Parameters Of Stochastic Differential Equations Research Articles

Fitting the Itô Stochastic differential equation to the COVID-19 data in Turkey

In this study, COVID-19 data in Turkey is investigated by Stochastic Differential Equation Modeling (SDEM). Firstly, parameters of SDE which occur in mentioned epidemic problem are estimated by using the maximum likelihood procedure. Then, we have obtained reasonable Stochastic Differential Equation (SDE) based on the given COVID-19 data. Moreover, by applying Euler-Maruyama Approximation Method trajectories of SDE are achieved. The performances of trajectories are established by Chi-Square criteria. The results are acquired by using statistical software R-Studio.These results are also corroborated by graphical representation.

A Comparative Study of Likelihood Approximations for Univariate Diffusions

Abstract Maximum likelihood estimation of the parameters of stochastic differential equations commonly used in finance requires numerical approximation of their transitional probability density functions. This article undertakes a comparative study of the accuracy of Hermite polynomial expansion approximations for univariate diffusions and checks how the accuracy of the existing methods responds to increasing the order of the approximation. It is found that one class of expansion dealing with irreducible diffusions is particularly problematic due to the need to evaluate a number of troublesome integrals. A Gaussian quadrature is introduced which resolves the problem and improves the reliability of the expansion. A simulation study demonstrates all the methods in action and provides insight into the practical aspects of using these expansions. An empirical application using data on the VIX indicates that the proposed method based on the Gaussian quadrature performs very well when applied to financial data.

Learning stochastic closures using ensemble Kalman inversion

Abstract Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore, researchers often seek simpler descriptions that describe complex phenomena without numerically resolving all the interacting components. Stochastic differential equations (SDEs) arise naturally as models in this context. The growth in data acquisition, both through experiment and through simulations, provides an opportunity for the systematic derivation of SDE models in many disciplines. However, inconsistencies between SDEs and real data at short time scales often cause problems, when standard statistical methodology is applied to parameter estimation. The incompatibility between SDEs and real data can be addressed by deriving sufficient statistics from the time-series data and learning parameters of SDEs based on these. Here, we study sufficient statistics computed from time averages, an approach that we demonstrate to lead to sufficient statistics on a variety of problems and that has the secondary benefit of obviating the need to match trajectories. Following this approach, we formulate the fitting of SDEs to sufficient statistics from real data as an inverse problem and demonstrate that this inverse problem can be solved by using ensemble Kalman inversion. Furthermore, we create a framework for non-parametric learning of drift and diffusion terms by introducing hierarchical, refinable parameterizations of unknown functions, using Gaussian process regression. We demonstrate the proposed methodology for the fitting of SDE models, first in a simulation study with a noisy Lorenz ’63 model, and then in other applications, including dimension reduction in deterministic chaotic systems arising in the atmospheric sciences, large-scale pattern modeling in climate dynamics and simplified models for key observables arising in molecular dynamics. The results confirm that the proposed methodology provides a robust and systematic approach to fitting SDE models to real data.

Hybrid estimators for stochastic differential equations from reduced data

We treat parametric inference for unknown parameters of stochastic differential equations from discrete observations from the viewpoint of computational cost. Following Kamatani et al. (Bull Inf Cybern 48:19–35, 2016) and Kaino and Uchida (Hybrid estimators for ergodic diffusion processes from thinned data, 2018), we present the asymptotic results of the multi-step estimators with the initial Bayes type estimators for both ergodic and non-ergodic diffusion type processes. The initial Bayes type estimators are constructed by means of both the reduced data and the thinned data obtained from the full data. Some examples and simulation results are also given.

Parameter estimation in SDEs via the Fokker–Planck equation: Likelihood function and adjoint based gradient computation

In this paper we consider the problem of identifying parameters in stochastic differential equations. For this purpose, we transform the originally stochastic and nonlinear state equation to a deterministic linear partial differential equation for the transition probability density. We provide an appropriate likelihood cost function for parameter fitting, and derive an adjoint based approach for the computation of its gradient.

Estimation of parameters in stochastic differential equations with two random effects

In this paper we investigate consistency and asymptotic normality of the posterior distribution of the parameters in the stochastic differential equations (SDE’s) with diffusion coefficients depending nonlinearly on a random variables  and  (the random effects).The distributions of the random effects  and  depends on unknown parameters which are to be estimated from the continuous observations of the independent processes . We propose the Gaussian distribution for the random effect  and the exponential distribution for the random effect   , we obtained an explicit formula for the likelihood function and find the estimators of the unknown parameters in the random effects.

Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient

We consider $N$ independent stochastic processes $(X_i(t), t\in [0,T_i])$, $i=1,\ldots, N$, defined by a stochastic differential equation with diffusion coefficients depending on a random variable $\phi_i$. The distribution of the random effect $\phi_i$ depends on unknown population parameters which are to be estimated from a discrete observation of the processes $(X_i)$. The likelihood generally does not have any closed form expression. Two estimation methods are proposed: one based on the Euler approximation of the likelihood and another based on estimations of the random effects. When the distribution of the random effects is Gamma, the asymptotic properties of the estimators are derived when both $N$ and the number of observations per subject tend to infinity. The estimators are computed on simulated data for several models and show good performances.

A penalized simulated maximum likelihood approach in parameter estimation for stochastic differential equations

We consider the problem of estimating parameters of stochastic differential equations (SDEs) with discrete-time observations that are either completely or partially observed. The transition density between two observations is generally unknown. We propose an importance sampling approach with an auxiliary parameter when the transition density is unknown. We embed the auxiliary importance sampler in a penalized maximum likelihood framework which produces more accurate and computationally efficient parameter estimates. Simulation studies in three different models illustrate promising improvements of the new penalized simulated maximum likelihood method. The new procedure is designed for the challenging case when some state variables are unobserved and moreover, observed states are sparse over time, which commonly arises in ecological studies. We apply this new approach to two epidemics of chronic wasting disease in mule deer.

Posterior inference on parameters of stochastic differential equations via non-linear Gaussian filtering and adaptive MCMC

This article is concerned with Bayesian estimation of parameters in non-linear multivariate stochastic differential equation (SDE) models occurring, for example, in physics, engineering, and financial applications. In particular, we study the use of adaptive Markov chain Monte Carlo (AMCMC) based numerical integration methods with non-linear Kalman-type approximate Gaussian filters for parameter estimation in non-linear SDEs. We study the accuracy and computational efficiency of gradient-free sigma-point approximations (Gaussian quadratures) in the context of parameter estimation, and compare them with Taylor series and particle MCMC approximations. The results indicate that the sigma-point based Gaussian approximations lead to better approximations of the parameter posterior distribution than the Taylor series, and the accuracy of the approximations is comparable to that of the computationally significantly heavier particle MCMC approximations.

Self-organized criticality as Witten-type topological field theory with spontaneously broken Becchi-Rouet-Stora-Tyutin symmetry

Here, a scenario is proposed, according to which a generic self-organized critical (SOC) system can be looked upon as a Witten-type topological field theory (W-TFT) with spontaneously broken Becchi-Rouet-Stora-Tyutin (BRST) symmetry. One of the conditions for the SOC is the slow driving noise, which unambiguously suggests Stratonovich interpretation of the corresponding stochastic differential equation (SDE). This, in turn, necessitates the use of Parisi-Sourlas-Wu stochastic quantization procedure, which straightforwardly leads to a model with BRST-exact action, i.e., to a W-TFT. In the parameter space of the SDE, there must exist full-dimensional regions where the BRST symmetry is spontaneously broken by instantons, which in the context of SOC are essentially avalanches. In these regions, the avalanche-type SOC dynamics is liberated from overwise a rightful dynamics-less W-TFT, and a Goldstone mode of Fadeev-Popov ghosts exists. Goldstinos represent moduli of instantons (avalanches) and being gapless are responsible for the critical avalanche distribution in the low-energy, long-wavelength limit. The above arguments are robust against moderate variations of the SDE's parameters and the criticality is "self-tuned." The proposition of this paper suggests that the machinery of W-TFTs may find its applications in many different areas of modern science studying various physical realizations of SOC. It also suggests that there may in principle exist a connection between some SOC's and the concept of topological quantum computing.

Approximation of transition densities of stochastic differential equations by saddlepoint methods applied to small-time Ito–Taylor sample-path expansions

Likelihood-based inference for parameters of stochastic differential equation (SDE) models is challenging because for most SDEs the transition density is unknown. We propose a method for estimating the transition density that involves expanding the sample path as an Ito---Taylor series, calculating the moment generating function of the retained terms in the Ito---Taylor expansion, then employing a saddlepoint approximation. We perform a numerical comparison with two other methods similarly based on small-time expansions and discuss the pros and cons of our new method relative to other approaches.

Parameter Estimation for a Bidimensional Partially Observed Ornstein-Uhlenbeck Process with Biological Application

We consider a bidimensional Ornstein-Uhlenbeck process to describe the tissue microvascularisation in anti-cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is the estimation of the SDE parameters. We use the main advantage of a one-dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion. We also propose a recursive computation of the gradient and hessian of the log-likelihood based on Kalman filtering, which allows to implement an easy numerical maximisation of the likelihood and the exact maximum likelihood estimator (MLE). Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the MLE. We show that this ARMA property can be generalised to a higher dimensional underlying Ornstein-Uhlenbeck diffusion. We compare this estimator with the one obtained by the well-known EM algorithm on simulated data.

The estimation of parameters for stochastic differential equations using neural networks

In this article artificial neural network (ANN) is used to estimate parameters of stochastic differential equations (SDEs) given the discrete output variables of the equations. Some techniques are used to improve ANN performance in data preparation and ANN training procedure. In particular, an increase in the number of Wiener processes used to create training data set raises the accuracy of estimates significantly. The analysis of the results suggests that ANN can predict parameters of SDEs accurately under certain noise level regimes using an appropriate network architecture.

An algorithm for the estimation of parameters in models with stochastic differential equations

An algorithm for the estimation of parameters of stochastic differential equations (SDEs) is presented. It is based on a nonlinear weighted least-squares formulation, in which the objective function is evaluated based on mean values of the measured variables predicted through an Euler discretisation of the SDEs and their integration by Monte-Carlo simulation. The problem is solved using a Levenberg–Marquardt algorithm. The presence of simulation noise is handled by choosing a convergence criterion based on the noise level and by ensuring that the optimality criterion is met for a large simulation size and hence a low noise level. In order to increase the reliability of the algorithm and to decrease its computational cost, stochastic sensitivity equations are derived. Furthermore, the number of trajectories used in the Monte-Carlo simulations is changed adaptively throughout the execution of the algorithm. This leads to a significant decrease in computational requirements. These concepts are illustrated on a simple example and a more complex model of polymer rheology. In all cases, parameter estimates close to the true parameter values are identified.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported.

On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential Equations*

Abstract. A method for estimating the parameters of stochastic differential equations (SDEs) by simulated maximum likelihood is presented. This method is feasible whenever the underlying SDE is a Markov process. Estimates are compared to those generated by indirect inference, discrete and exact maximum likelihood. The technique is illustrated with reference to a one‐factor model of the term structure of interest rates using 3‐month US Treasury Bill data.

The Application of Moment Method for Parameter Identification of Stochastic Systems

A moment method for parameter identification of stochastic systems is examined. Parameters of stochastic differential equations are defined from a solution of moment equations. Simple formulae are obtained for parameters of second order stochastic differential equation with linear damping and nonlinear restriction. These formulae are derived from a stationary solution of moment equations for first, second, third and fourth order moments. Some calculations by these formulae are presented.

ESTIMATION IN CONTINUOUS-TIME STOCHASTIC VOLATILITY MODELS USING NONLINEAR FILTERS

The stylized facts of stock prices, interest and exchange rates have led econometricians to propose stochastic volatility models in both discrete and continuous time. However, the volatility as a measure of economic uncertainty is not directly observable in the financial markets. The objective of the continuous-discrete filtering problem considered here is to obtain estimates of the stock price and, in particular, the volatility using discrete-time observations of the stock price. Furthermore, the nonlinear filter acts as an important part of a proposed method for maximum likelihood for estimating embedded parameters in stochastic differential equations. In general, only approximate solutions to the continuous-discrete filtering problem exist in the form of a set of ordinary differential equations for the mean and covariance of the state variables. In the present paper the small-sample properties of a second order filter is examined for some bivariate stochastic volatility models and the new combined parameter and state estimation method is applied to US stock market data.

Estimating the parameters of stochastic differential equations

Two maximum likelihood methods for estimating the parameters of stochastic differential equations (SDEs) from time-series data are proposed. The first is that of simulated maximum likelihood in which a nonparametric kernel is used to construct the transitional density of an SDE from a series of simulated trials. The second approach uses a spectral technique to solve the Kolmogorov equation satisfied by the transitional probability density. The exact likelihood function for a geometric random walk is used as a benchmark against which the performance of each method is measured. Both methods perform well with the spectral method returning results which are practically identical to those derived from the exact likelihood. The technique is illustrated by modelling interest rates in the UK gilts market using a fundamental one-factor term-structure equation for the instantaneous rate of interest.