Stochastic Volatility Models Based on OU-Gamma Time Change: Theory and Estimation

We consider stochastic volatility models that are defined by an Ornstein–Uhlenbeck (OU)-Gamma time change. These models are most suitable for modeling financial time series and follow the general framework of the popular non-Gaussian OU models of Barndorff-Nielsen and Shephard. One current problem of these otherwise attractive nontrivial models is, in general, the unavailability of a tractable likelihood-based statistical analysis for the returns of financial assets, which requires the ability to sample from a nontrivial joint distribution. We show that an OU process driven by an infinite activity Gamma process, which is an OU-Gamma process, exhibits unique features, which allows one to explicitly describe and exactly sample from relevant joint distributions. This is a consequence of the OU structure and the calculus of Gamma and Dirichlet processes. We develop a particle marginal Metropolis–Hastings algorithm for this type of continuous-time stochastic volatility models and check its performance using simulated data. For illustration we finally fit the model to S&P500 index data.     

Inference for a change-point problem under an OU setting with unequal and unknown volatilities

An Ornstein–Uhlenbeck (OU) process is employed as a versatile model to capture the mean-reverting and stochastic evolution of many variables in various fields of applications including finance and economics. Within the OU setting, we develop a new estimation method to determine the unknown change-point location under the assumption that the volatilities before and after the change point in a time series are unequal. Our method hinges on the concept of a weighted least sum of squared errors approach and enhanced by a fusion of an iterative algorithm. The consistency of the change-point estimator is established. This article highlights a numerical implementation on simulated and observed financial market data demonstrating the significant flexibility and accuracy of our proposed modelling and estimation method. The Canadian Journal of Statistics 48: 62–78; 2020 © 2019 Statistical Society of Canada     

Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics

Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.     

Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility-covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility-covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.     

Continuous Time Wishart Process for Stochastic Risk

Risks are usually represented and measured by volatility–covolatility matrices. Wishart processes are models for a dynamic analysis of multivariate risk and describe the evolution of stochastic volatility–covolatility matrices, constrained to be symmetric positive definite. The autoregressive Wishart process (WAR) is the multivariate extension of the Cox, Ingersoll, Ross (CIR) process introduced for scalar stochastic volatility. As a CIR process it allows for closed-form solutions for a number of financial problems, such as term structure of T-bonds and corporate bonds, derivative pricing in a multivariate stochastic volatility model, and the structural model for credit risk. Moreover, the Wishart dynamics are very flexible and are serious competitors for less structural multivariate ARCH models.     

A UNIFIED FRAMEWORK FOR MODELLING WILDLIFE POPULATION DYNAMICS†

This paper proposes a unified framework for defining and fitting stochastic, discrete‐time, discrete‐stage population dynamics models. The biological system is described by a state‐space model, where the true but unknown state of the population is modelled by a state process, and this is linked to survey data by an observation process. All sources of uncertainty in the inputs, including uncertainty about model specification, are readily incorporated. The paper shows how the state process can be represented as a generalization of the standard Leslie or Lefkovitch matrix. By dividing the state process into subprocesses, complex models can be constructed from manageable building blocks. The paper illustrates the approach with a model of the British grey seal metapopulation, using sequential importance sampling with kernel smoothing to fit the model.     

Integrated OU Processes and Non-Gaussian OU-based Stochastic Volatility Models

Abstract. In this paper, we study the detailed distributional properties of integrated non-Gaussian Ornstein–Uhlenbeck (intOU) processes. Both exact and approximate results are given. We emphasize the study of the tail behaviour of the intOU process. Our results have many potential applications in financial economics, as OU processes are used as models of instantaneous variance in stochastic volatility (SV) models. In this case, an intOU process can be regarded as a model of integrated variance. Hence, the tail behaviour of the intOU process will determine the tail behaviour of returns generated by SV models.     

Estimation and Prediction of Functional Autoregressive Processes

We present a generalization of some previous works (Bosq, Mourid, Pumo) about the functional forecast of a Banach autoregressive processes. We are mainly concerned with order p , p >1, autoregressive processes which appear to be a natural extension of the well-known R d -valued autoregressive processes to a functional framework. This modelization provides an new approach for estimating and for predicting a continuous time stochastic process over an entire time interval. Using results from [12] we prove asymptotic properties of estimators of the parameters and predictors which are based upon a principal component decomposition of a Hilbert-Schmidt operator with unknown eigenvectors.     

Improved Simulation Techniques for First Exit Time of Neural Diffusion Models

We consider the fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) of two one-dimensional neural diffusion models, represented by the Ornstein–Uhlenbeck (OU) process and a stochastic space-clamped FitzHugh–Nagumo (FHN) system. The numerical methods are described and the convergence rates for the MFET analyzed. A boundary test improves the rate of convergence from order one-half to order 1. We show how to apply the multi-level Monte Carlo (MLMC) method to an Euler time-stepping method with boundary test and this improves the Monte Carlo computation of the MFET.     

On two conjectures about perturbations of the stochastic growth rate

The stochastic growth rate describes long-run growth of a population that lives in a fluctuating environment. Perturbation analysis of the stochastic growth rate provides crucial information for population managers, ecologists and evolutionary biologists. This analysis quantifies the response of the stochastic growth rate to changes in demographic parameters. A form of this analysis deals with changes that only occur in some environmental states. Caswell put forth two conjectures about environment-specific perturbations of the stochastic growth rate. The conjectures link the stationary distribution of the stochastic environmental process with the magnitude of some environment-specific perturbations. This note disproves one conjecture and proves the other.     

Multi-level Monte Carlo methods for the approximation of invariant measures of stochastic differential equations

We develop a framework that allows the use of the multi-level Monte Carlo (MLMC) methodology (Giles in Acta Numer. 24:259–328, 2015. https://doi.org/10.1017/S096249291500001X) to calculate expectations with respect to the invariant measure of an ergodic SDE. In that context, we study the (over-damped) Langevin equations with a strongly concave potential. We show that when appropriate contracting couplings for the numerical integrators are available, one can obtain a uniform-in-time estimate of the MLMC variance in contrast to the majority of the results in the MLMC literature. As a consequence, a root mean square error of $$\mathcal {O}(\varepsilon )$$ is achieved with $$\mathcal {O}(\varepsilon ^{-2})$$ complexity on par with Markov Chain Monte Carlo (MCMC) methods, which, however, can be computationally intensive when applied to large datasets. Finally, we present a multi-level version of the recently introduced stochastic gradient Langevin dynamics method (Welling and Teh, in: Proceedings of the 28th ICML, 2011) built for large datasets applications. We show that this is the first stochastic gradient MCMC method with complexity $$\mathcal {O}(\varepsilon ^{-2}|\log {\varepsilon }|^{3})$$, in contrast to the complexity $$\mathcal {O}(\varepsilon ^{-3})$$ of currently available methods. Numerical experiments confirm our theoretical findings.     

Some Nonparametric Asymptotic Results for a Class of Stochastic Processes

This article provides a solution of a generalized eigenvalue problem for integrated processes of order 2 in a nonparametric framework. Our analysis focuses on a pair of random matrices related to such integrated process. The matrices are constructed considering some weight functions. Under asymptotic conditions on such weights, convergence results in distribution are obtained and the generalized eigenvalue problem is solved. Differential equations and stochastic calculus theory are used.     

概述随机非线性动力系统在经济周期研究中的运用

非线性动力学为经济周期的动态分析提供了全新的思路和方法,打破了传统的均衡线性分析的范式。考虑到复杂经济系统中本质的表现为非线性,而且还不可避免地存在随机噪声。因此,为了深入地探究经济周期的动力学形成机理,将随机非线性动力系统引入到经济周期问题的研究中。通过研究随机模型的稳定性、分岔、混沌和随机最优控制,实现对宏观经济动态演化和运行的评价、监测与控制。这不仅拓宽了随机动力学在宏观经济领域中的应用,而且也为宏观经济运行的研究提供了一个全新的思路和方法。     

Analysing the interevent time distribution to identify seismicity phases: a Bayesian nonparametric approach to the multiple-changepoint problem

In the study of earthquakes, several aspects of the underlying physical process, such as the time non-stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple-changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.     

Describing the Dynamics of Attention to TV Commercials: A Hierarchical Bayes Analysis of the Time to Zap an Ad

This paper provides insights into the dynamics of attention to TV commercials via an analysis of the length of time that commercials are viewed before being 'zapped'. The model, which incorporates a flexible baseline hazard rate and captures unobserved heterogeneity across both consumers and commercials using a hierarchical Bayes approach, is estimated on two datasets in which commercial viewing is captured by a passive online device that continually monitors a household's TV viewing. Consistent with previous findings in psychology about the nature of attentional engagement in TV viewing, baseline hazard rates are found to be non-monotonic. In addition, the data show considerable ad-to-ad and household-to-household heterogeneity in zapping behavior. While one of the datasets contains some information on characteristics of the ads, these data do not reveal any firm links between the ad heterogeneity and the ad characteristics. A number of methodological and computational issues arise in the hierarchical Bayes analysis.     

Local quantile regression

Quantile regression is a technique to estimate conditional quantile curves. It provides a comprehensive picture of a response contingent on explanatory variables. In a flexible modeling framework, a specific form of the conditional quantile curve is not a priori fixed. This motivates a local parametric rather than a global fixed model fitting approach. A nonparametric smoothing estimator of the conditional quantile curve requires to balance between local curvature and stochastic variability. In this paper, we suggest a local model selection technique that provides an adaptive estimator of the conditional quantile regression curve at each design point. Theoretical results claim that the proposed adaptive procedure performs as good as an oracle which would minimize the local estimation risk for the problem at hand. We illustrate the performance of the procedure by an extensive simulation study and consider a couple of applications: to tail dependence analysis for the Hong Kong stock market and to analysis of the distributions of the risk factors of temperature dynamics.     

随机波动HJM框架下信用利差模型及实证研究

将随机波动引入到具有违约风险的HJM模型中,基于无套利条件推导得出随机波动HJM模型框架下信用利差的漂移项限制条件,从而构建了随机波动HJM框架下的信用利差模型,并基于波动率结构的适当设定对模型进行马尔科夫简化,进而利用该模型对中国可违约债券市场信用利差的动态特性进行实证分析。结果发现：短期信用利差的动态特性具有显著的随机波动特征,而随机波动HJM框架下的信用利差模型可以很好地刻画中国可违约债券市场信用利差的动态特性。     

Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop efficient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.     

Piecewise deterministic Markov processes applied to fatigue crack growth modelling

In this paper, we use a particular piecewise deterministic Markov process (PDMP) to model the evolution of a degradation mechanism that may arise in various structural components, namely, the fatigue crack growth. We first derive some probability results on the stochastic dynamics with the help of Markov renewal theory: a closed-form solution for the transition function of the PDMP is given. Then, we investigate some methods to estimate the parameters of the dynamical system, involving Bogolyubov's averaging principle and maximum likelihood estimation for the infinitesimal generator of the underlying jump Markov process. Numerical applications on a real crack data set are given.     

Stochastic variational inference for large-scale discrete choice models using adaptive batch sizes

Discrete choice models describe the choices made by decision makers among alternatives and play an important role in transportation planning, marketing research and other applications. The mixed multinomial logit (MMNL) model is a popular discrete choice model that captures heterogeneity in the preferences of decision makers through random coefficients. While Markov chain Monte Carlo methods provide the Bayesian analogue to classical procedures for estimating MMNL models, computations can be prohibitively expensive for large datasets. Approximate inference can be obtained using variational methods at a lower computational cost with competitive accuracy. In this paper, we develop variational methods for estimating MMNL models that allow random coefficients to be correlated in the posterior and can be extended easily to large-scale datasets. We explore three alternatives: (1) Laplace variational inference, (2) nonconjugate variational message passing and (3) stochastic linear regression. Their performances are compared using real and simulated data. To accelerate convergence for large datasets, we develop stochastic variational inference for MMNL models using each of the above alternatives. Stochastic variational inference allows data to be processed in minibatches by optimizing global variational parameters using stochastic gradient approximation. A novel strategy for increasing minibatch sizes adaptively within stochastic variational inference is proposed.