General Diffusion Model Research Articles

3D modeling of overall adsorption rate of acetaminophen on activated carbon pellets

In this work the overall adsorption rate of acetaminophen on activated carbon pellets (ACP) was analyzed in deep. The concentration decay curves were interpreted by a 3D diffusional model because the intraparticle diffusion of acetaminophen inside ACP in radial and axial directions are important in this form of activated carbon. The 3D diffusional model considers the external mass transfer, intraparticle diffusion (pore volume and surface diffusion) and the adsorption on an active site. The results demonstrated that the application of 3D diffusional model based on pore volume diffusion interprets clearly the kinetic curves, however values of effective diffusion coefficient (Dep) higher than molecular diffusivity are obtained indicating superdiffusion phenomenon. On the other hand, the application of a general diffusion model evidenced that during the whole time interval, the acetaminophen diffuses consecutively by surface diffusion followed by pore volume diffusion. In short times the surface flux is higher than the pore volume flux, but at higher intervals of time, the relevance of both fluxes reverts. Finally, from 3D simulation it is clear that at longer times, the solute mainly enters from the solution through the pellet borders, and acetaminophen desorption signs are even evident at the center of the pellet covers, due to the inverse concentration gradient established between the pellet and the solution.

Harnack's inequality for a space–time fractional diffusion equation and applications to an inverse source problem

In this paper, we focus on a space–time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.

Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior

Continuous time Markov chain (CTMC) approximation is an intuitive and powerful method for pricing options in general Markovian models. This paper analyzes how grid design affects the convergence behavior of barrier and European options in general diffusion models. Using the spectral method, we obtain sharp estimates for the convergence rate of option price for non-uniform grids. We propose to calculate an option’s delta and gamma by taking central difference of option prices on the grid. For this simple method, we prove that, surprisingly, delta and gamma converge at the same rate as option price does. Our analysis allows us to develop principles that are sufficient and necessary for designing nonuniform grids that can achieve second order convergence for option price, delta and gamma. Based on these principles, we propose a novel class of non-uniform grids, which ensures that convergence is not only second order, but also smooth. This further allows extrapolation to be applied to achieve even higher convergence rate. Our grids enable the CTMC approximation method to price and hedge a large number of options with different strikes fast and accurately. Applicability of our results to jump models is discussed through numerical examples.

Solving Multidimensional Fractional Fokker--Planck Equations via Unbiased Density Formulas for Anomalous Diffusion Processes

Fractional Fokker--Planck equations are generalizations of ordinary Fokker--Planck equations that include fractional derivative operators to encapsulate trapping effects in the anomalously diffusing population they describe. They do not admit analytical solutions in general and there are significant drawbacks to current numerical approximation methods. Using Malliavin calculus, we establish formulas in the form of a mathematical expectation for the probability density functions of anomalous diffusion processes, which are known to solve fractional Fokker--Planck equations. These density formulas lend themselves easily to computation via Monte Carlo sampling and are easy to implement. Moreover, they have the capability to deal with a general model of anomalous diffusion, including the case where the parent process of the anomalous diffusion has a Levy jump component. The numerical solutions obtained are inherently unbiased, which allows them to form the basis for reliable inferences about the population dyn...

Fisher–Wright model with deterministic seed bank and selection

Seed banks are common characteristics to many plant species, which allow storage of genetic diversity in the soil as dormant seeds for various periods of time. We investigate an above-ground population following a Fisher–Wright model with selection coupled with a deterministic seed bank assuming the length of the seed bank is kept constant and the number of seeds is large. To assess the combined impact of seed banks and selection on genetic diversity, we derive a general diffusion model. The applied techniques outline a path of approximating a stochastic delay differential equation by an appropriately rescaled stochastic differential equation. We compute the equilibrium solution of the site-frequency spectrum and derive the times to fixation of an allele with and without selection. Finally, it is demonstrated that seed banks enhance the effect of selection onto the site-frequency spectrum while slowing down the time until the mutation–selection equilibrium is reached.

Nonrecursive Separation of Risk and Time Preferences

Recursive utility disentangles preferences with respect to time and risk by recursively building up a value function of local increments. This involves certainty equivalents of indirect utility. Instead we disentangle preferences with respect to time and risk by building up a value function as a non-linear aggregation of certainty equivalents of direct utility of consumption. This entails time-consistency issues which are dealt with by looking for an equilibrium control and an equilibrium value function rather than a classical optimal control and a classical optimal value function. We characterize the solution in a general diffusive incomplete market model and find that, in certain special cases of utmost interest, the characterization coincides with what would arise from a recursive utility approach. But also importantly, in other cases, it does not: The two approaches are fundamentally different but match, exclusively but importantly, in the mathematically special case of homogeneity of the value function.

Optimal financing and dividend distribution in a general diffusion model with regime switching

Abstract We study the optimal financing and dividend distribution problem with restricted dividend rates in a diffusion type surplus model, where the drift and volatility coefficients are general functions of the level of surplus and the external environment regime. The environment regime is modeled by a Markov process. Both capital injection and dividend payments incur expenses. The objective is to maximize the expectation of the total discounted dividends minus the total cost of the capital injection. We prove that it is optimal to inject capital only when the surplus tends to fall below 0 and to pay out dividends at the maximal rate when the surplus is at or above the threshold, dependent on the environment regime.

Sensitivity analysis of the optimal exercise boundary of the American put option

Abstract We consider the American put problem in a general one-dimensional diffusion model. The risk-free interest rate is constant, and volatility is assumed to be a function of time and stock price. We use the well-known parabolic obstacle problem and establish the continuity estimate of the optional exercise boundaries of the American put option with respect to the local volatilities, which may be considered as a generalization of the Achdou results [1].

Validity of the second Fick's law for modeling ion-exchange diffusion in non-crystalline viscoelastic media (glasses)

It is shown that, in general case, the diffusion equation (or the second Fick's law) does not provide an adequate description of ion-exchange transport phenomena in viscoelastic media, including glassy or any other non-crystalline media. In this connection the general phenomenological model of ion-exchange diffusion in viscoelastic media has been developed. A theoretical analysis of the model shows that, in the case of a linear dependence of medium density on the concentration of diffusing ions, the necessary and sufficient condition of the absolute validity of the diffusion equation in viscoelastic media is Φ ≫ 1, where Φ = τD/τR is the dimensionless value (or criterion of similarity), with τD = L2/D being the characteristic time of diffusion and τR = η/G being the characteristic time of stress relaxation, where L, D, η, and G are the characteristic length of diffusion, the diffusivity, the viscosity, and the shear modulus, respectively. The value of 1/Φ characterizes the accuracy which is provided if the second Fick's law is used in the simulation of ion-exchange diffusion in viscoelastic media. We have demonstrated the applicability of this criterion experimentally. Our experimental studies on ion-exchange diffusion in an oxide glass (typical viscoelastic media) have shown that under the condition the Φ > 105 the experimental concentration profiles are close to those predicted by the second Fick's law to within an accuracy of 1%.

A continuum model of deformation, transport and irreversible changes in atomic structure in amorphous lithium–silicon electrodes

Recent experiments and atomic scale computations indicate that the standard continuum models of diffusion in stressed solids do not accurately describe transport, deformation and stress in Li–Si alloys. We suggest that this is because classical models do not account for the irreversible changes in atomic structure of Si that are known to occur during a charge–discharge cycle. A more general model of diffusion in an amorphous solid is described, which permits unoccupied Si lattice sites to be created or destroyed. This may occur as a thermally activated process; or as a result of irreversible plastic deformation under stress. The model predicts a range of phenomena observed in experiment that cannot be captured using classical models, including irreversible changes in volume resulting from a charge–discharge cycle, asymmetry between the tensile and compressive yield stress, and a slow evolution in mechanical and electrochemical response over many charge–discharge cycles.

A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases.

Explicit form of approximate transition probability density functions of diffusion processes

A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black–Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.

Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay

In this paper, we examine the asymptotic behaviors of a diffusive delayed consumer-resource model subject to homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of juvenile consumers to their maturity, and the predation is of a general type of functional response. We constructthe threshold dynamics of the persistence and extinction of the consumer. Moreover, we establish the sufficient conditions for the global attractivity of the semitrivial and coexistence equilibria. Finally, we apply our results to the specific consumer-resourcemodels with Beddington-DeAngelis, Crowley-Martin, and ratio-dependent type of functional responses.

ESTIMATING RESIDUAL HEDGING RISK WITH LEAST-SQUARES MONTE CARLO

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Monte Carlo algorithm to estimate residual hedging risk. The algorithm approximates the variance minimal hedging strategy within general diffusion models. Moreover, the algorithm produces both high-biased and low-biased estimators for the residual hedging error variance, thus providing an intrinsic criterion for the quality of the approximation. In a number of examples we show that the algorithm delivers accurate hedging error characteristics within seconds.

Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models

In this paper we propose a closed-form asymptotic expansion approach to pricing discretely monitored Asian options in general one-dimensional diffusion models. Our expansion is a small-time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval. This expansion method is distinguished from many other pricing-oriented expansion algorithms in the literature because of two appealing features. First, we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way. Second, the convergence of the expansion is proved rigorously under some regularity conditions. Numerical experiments suggest that the closed-form expansion formula with only a few terms (e.g., four terms up to the third order) is accurate, fast, and easy to implement for a broad range of diffusion models, even including those violating the regularity conditions.

CO2 long-term diffusive leakage into biosphere in geological carbon storage

Prediction of CO2 leakage into biosphere is very important for risk assessment in geological carbon storage projects. Underground CO2 can be transported into biosphere through short term leakage due to fractures of wellbores or cap rocks, which has been extensively investigated, and long term leakage due to diffusion, which has few relevant studies. This paper presents a diffusive model for CO2 gradual leakage into biosphere during a long period after CO2 injection. First, the paper describes a general diffusive model with long term secondary trapping effects for CO2 fluxes from underground into biosphere. Secondly, a simplified one-dimensional model is presented and solved for the CO2 concentrations in groundwater. The results show that the groundwater CO2 concentration will reach the maximum value at about 50th year after CO2 injection and then slowly decrease due to secondary trapping effects. Moreover, the partition coefficient is the dominant parameter for predicting the groundwater CO2 concentration while the convective mass transfer coefficient plays an insignificant role.

EFFECT OF REPEAT PURCHASE AND DYNAMIC MARKET SIZE ON DIFFUSION OF AN INNOVATIVE TECHNOLOGICAL CONSUMER PRODUCT IN A SEGMENTED MARKET

This study develops diffusion models for technological consumer products under the marketing environment when a product is marketed in a segmented market and observes two distinctive promotional strategies of mass and differentiated promotion; an under explored study area. Mass promotion strategy creates a spectrum effect in market with an aim to create wider product awareness and influence the market size. Whereas the differentiated promotion strategy plays major role in external influence component in the respective segment and target for adoption by the current potential segment. Previous studies on segmented diffusion models assumed only first time purchase and constant market size which may yield underestimated results and fail to give appropriate insight of the diffusion process. The study develops and validates generalized diffusion models for segmented market incorporating the repurchase behaviour of the adopter population and dynamic potential market size considerations. Performance of the proposed models is analysed on real life data for a new product marketed in four segments and compared with the previous study.

Linear Multistep Schemes for BSDEs

We study the convergence rate of a class of linear multistep methods for backward stochastic differential equations (BSDEs). We show that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property. Coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence. Contrary to the analysis performed in [W. Zhao, G. Zhang, and L. Ju, SIAM J. Numer. Anal., 48 (2010), pp. 1369--1394], we consider general diffusion models and BSDEs with driver depending on $z$. The class of methods we consider contains well-known methods from the ODE framework as Nystrom, Milne, or Adams methods. Finally, we provide a numerical illustration of the convergence of some methods.

A generalized diffusion model for growth of nanoparticles synthesized by colloidal methods

A nanoparticle growth model is developed to predict and guide the syntheses of monodisperse colloidal nanoparticles in the liquid phase. The model, without any a priori assumptions, is based on the Fick’s law of diffusion, conservation of mass and the Gibbs–Thomson equation for crystal growth. In the limiting case, this model reduces to the same expression as the currently accepted model that requires the assumption of a diffusion layer around each nanoparticle. The present growth model bridges the two limiting cases of the previous model i.e. complete diffusion controlled and adsorption controlled growth of nanoparticles. Specifically, the results show that a monodispersion of nanoparticles can be obtained both with fast monomer diffusion and with surface reaction under conditions of small diffusivity to surface reaction constant ratio that results is growth ‘focusing’. This comprehensive description of nanoparticle growth provides new insights and establishes the required conditions for fabricating monodisperse nanoparticles critical for a wide range of applications.

A General Stochastic Information Diffusion Model in Social Networks Based on Epidemic Diseases

Social networks are an important infrastructure for information, viruses and innovations propagation. Since users behavior has influenced by other users activity, some groups of people would be made regard to similarity of users interests. On the other hand, dealing with many events in real worlds, can be justified in social networks; spreading disease is one instance of them. People manner and infection severity are more important parameters in dissemination of diseases. Both of these reasons derive, whether the diffusion leads to an epidemic or not. SIRS is a hybrid model of SIR and SIS disease models to spread contamination. A person in this model can be returned to susceptible state after it removed. According to communities which are established on the social network, we use the compartmental type of SIRS model. During this paper, a general compartmental information diffusion model would be proposed and extracted some of the beneficial parameters to analyze our model. To adapt our model to realistic behaviors, we use Markovian model, which would be helpful to create a stochastic manner of the proposed model. In the case of random model, we can calculate probabilities of transaction between states and predicting value of each state. The comparison between two mode of the model shows that, the prediction of population would be verified in each state.