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Statistical Inference for Multivariate Stochastic Differential Equations

Liu, Ge
http://rave.ohiolink.edu/etdc/view?acc_num=osu1562966204796479

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Statistics.
Multivariate stochastic differential equations (MVSDEs) are commonly used in many applications in fields such as biology, economics, mathematical finance, oceanography and many other scientific areas. Statistical inference based on discretely observed data requires estimating the transition density which is unknown for most models. Typically, one would estimate the transition density and use the approximation for statistical inference. However, many estimation methods will fail when the observations are too sparse or when the SDE models have a hierarchical structure. Making statistical inference for such models is also computationally demanding. We aim to implement an approximation method to make accurate and reliable statistical inference while taking the computation complexity into consideration. In this dissertation, we compare several approximation methods to estimate the transition density of MVSDEs and propose to use the data imputation method. We perform a thorough analysis of the data imputation strategy, in terms of where to impute the data and the amount of data imputation needed. We design data imputation strategies for a univariate SDE model and a MVSDE model. The strategy is generalized to be applicable to general MVSDE models that do not have explicitly known solutions. To demonstrate the data imputation approximation method, we study simulated data from the multivariate Ornstein-Uhlenbeck (MVOU) model and a latent hierarchical model. We explore the posterior distribution of the MVSDE model parameters in a Bayesian approach. In the Bayesian Markov Chain Monte Carlo algorithm we use data augmentation to understand how the approximation of the transition density affects the inference procedure. We give practical guidelines on balancing the computational demands with the need to provide reliable and accurate posterior inference. Simulations are used to evaluate these guidelines with two MVSDE models, one fully observed and one partially observed. We deliver robust posterior inference on the parameters of these MVSDE models. For illustration, we apply the methods to the analysis of oceanography float observations. In addition, we develop an exact sampling algorithm to estimate the transition density of the latent hierarchical MVSDE model. The approximated transition density is used in the pseudo-marginal Markov Chain Monte Carlo algorithm to explore the posterior distributions of the parameters of the latent MVSDE model. The exact sampling approximation allows us to obtain a less biased inference for the latent model compared to other approximation methods. Another contribution of this dissertation is that we use the Hellinger metric to measure the accuracy of the approximation of a probability density, either the transition density of the MVSDE process or the posterior densities of the parameters of the MVSDEs. We show that the Hellinger metric, evaluated exactly or empirically, is a great tool to assess the accuracy of the approximate densities.
Peter Craigmile (Advisor)
Radu Herbei (Advisor)
172 p.
Statistics
multivariate stochastic differential equations; data imputation; Bayesian data augmentation method; Bayesian MCMC; pseudo marginal MCMC; stochastic process;

Recommended Citations

Citations

  • Liu, G. (2019). Statistical Inference for Multivariate Stochastic Differential Equations [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562966204796479

    APA Style (7th edition)

  • Liu, Ge. Statistical Inference for Multivariate Stochastic Differential Equations. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1562966204796479.

    MLA Style (8th edition)

  • Liu, Ge. "Statistical Inference for Multivariate Stochastic Differential Equations." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562966204796479

    Chicago Manual of Style (17th edition)

osu1562966204796479
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This open access ETD is published by The Ohio State University and OhioLINK.