Non-homogeneous Diffusion Process Research Articles

Inference with Non-Homogeneous Lognormal Diffusion Processes Conditioned on Nearest Neighbor

In this work, we approach the forecast problem for a general non-homogeneous diffusion process over time with a different perspective from the classical one. We study the main characteristic functions as mean, mode, and α-quantiles conditioned on a future time, not conditioned on the past (as is normally the case), and we observe the specific formula in some interesting particular cases, such as Gompertz, logistic, or Bertalanffy diffusion processes, among others. This study aims to enhance classical inference methods when we need to impute data based on available information, past or future. We develop a simulation and obtain a dataset that is closer to reality, where there is no regularity in the number or timing of observations, to extend the traditional inference method. For such data, we propose using characteristic functions conditioned on the past or the future, depending on the closest point at which we aim to perform the imputation. The proposed inference procedure greatly reduces imputation errors in the simulated dataset.

Carbon nanotube-loaded copper-nickel ferrite activated persulfate system for adsorption and degradation of oxytetracycline hydrochloride

In this study, a new composite (MWCNTs-CuNiFe2O4) prepared by loading magnetic CuNiFe2O4 particles onto carboxylated carbon nanotubes (MWCNTs) through co-precipitation was applied to remove oxytetracycline hydrochloride (OTC-HCl) in solution. The magnetic properties of this composite could address of the issue of difficulty associated with the separation of MWCNTs from mixtures when applied as an adsorbent. In addition to the good adsorption properties recorded for MWCNTs-CuNiFe2O4 towards OTC-HCl, this developed composite could be used to activate potassium persulfate (KPS) for an efficient degradation of OTC-HCl. The MWCNTs-CuNiFe2O4 was systematically characterized using Vibrating Sample Magnetometer (VSM), Electron Paramagnetic Resonance (EPR) and X-ray Photoelectron Spectroscopy (XPS). The influence of dose of MWCNTs-CuNiFe2O4, the initial pH, the amount of KPS and the reaction temperature on the adsorption and degradation of OTC-HCl by MWCNTs-CuNiFe2O4 were discussed. The adsorption and degradation experiments showed that MWCNTs-CuNiFe2O4 exhibited an adsorption capacity of 270 mg·g−1 for OTC-HCl with the removal efficiency 88.6% at 303 K (at an initial pH 3.52, 5 mg KPS, 10 mg composite, 10 mL reaction concentration 300 mg·L−1 of OTC-HCl). The Langmuir and Koble-Corrigan models were used to describe the equilibrium process while the Elovich equation and Double constant model were suitable to describe the kinetic process. The adsorption process was based on single-molecule layer reaction and non-homogeneous diffusion process. The mechanisms of adsorption were complexation and hydrogen bond whereas active species such as SO4‧-, ‧OH and 1O2 were confirmed to have played a major role in the degradation of OTC-HCl. The composite was also found to be very stable with good reusability property. These results confirm the good potential associated with the use of MWCNTs-CuNiFe2O4/KPS system for the removal of some typical pollutants from wastewater.

Sequential estimation for nonhomogeneous Ornstein-Uhlenbeck processes

ABSTRACTIn this paper, we propose a stochastic process, which is a class of nonhomogeneous diffusion process from the perspective of the corresponding nonlinear stochastic differential equation. The parameter included in the drift term are estimated by sequential maximum likelihood methodology on the basis of continuous sampling of the process. The sequential estimators are proved to be closed, unbiased, strongly consistent, normally distributed, and optimal in the mean square sense.

Estimating a non-homogeneous Gompertz process with jumps as model of tumor dynamics

A non-homogeneous stochastic model based on a Gompertz-type diffusion process with jumps is proposed to describe the evolution of a solid tumor subject to an intermittent therapeutic program. Each therapeutic application, represented by a jump in the process, instantly reduces the tumor size to a fixed value and, simultaneously, increases the growth rate of the model to represent the toxicity of the therapy. This effect is described by introducing a time-dependent function in the drift of the process. The resulting model is a combination of several non-homogeneous diffusion processes characterized by different drifts, whose transition probability density function and main characteristics are studied. The study of the model is performed by distinguishing whether the therapeutic instances are fixed in advance or guided by a strategy based on the mean of the first-passage-time through a control threshold. Simulation studies are carried out for different choices of the parameters and time-dependent functions involved.

Generalized fixed-effects and mixed-effects parameters height–diameter models with diffusion processes

Statistical models using stochastic differential equations (SDEs) to describe dynamically evolving natural systems are appearing in the scientific literature with some regularity in recent years. In this study, the SDE mixed-effects parameter models based on a Vasicek non-homogeneous diffusion process are formulated. The breast height diameter-dependent drift function additionally depends on deterministic function that describes the dynamic of certain exogenous stand variables (crown height, ch, crown width, cw, mean breast height diameter, d0, mean height, h0, age, A, soil fertility index, SFI, stocking level, S) versus breast height diameter. The mixed-effects parameters SDE models included a random parameter that affected the models asymptote. The parameter estimators are evaluated by maximum likelihood procedure. The objective of the research was to develop a generalized mixed-effects parameters SDE height–diameter models and to illustrate issues using dataset of Scots pine trees (Pinus sylvestris L.) in Lithuania with the breast height diameter outside the bark larger than 0 cm. The parameters of all used models were estimated using an estimation dataset and were evaluated using a validation dataset. The new developed height–diameter models are an improvement over exogenous stand variables, in that it can be calibrated to a new stand with observed height–diameter pairs, thus improving height prediction.

A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities

Incomplete financial markets are considered, defined by a multi-dimensional non-homogeneous diffusion process, being the direct sum of an Ito process (the price process), and another non-homogeneous diffusion process (the exogenous process, representing exogenous stochastic sources). The drift and the diffusion matrix of the price process are functions of the time, the price process itself and the exogenous process. In the context of such markets and for power utility functions, it is proved that the stochastic control problem consisting of optimizing the expected utility of the terminal wealth, has a classical solution (i.e. \(C^{1,2}\)). This result paves the way to a study of the optimal portfolio problem in incomplete forward variance stochastic volatility models, along the lines of Ekeland and Taflin [7].

A numerical and experimental validation of the room acoustics diffusion theory inside long rooms

The paper focuses on the validation of the recently proposed room-acoustics diffusion theory by means of numerical simulations and experimental measurements. The analysis aims to verify the equation underlying the theory (Fick’s law of diffusion) which relates the energy density gradient and the sound intensity inside a room through a constant diffusion coefficient. In this work, the acoustic quantities are numerically/experimentally derived under stationary conditions, and their ratio is employed to estimate the effective value of the diffusion coefficient inside long rooms. The numerical study was carried out with particle-tracing simulations. The measurements were performed with a Microflown® three-dimensional sound intensity probe inside a 1:16 scale model of a long room, varying the absorption and the scattering properties at the boundaries. A comparison between numerical and experimental results is carried out with a least-square algorithm, showing a fair agreement between the diffusion coefficients estimated with the two methods. The results lead to the conclusion that the reverberant sound field inside long rooms can be described by a non-homogeneous diffusion process: the local diffusion coefficient is not a constant inside the room but increases with the distance from the source and depends on the acoustical properties of the room boundaries.

A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals

The aim of this work is to characterize one‐dimensional homogeneous diffusion process, under the assumption that marginal density of the process is Gaussian. The method considers the forward Kolmogorov equation and Fourier transform operator approach. The result establishes the necessary characteristic equation between drift and diffusion coefficients for homogeneous and nonhomogeneous diffusion processes. The equation for homogeneous diffusion process leads to characterize the possible diffusion processes that can exist. Two well‐known examples using the necessary characteristic equation are also given.

Detection, modelling and estimation of non-linear trends by using a non-homogeneous Vasicek stochastic diffusion. Application to CO2 emissions in Morocco

This paper proposes a new stochastic model, based on a Vasicek non-homogeneous diffusion process, in which the non-linear trend coefficient (drift) depends on deterministic functions that describe the dynamic evolution of certain exogenous variables. After studying its probabilistic characteristics, and in particular the transition probability density and trend function, the associated stochastic inference based on discrete sampling in time is established using maximum likelihood methodology. This model is applied to detect, estimate and model the non-linear trend present in data corresponding to CO2 emissions in Morocco. Energy and financial variables that affect the behaviour of this trend are also detected, and substantial improvement provided by this non-homogeneous model with respect to its corresponding homogeneous version, is confirmed.

On the diffusion of a fast molecule

We consider the motion of a heavy particle in interaction with an infinite ideal gas of slow atoms. We prove that the velocity of the heavy particle is, in a suitable limit, modeled by a deterministic process. We also treat the process of rescaled velocity fluctuations around a certain deterministic motion and show that this is appropriately modeled by a nonhomogeneous diffusion process.