical species) can be
marginalized in order to obtain this decoupled model. We derive its
corresponding process- and master equations and show how stochastic simulations
can be performed. Using several case studies, we demonstrate the significance
of the approach. For instance, we exemplarily formulate and solve a marginal
master equation describing the protein translation and degradation in a
fluctuating environment.Comment: 7 pages, 4 figures, Appendix attached as SI.pdf, under submissio
Explicit computations for some Markov modulated counting processes
In this paper we present elementary computations for some Markov modulated
counting processes, also called counting processes with regime switching.
Regime switching has become an increasingly popular concept in many branches of
science. In finance, for instance, one could identify the background process
with the `state of the economy', to which asset prices react, or as an
identification of the varying default rate of an obligor. The key feature of
the counting processes in this paper is that their intensity processes are
functions of a finite state Markov chain. This kind of processes can be used to
model default events of some companies.
Many quantities of interest in this paper, like conditional characteristic
functions, can all be derived from conditional probabilities, which can, in
principle, be analytically computed. We will also study limit results for
models with rapid switching, which occur when inflating the intensity matrix of
the Markov chain by a factor tending to infinity. The paper is largely
expository in nature, with a didactic flavor
Filters and smoothers for self-exciting Markov modulated counting processes
We consider a self-exciting counting process, the parameters of which depend
on a hidden finite-state Markov chain. We derive the optimal filter and
smoother for the hidden chain based on observation of the jump process. This
filter is in closed form and is finite dimensional. We demonstrate the
performance of this filter both with simulated data, and by analysing the
`flash crash' of 6th May 2010 in this framework
Filtering and forecasting commodity futures prices under an HMM framework
We propose a model for the evolution of arbitrage-free futures prices under a regime-switching framework. The estimation of model parameters is carried out using the hidden Markov filtering algorithms. Comprehensive numerical experiments on real financial market data are provided to illustrate the effectiveness of our algorithm. In particular, the model is calibrated with data from heating oil futures and its forecasting performance as well as statistical validity is investigated. The proposed model is parsimonious, self-calibrating and can be very useful in predicting futures prices. © 2013 Elsevier B.V
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