An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure

EM algorithm for Markov chains observed via Gaussian noise and point process information: Theory and case studies

In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations

Steady-state simulation of reflected Brownian motion and related stochastic networks

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

HMM filtering and parameter estimation of an electricity spot price model

In this paper we develop a model for electricity spot price dynamics. The spot price is assumed to follow an exponential Ornstein-Uhlenbeck (OU) process with an added compound Poisson process, therefore the model allows for mean-reversion and possible jumps. A sinusoidal factor is also introduced to capture the seasonality component of prices. The mean-reverting level, speed of adjustment and volatility of the OU process as well as the mean and variance of the normally distributed jump sizes of the compound Poisson process are all modulated by a hidden Markov chain in discrete time. The parameters are able to switch between different economic regimes representing various levels of supply and demand. Through the application of reference probability technique, adaptive filters are derived, which in turn, provide optimal estimates for the state of the Markov chain and related quantities of the observation process. The EM algorithm is applied to find optimal estimates of the model parameters in terms of the recursive filters. Since the parameters are updated everytime a new information is available, the model is self-calibrating. We implement the model on a deseasonalized series of daily spot electricity prices from the Nordic exchange Nord Pool. On the basis of one-step ahead forecasts, we found that the model is able to capture the stylised features of Nord Pool spot prices

The continuous-time hidden Markov model based on discretization. Properties of estimators and applications

This work was jointly supported by the Spanish Ministry of Science and Innovation-State Research Agency through grants numbered PID2020-120217RB-I00 and PID2021-123737NB-I00, and the IMAG-Maria de Maeztu grant CEX2020-001105-/AEI/10.13039/501100011033; and by the Spanish Junta de Andalucia through grant number B-FQM-284-UGR20.In this paper we consider continuous-time hidden Markov processes (CTHMM). The model considered is a two-dimensional stochastic process (Xt, Yt) , with Xt an unobserved (hidden) Markov chain defined by its generating matrix and Yt an observed process whose distribution law depends on Xt and is called the emission function. In general, we allow the process Yt to take values in a subset of the q-dimensional real space, for some q. The coupled process (Xt, Yt) is a continuous-time Markov chain whose generator is constructed from the generating matrix of X and the emission distribution. We study the theoretical properties of this two-dimensional process using a formulation based on semi-Markov processes. Observations of the CTHMM are obtained by discretization considering two different scenarii. In the first case we consider that observations of the process Y are registered regularly in time, while in the second one, observations arrive at random. Maximum-likelihood estimators of the characteristics of the coupled process are obtained in both scenarii and the asymptotic properties of these estimators are shown, such as consistency and normality. To illustrate the model a real-data example and a simulation study are considered.Universidad de Granada/CBUAIMAG-Maria de Maeztu CEX2020-001105-/AEI/10.13039/501100011033Spanish Junta de Andalucia B-FQM-284-UGR20Spanish Ministry of Science and Innovation-State Research Agency PID2020-120217RB-I00, PID2021-123737NB-I0

The continuous-time hidden Markovmodel based on discretization. Properties of estimators and applications

Universidad de Granada/CBU