Multivariate Diffusion Process Research Articles

Numerics with coordinate transforms for efficient Brownian dynamics simulations

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.

Transition density function expansion methods for portfolio optimization

AbstractIn this study, we introduce transition density function expansion methods inspired from Yang et al. (J Econom. 2019;209(2):256–288.) to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Utilizing Bellman's dynamic programming principle, we initially recast the conditional expectation via the transition density function pertinent to the diffusion process. Subsequently, we employ the Itô‐Taylor expansion and Delta expansion techniques to the transition density function associated with the multivariate diffusion process, facilitated by a quasi‐Lamperti transformation, aiming to derive explicit recursive expressions for expansion coefficient functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies, achieved through discretization methodologies with rigorous proof of expansion convergence in portfolio optimization. Both theoretical and practical demonstrations are presented to validate the convergence of these approximate techniques in addressing stochastic control challenges. To underscore the efficiency and precision of our proposed methods, we apply them to portfolio selection problems within several benchmark models, and highlight the reduced complexity in comparison to the current methodologies.

Augmented pseudo-marginal Metropolis\u2013Hastings for partially observed diffusion processes

We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying Itô stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis–Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. We therefore propose a novel augmentation scheme, that allows for conditioning on values of the latent process at the observation times, completely avoiding the need for resampling steps. We integrate over the uncertainty at the observation times with an additional Gibbs step. Connections between the resulting pseudo-marginal scheme and existing inference schemes for diffusion processes are made, giving a unified inference framework that encompasses Gibbs sampling and pseudo marginal schemes. The methodology is applied in three examples of increasing complexity. We find that our approach offers substantial increases in overall efficiency, compared to competing methods

Continuous-discrete smoothing of diffusions

Suppose X is a multivariate diffusion process that is observed discretely in time. At each observation time, a transformation of the state of the process is observed with noise. The smoothing problem consists of recovering the path of the process, consistent with the observations. We derive a novel Markov Chain Monte Carlo algorithm to sample from the exact smoothing distribution. The resulting algorithm is called the Backward Filtering Forward Guiding (BFFG) algorithm. We extend the algorithm to include parameter estimation. The proposed method relies on guided proposals introduced in [53]. We illustrate its efficiency in a number of challenging problems.

Modelling and inference for the movement of interacting animals

Abstract Statistical modelling of animal movement data is a rapidly growing area of research. Typically though, these models have been developed for analysing the tracks of individual animals and we lose sight of the impact animals have on each other with regards to their movement behaviours. We aim to develop a model with a flexible social framework that allows us to capture that information. Our approach is based on the concept of social hierarchies, and this is embedded in a multivariate diffusion process which models the movement of a group of animals. The possibility of switching between behavioural states facilitates dynamic social behaviours and we augment the observed data with sampled state switching times in order to model the animals' behaviour naturally in continuous time. In addition, this enables us to carry out exact inference in a Bayesian setting with the benefits of being able to handle regular, irregular and missing data. All movement and behaviour parameters are estimated with Markov chain Monte Carlo methods. We examine the capability of our model with simulated data before fitting it to GPS locations of five wild olive baboons Papio anubis. The results enable us to identify which animals are influencing the movement of others and when, which provides both a dynamic and long‐term static insight into the group's social behaviours. Our model offers a flexible method in continuous time with which to model the network of social interactions within animal movement. Doing so avoids the limitations caused by a discrete‐time approach and it allows us to capture rich information with regards to a group's social structure, leading to constructive applications in conservation and management decisions. However, currently it is a computationally expensive task to fit the model to data, which in turns limits extending the model to more fruitful but complex cases such as heterogeneity in space or individual characteristics. Furthermore, our social hierarchy approach assumes all relevant animals are tracked and that any interactions have some ordering, both of which narrow the scope within which this approach is appropriate.

Common‐factor stochastic volatility modelling with observable proxy

AbstractMulti‐asset modelling is of fundamental importance to financial applications such as risk management and portfolio selection. In this article, we propose a multivariate stochastic volatility modelling framework with a parsimonious and interpretable correlation structure. Building on well‐established evidence of common volatility factors among individual assets, we consider a multivariate diffusion process with a common‐factor structure in the volatility innovations. Upon substituting an observable market proxy for the common volatility factor, we markedly improve the estimation of several model parameters and latent volatilities. The model is applied to a portfolio of several important constituents of the S&P500 in the financial sector, with the VIX index as the common‐factor proxy. We find that the prediction intervals for asset forecasts are comparable to those of more complex dependence models, but that option‐pricing uncertainty can be greatly reduced by adopting a common‐volatility structure. The Canadian Journal of Statistics 48: 36–61; 2020 © 2020 Statistical Society of Canada

Adaptive invariant density estimation for ergodic diffusions over anisotropic classes

Consider some multivariate diffusion process $\mathbf{X}=(X_{t})_{t\geq0}$ with unique invariant probability measure and associated invariant density $\rho$, and assume that a continuous record of observations $X^{T}=(X_{t})_{0\leq t\leq T}$ of $\mathbf{X}$ is available. Recent results on functional inequalities for symmetric Markov semigroups are used in the statistical analysis of kernel estimators $\widehat{\rho}_{T}=\widehat{\rho}_{T}(X^{T})$ of $\rho$. For the basic problem of estimation with respect to $\mathrm{sup}$-norm risk under anisotropic Holder smoothness constraints, the proposed approach yields an adaptive estimator which converges at a substantially faster rate than in standard multivariate density estimation from i.i.d. observations.

A generalized dynamic discrete choice model for green vehicle adoption

Much is happening in the automotive industry and new models are in the market or are expected to be available soon. At the same time, environmental awareness, new regulations for increased fuel efficiency, and the need to diminish greenhouse gas emissions make small vehicles and alternative fuel vehicles more competitive. As a consequence, vehicle characteristics and consumer decisions will change rapidly in the short and medium run. Accounting for the dynamic of the problem is important to correctly forecast green vehicle acceptance and to evaluate eco-friendly policies. This paper proposes a generalized dynamic discrete choice approach that models purchase behavior and forecasts future preferences in a finite time horizon setting. The framework allows one-time purchases, repeated purchases, univariate and multivariate diffusion processes that capture the evolution of vehicle characteristics and dynamics in the market conditions. The models proposed are estimated using stated preference data collected in Maryland. Results show that the formulation with repeated purchases successfully captures changes in the market shares, and that the multivariate diffusion process adopted to model the evolution of fuel prices further improves both model fit and the ability to recover peaks in demand. The estimated coefficients have been applied to test different policy scenarios, including changes in fuel prices, vehicle purchase prices, and improvements of car characteristics. These policies have a high impact on the adoption of electric cars and on their diffusion in the marketplace.

Bayesian estimation of incompletely observed diffusions

We present a general framework for Bayesian estimation of incompletely observed multivariate diffusion processes. Observations are assumed to be discrete in time, noisy and incomplete. We assume the drift and diffusion coefficient depend on an unknown parameter. A data-augmentation algorithm for drawing from the posterior distribution is presented which is based on simulating diffusion bridges conditional on a noisy incomplete observation at an intermediate time. The dynamics of such filtered bridges are derived and it is shown how these can be simulated using a generalised version of the guided proposals introduced in Schauer, Van der Meulen and Van Zanten (2017, Bernoulli 23(4A)).

ON THE FUNCTIONAL ESTIMATION OF MULTIVARIATE DIFFUSION PROCESSES

We propose a nonparametric estimation theory for the occupation density, the drift vector, and the diffusion matrix of multivariate diffusion processes. The estimators are sample analogues to infinitesimal conditional expectations constructed as Nadaraya-Watson kernel averages. Mild assumptions are imposed on the statistical properties of the multivariate system to obtain limiting results. Harris recurrence is all that we require to show consistency and asymptotic (mixed) normality of the proposed functional estimators. The identification method and asymptotic theory apply to both stationary and nonstationary multivariate diffusion processes of the recurrent type.

Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators

In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.

A Generalized Dynamic Discrete Choice Model for Green Vehicle Adoption

Abstract: Much is happening in the automotive industry and new models are in the market or are expected to be available soon. At the same time, environmental awareness, new regulations for increased fuel efficiency, and the need to diminish greenhouse gas emissions make small vehicles and alternative fuel vehicles more competitive. As a consequence, vehicle characteristics and consumer decisions will change rapidly in the short and medium run. Accounting for the dynamic of the problem is important to correctly forecast green vehicle acceptance and to evaluate eco-friendly policies. This paper proposes a generalized dynamic discrete choice approach that models purchase behavior and forecasts future preferences in a finite time horizon setting. The framework allows one-time purchases, repeated purchases, univariate and multivariate diffusion processes that capture the evolution of vehicle characteristics and dynamics in the market conditions. The models proposed are estimated using stated preference data collected in Maryland. Results show that the formulation with repeated purchases successfully captures changes in the market shares, and that the multivariate diffusion process adopted to model the evolution of fuel prices further improves both model fit and the ability to recover peaks in demand. The estimated coefficients have been applied to test different policy scenarios, including changes in fuel prices, vehicle purchase prices, and improvements of car characteristics. These policies have a high impact on the adoption of electric cars and on their diffusion in the marketplace.

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

ABSTRACTStatistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.

Improved bridge constructs for stochastic differential equations

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Itô stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler–Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.

Exact adaptive pointwise drift estimation for multidimensional ergodic diffusions

The problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated. We propose an estimator which is sharp adaptive on scales of Sobolev smoothness classes. The analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values—such as the geometry of the diffusion coefficient—of the prototypical drift estimation problem for a large class of multidimensional diffusion processes. We further sketch generalizations of our results to arbitrary diffusions satisfying suitable Bernstein-type inequalities.

Thermal fluctuation statistics in a molecular motor described by a multidimensional master equation

We present a theoretical investigation of thermal fluctuation statistics in a molecular motor. Energy transfer in the motor is described using a multidimensional discrete master equation with nearest-neighbor hopping. In this theory, energy transfer leads to statistical correlations between thermal fluctuations in different degrees of freedom. For long times, the energy transfer is a multivariate diffusion process with constant drift and diffusion. The fluctuations and drift align in the strong-coupling limit enabling a one-dimensional description along the coupled coordinate. We derive formal expressions for the probability distribution and simulate single trajectories of the system in the near- and far-from-equilibrium limits both for strong and weak coupling. Our results show that the hopping statistics provide an opportunity to distinguish different operating regimes.

Maximum-likelihood estimation for diffusion processes via closed-form density expansions

This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of Watanabe [Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992) 139-159] on analysis of the generalized random variables under some standard sufficient conditions.

Preliminary test and estimation in some multifactor diffusion processes

We consider the general inference problem of the drift parameters matrices of m independent multivariate diffusion processes. In particular, we develop preliminary test and Stein-type estimation theory for m-parameters matrices which are suspected to satisfy some constraints. By using asymptotic distributional quadratic risk criterion, we analyze the relative performance of the pretest and shrinkage estimators. Further, we generalize some inference problems recently studied for a single multifactor Vasicek model. Finally, we carry out some simulation studies whose results corroborate our theoretical findings for observation periods of small and moderate lengths of time.

ADAPTIVE LASSO-TYPE ESTIMATION FOR MULTIVARIATE DIFFUSION PROCESSES

The least absolute shrinkage and selection operator (LASSO) is a widely used statistical methodology for simultaneous estimation and variable selection. It is a shrinkage estimation method that allows one to select parsimonious models. In other words, this method estimates the redundant parameters as zero in the large samples and reduces variance of estimates. In recent years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed multivariate diffusion processes. We prove oracle properties and also derive the asymptotic distribution of the LASSO estimator. This is a nontrivial extension of previous results by Wang and Leng (2007,Journal of the American Statistical Association, 102(479), 1039–1048) on LASSO estimation because of different rates of convergence of the estimators in the drift and diffusion coefficients. We perform simulations and real data analysis to provide some evidence on the applicability of this method.

Shrinkage strategies in some multiple multi-factor dynamical systems

In this paper, we are interested in estimation problem for the drift parameters matrices of m independent multivariate diffusion processes. More specifically, we consider the case where the m -parameters matrices are supposed to satisfy some uncertain constraints. Given such an uncertainty, we develop shrinkage estimators which improve over the performance of the maximum likelihood estimator (MLE). Under an asymptotic distributional quadratic risk criterion, we study the relative dominance of the established estimators. Further, we carry out simulation studies for observation periods of small and moderate lengths of time that corroborate the theoretical finding for which shrinkage estimators outperform over the MLE. The proposed method is useful in model assessment and variable selection.