Inference for stochastic volatility models using time change transformations

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

Impact of global liquidity on Indian financial markets and monetary policy outcomes: An ARDL approach

References and Further Reading

Suppose that one wishes to sample from the density π(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution π(u|x) can be defined, giving the joint distribution π(x, u) = π(x)π(u|x). A MCMC scheme that samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches. The revolutionary algorithm of Swendsen and Wang is one such example. Besides reviewing the Swendsen-Wang algorithm and its generalizations, this article introduces a new auxiliary variable method called partial decoupling. Two applications in Bayesian image analysis are considered: a binary classification problem in which partial decoupling out performs Swendsen-Wang and single-site Metropolis methods, and a positron emission tomography (PET) reconstruction that uses the gray level prior of Geman and McClure. A generalized Swendsen–Wang algorithm is developed for this problem, which reduces the computing time to the point where MCMC is a viable method of posterior exploration.

Where several different kinds of geophysical datasets have been acquired from a particular region, each of these can contain valuable information about the Earth, which may not be present in the other datasets. Jointly determining a common model, therefore, often gives a more thorough and more constrained description of the Earth structure than considering each dataset individually. For example, a seismic velocity inversion is only weakly constrained by first arrival seismic refraction data, but considering it alongside Magneto‐Telluric (MT) and gravity data can greatly assist in the constraint (Jegen‐Kulcsar et al., 2009). Strategies for joint inversion are therefore an active area of research. To date, most schemes for accomplishing this have been deterministic in nature. Using a deterministic technique often means that it is conceptually difficult to include prior beliefs about the system under determination, uncertainties both in measurement and the relationship between the different physical quantities (velocity, resistivity, density), and the discrepancy between the model and the real Earth. Statistical strategies such as MCMC (Markov Chain Monte Carlo) model searches exist for assessing this kind of problem, but the number of potentially computationally expensive forward model runs required to effectively sample the whole model space and thus achieve a meaningful result is normally prohibitively high (> 105), even for simple 1D models, so such schemes are not generally implemented. However, a technique known as emulation is used in various scientific fields eg. cosmology (Vernon and Goldstein, 2009), whereby computationally expensive forward modelling code (a simulator) is approximated by an uncertainty‐calibrated computationally cheap function. Here we apply emulation to the problem of stochastic joint model determination. We show that emulation can be used to quickly exclude large areas of implausible model space, allowing fast updating of beliefs about an Earth structure. It thus provides a means by which the input model space for a deterministic inversion or MCMC scheme can be greatly reduced. We also show how an emulator can, by itself, effectively constrain a region of the Earth. We demonstrate the concept using a 1D model.

Ordinal stochastic volatility (OSV) models were recently developed and fitted by Müller & Czado (2009) to account for the discreteness of financial price changes, while allowing for stochastic volatility (SV). The model allows for exogenous factors both on the mean and volatility level. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is followed to facilitate estimation in these parameter driven models. In this paper the applicability of the OSV model to financial stocks with different levels of trading activity is investigated and the influence of time between trades, volume, day time and the number of quotes between trades is determined. In a second focus we compare the performance of OSV models and SV models. The analysis shows that the OSV models which account for the discreteness of the price changes perform quite well when applied to such data sets.

In this paper, we consider a jump-diffusion two-factor model which stochastic volatility to obtain the yield curves efficiently. As this is a jump-diffusion model, the estimation of the market prices of risk is not possible unless a closed form solution is known for the model. Then, we obtain some results that allow us to estimate all the risk-neutral functions, which are necessary to obtain the yield curves, directly from data in the markets. As the market prices of risk are included in the risk-neutral functions, they can also be obtained. Finally, we use US Treasury Bill data, a nonparametric approach, numerical differentiation and Monte Carlo simulation approach to obtain the yield curves. Then, we show the advantages of considering the volatility as second stochastic factor and our approach in an interest rate model.

This chapter provides a step-by-step review of Monte Carlo (MC) methods for filtering in general nonlinear and non-Gaussian dynamic models, also known as state-space models or hidden Markov models. The chapter is organized as follows. Section 11.2 introduces the basic notation, results, and references for the general class of Gaussian dynamic linear models (DLM), the AR(1) plus noise model, and the standard stochastic volatility model with AR(1) dynamics. Sections 11.3 and 11.4 discuss particle filters for state learning with fixed parameters (also known as pure filtering) and particle filters for state and parameter learning, respectively. Section 11.5 deals with general issues, such as MC error, sequential model checking, particle smoothing, and the interaction between particle filters and Markov chain Monte Carlo (MCMC) schemes.

Variational Bayes approximation of factor stochastic volatility models

In this paper, we compare the forecast ability of GARCH(1,1) and stochastic volatility models for interest rates. The stochastic volatility is estimated using Markov chain Monte Carlo methods. The comparison is based on daily data from 1994 to 1996 for the ten year swap rates for Deutsch Mark, Japanese Yen, and Pound Sterling. Various forecast horizons are considered. It turns out that forecasts based on stochastic volatility models are in most cases superiour to those obtained by GARCH(1,1) models.

We investigate the spread of American foulbrood (AFB), a disease caused by the bacterium Paenibacillus larvae, that affects bees and can be extremely damaging to beehives. Our dataset comes from an inspection period carried out during an AFB epidemic of honeybee colonies on the island of Jersey during the summer of 2010. The data include the number of hives of honeybees, location and owner of honeybee apiaries across the island. We use a spatial SIR model with an underlying owner network to simulate the epidemic and characterize the epidemic using a Markov chain Monte Carlo (MCMC) scheme to determine model parameters and infection times (including undetected ‘occult’ infections). Likely methods of infection spread can be inferred from the analysis, with both distance- and owner-based transmissions being found to contribute to the spread of AFB. The results of the MCMC are corroborated by simulating the epidemic using a stochastic SIR model, resulting in aggregate levels of infection that are comparable to the data. We use this stochastic SIR model to simulate the impact of different control strategies on controlling the epidemic. It is found that earlier inspections result in smaller epidemics and a higher likelihood of AFB extinction.

On leverage in a stochastic volatility model

Airport air traffic is one of the most important and hardest one among all the airport data forecasts. In this paper, the Markov Chain Monte Carlo (MCMC) method of applied theory of statistics has been introduced into the aviation sector, and the discussion on airport air traffic forecast has been conducted taking Shanghai airport as the application background. MCMC method is designed to solve the problem parameters expectations where some baysian inference can not be directly calculated. This paper conducts the iterative computation using WinBUGS, EViews is used to estimate the initial iteration parameter of the regression model, and it is on this basis that the regression equation under the MCMC method is obtained.

The preconditioned Crank--Nicolson (pCN) method is a Markov chain Monte Carlo (MCMC) scheme, specifically designed to perform Bayesian inferences in function spaces. Unlike many standard MCMC algorithms, the pCN method can preserve the sampling efficiency under the mesh refinement, a property referred to as being dimension independent. In this work we consider an adaptive strategy to further improve the efficiency of pCN. In particular we develop a hybrid adaptive MCMC method: the algorithm performs an adaptive Metropolis scheme in a chosen finite dimensional subspace and a standard pCN algorithm in the complement space of the chosen subspace. We show that the proposed algorithm satisfies certain important ergodicity conditions. Finally with numerical examples we demonstrate that the proposed method has competitive performance with existing adaptive algorithms.

Patz and Junker (1999) describe a general Markov chain Monte Carlo (MCMC) strategy, based on Metropolis-Hastings sampling, for Bayesian inference in complex item response theory (IRT) settings. They demonstrate the basic methodology using the two-parameter logistic (2PL) model. In this paper we extend their basic MCMC methodology to address issues such as non-response, designed missingness, multiple raters, guessing behavior and partial credit (polytomous) test items. We apply the basic MCMC methodology to two examples from the National Assessment of Educational Progress 1992 Trial State Assessment in Reading: (a) a multiple item format (2PL, 3PL, and generalized partial credit) subtest with missing response data; and (b) a sequence of rated, dichotomous short-response items, using a new IRT model called the generalized linear logistic test model (GLLTM).