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21
Bayesian inference for nonlinear multivariate diffusion models observed with error
 Computational Statistics and Data Analysis
, 2008
"... Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood based i ..."
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Cited by 32 (7 self)
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Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood based inference can be problematic as closed form transition densities are rarely available. One widely used solution involves the introduction of latent data points between every pair of observations to allow an EulerMaruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. In this paper, we explore an MCMC scheme whose performance is not adversely affected by the number of latent values. We illustrate the methodology by estimating parameters governing an autoregulatory gene network, using partial and discrete data that is subject to measurement error.
Bayesian sequential inference for nonlinear multivariate diffusions
 Statistics and Computing
, 2006
"... In this paper, we adapt recently developed simulationbased sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m −1 latent data points between every pair of observations. Sequential ..."
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Cited by 28 (3 self)
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In this paper, we adapt recently developed simulationbased sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m −1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters online. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. shortterm interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. shortterm interest rate. 1
Optimal filtering of jump diffusions: extracting latent states from asset prices”, Working Paper, http://wwwstat.wharton.upenn.edu/ stroud/pubs.html
, 2006
"... This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing mo ..."
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Cited by 20 (5 self)
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This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines timediscretization schemes with Monte Carlo methods to compute the optimal filtering distribution. Our approach is very general, applying in multivariate jumpdiffusion models with nonlinear characteristics and even nonanalytic observation equations, such as those that arise when option prices are available. We provide a detailed analysis of the performance of the filter, and analyze four applications: disentangling jumps from stochastic volatility, forecasting realized volatility, likelihood based model comparison, and filtering using both option prices and underlying returns. 2 1
Particle filters for partially observed diffusions
, 2006
"... In this paper we introduce novel particle filters for a class of partiallyobserved continuoustime dynamic models where the signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the comp ..."
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Cited by 19 (4 self)
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In this paper we introduce novel particle filters for a class of partiallyobserved continuoustime dynamic models where the signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Unlike available methods, our particle filters do not require approximations of the transition and/or the observation density using timediscretisations. Instead, they build on recent methodology for the exact simulation of diffusion process and the unbiased estimation of the transition density as described in the recent article Beskos et al. (2005c). In particular, we require the Generalised Poisson Estimator, which is a substantial generalisation of the Poisson Estimator (Beskos et al., 2005c), and it is introduced in this paper. Thus, our filters avoid the systematic biases caused by timediscretisations and they have significant computational advantages over alternative continuoustime filters. These advantages are supported by a central limit theorem which is established in this paper. Keywords: Continuoustime filtering, Exact Algorithm, Central Limit Theorem, Cox Process 1
SEQUENTIALLY INTERACTING MARKOV CHAIN Monte Carlo Methods
 SUBMITTED TO THE ANNALS OF STATISTICS
, 2008
"... We introduce a novel methodology for sampling from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. These problems are usually addressed using Sequential Monte Carlo (SMC) methods. The alternative Sequentially Interacting Markov Chain Monte ..."
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Cited by 10 (4 self)
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We introduce a novel methodology for sampling from a sequence of probability distributions of increasing dimension and estimating their normalizing constants. These problems are usually addressed using Sequential Monte Carlo (SMC) methods. The alternative Sequentially Interacting Markov Chain Monte Carlo (SIMCMC) scheme proposed here works by generating interacting nonMarkovian sequences which behave asymptotically like independent MetropolisHastings (MH) Markov chains with the desired limiting distributions. Contrary to SMC methods, this scheme allows us to iteratively improve our estimates in an MCMClike fashion. We establish convergence of the algorithm under realistic verifiable assumptions and demonstrate its performance on several examples arising in Bayesian time series analysis.
A General Framework for the Parametrization of Hierarchical Models
, 708
"... Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the statespace expansion technique. We also describe partially noncentered methods, and demonstrate their use in constructing robust Gibbs sampler algorithms whose convergence properties are not overly sensitive to the data. Key words and phrases: Parametrization, hierarchical models, latent stochastic processes, MCMC.
On simulated likelihood of discretely observed diffusion processes and comparison to closed form approximation
 Journal of Computational and Graphical Statistics
, 2007
"... This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closedform approach. For the case of constant volatility, we give a theoretical justifica ..."
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Cited by 6 (0 self)
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This article focuses on two methods to approximate the loglikelihood function for univariate diffusions: 1) the simulation approach using a modified Brownian bridge as the importance sampler; and 2) the recent closedform approach. For the case of constant volatility, we give a theoretical justification of the modified Brownian bridge sampler by showing that it is exactly a Brownian bridge. We also discuss computational issues in the simulation approach such as accelerating numerical variance stabilizing transformation, computing derivatives of the simulated loglikelihood, and choosing initial values of parameter estimates. The two approaches are compared in the context of financial applications with annualized parameter values, where the diffusion model has an unknown transition density and has no analytical variance stabilizing transformation. The closedform expansion, particularly the secondorder closedform, is found to be computationally efficient and very accurate when the observation frequency is monthly or higher. It is more accurate in the center than in the tail of the transition density. The simulation approach combined with the variance stabilizing transformation is found to be more reliable than the closedform approach when the observation frequency is lower. Both methods performs better when the volatility level is lower, but the simulation method is more robust to the volatility nature of the diffusion model. When applied to two well known datasets of daily observations, the two methods yield similar parameter estimates in both datasets but slightly different loglikelihood in the case of higher volatility.
Importance sampling techniques for estimation of diffusion models
 In
, 2012
"... This article develops a class of Monte Carlo (MC) methods for simulating conditioned diffusion sample paths, with special emphasis on importance sampling schemes. We restrict attention to a particular type of conditioned diffusions, the socalled diffusion bridge processes. The diffusion bridge is t ..."
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Cited by 4 (2 self)
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This article develops a class of Monte Carlo (MC) methods for simulating conditioned diffusion sample paths, with special emphasis on importance sampling schemes. We restrict attention to a particular type of conditioned diffusions, the socalled diffusion bridge processes. The diffusion bridge is the process obtained by conditioning a diffusion to start and finish at specific values at two consecutive times t0 < t1. Diffusion bridge simulation is a highly nontrivial problem. At an even more elementary level unconditional simulation of diffusions, that is without fixing the value of the process at t1, is difficult. This is a simulation from the transition distribution of the diffusion which is typically intractable. This intractability stems from the implicit specification of the diffusion as a solution of a stochastic differential equation (SDE). Although the unconditional simulation can be carried out by various approximate schemes based on discretizations of the SDE, it is not feasible to devise similar schemes for diffusion bridges in general. This has motivated active research in the last 15 years or so for the development of MC methodology for diffusion bridges. The research in this direction has been fuelled by the fundamental role that diffusion bridge simulation plays in the statistical inference for diffusion processes. Any statistical analysis which requires the transition density of the process is halted whenever the latter is not explicitly available, which is typically the case. Hence it is challenging to fit diffusion models employed in applications to the incomplete data typically available. An interesting possibility is to approximate the intractable transition density using an appropriate MC scheme and carry
Sampling Conditioned Diffusions
, 2007
"... For many practical problems it is useful to be able to sample conditioned diffusions on a computer (e.g. in filtering/smoothing to sample from the conditioned distribution of the unknown signal given the known observations). We present a recently developed, SPDEbased method to tackle this problem. ..."
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Cited by 3 (2 self)
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For many practical problems it is useful to be able to sample conditioned diffusions on a computer (e.g. in filtering/smoothing to sample from the conditioned distribution of the unknown signal given the known observations). We present a recently developed, SPDEbased method to tackle this problem. The method is an infinite dimensional generalisation of the Langevin sampling technique. 1