Hybrid method of moments with interpolation closure–Taylor-series expansion method of moments scheme for solving the Smoluchowski coagulation equation

This paper presents a hybrid method of moments with interpolation closure–Taylor-series expansion method of moments (MoMIC–TEMoM) scheme for solving the Smoluchowski coagulation equation. In the proposed scheme, the exponential function, which arises in the conversion from a particle size distribution space to a space of moments, is expressed in an additive form using the third-order Taylor-series expansion; the implicit moments are approximated using two Lagrange interpolation functions, namely the newly defined normalized moment function and the normalized moment function defined by Frenklach and Harris (1987). The new hybrid scheme allows implementation of the method of moments with an arbitrary type of moment sequence, and it overcomes the shortcomings of the Taylor-series expansion moment method proposed by Frenklach and Harris. The proposed scheme is verified with three aerosol dynamics, namely Brownian coagulation in the free molecular regime, Brownian coagulation in the continuum-slip regime, and turbulence coagulation. The results reveal that the hybrid MoMIC–TEMoM scheme has similar accuracy to currently recognized methods including the quadrature method of moments, MoMIC, and TEMoM, and its accuracy can be further enhanced as the fractional moment sequence type is used for Brownian coagulation in the free molecular regime. Thus, the proposed scheme is a reliable for solving the Smoluchowski coagulation equation.     

Basket options valuation for a local volatility jump–diffusion model with the asymptotic expansion method

In this paper we discuss the basket options valuation for a jump–diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expansion method to approximate the conditional expectation of the stochastic variance associated with the basket value process. The numerical tests show that the suggested method is fast and accurate in comparison with the Monte Carlo and other methods in most cases.     

Thoughts on modeling complexity

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This simple model has been applied to all manner of phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model beyond that of mimicking simple statistics to include long‐time memory in the dynamics of complex phenomena. We show that complexity can give rise to fractional‐difference stochastic processes whose continuum limit is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. Furthermore, the index of the inverse power‐law spectrum in many complex processes can be related to the fractional derivative index in the fractional Langevin equation. This fractional stochastic model suggests that a scaling process guides the dynamics of many complex phenomena. The alternative to the fractional Langevin equation is a fractional diffusion equation describing the evolution of the probability density for certain kinds of anomalous diffusion. © 2006 Wiley Periodicals, Inc. Complexity 11: 33–43, 2006     

Two-component aerosol dynamic simulation using differentially weighted operator splitting Monte Carlo method

A differentially weighted operator splitting Monte Carlo (DWOSMC) method is further developed to study multi-component aerosol dynamics. The proposed method involves the use of an excellent combination of stochastic and deterministic approaches. Component-related particle volume density distributions are examined, and the computational accuracy and efficiency of the two-component DWOSMC method is verified against a sectional method. For the one-component aerosol system, the sectional method is more computationally efficient than the DWOSMC method, while for two-component aerosol systems, the DWOSMC method proves to be much more computationally efficient than the sectional method. Using this newly developed multi-component DWOSMC method, compositional distributions of particles can be obtained to determine simultaneous coagulation and condensation processes that occur in different regimes of two-component aerosol systems.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

On the construction of non-affine jump-diffusion models

We describe a method for construction of jump analogues of certain one-dimensional diffusion processes satisfying solvable stochastic differential equations. The method is based on the reduction of the original stochastic differential equations to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method by constructing jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.     

Modelling lumped-parameter sorption kinetics and diffusion dynamics of odour-causing VOCs to dust particles

An analytical algorithm is presented for fast simulation of the adsorption kinetics and diffusion dynamics of odour-causing volatile organic compounds (VOC-odour) which originate in the stored swine manure to airborne dust particles in a ventilated airspace. The model is an extension to the well-known lumped-parameter model (LPM) that incorporates a Langmuir–Hinshelwood (LH) kinetic concept dependent on VOC-odour concentration with diffusion limitation. The basic idea behind the model implementation is to couple the calculations of the two major processes in the VOC-odour/dust particle system: VOC-odour diffusion based on the homogeneous surface diffusion model (HSDM) and surface reaction based on the LH kinetics in an LPM scheme. The LPM employs Laplace transforms and gamma distributions of the rate coefficient to produce a lumped-parameter gamma model (LPGM) for kinetic equation of VOC-odour adsorption to airborne dust particles, whereas the HSDM incorporates the age and size distributions of airborne dust for evaluating the dust-borne VOC-odour dynamics. The integrate assessment of VOC-odour sorption kinetics and diffusion dynamics allows to relate the adsorption rate coefficient, reaction order, and surface effective diffusivity in a complex VOC-odour/dust particle system. The LPGM fitted well with the data obtained numerically from HSDM and successfully determined the adsorption rate coefficient and reaction order for each sorption process.     

Stochastic exponential robust stability of interval neural networks with reaction–diffusion terms and mixed delays

The stochastic exponential robust stability is considered for a class of delayed neural networks with reaction–diffusion terms and Markov jumping parameters in this paper. It is assumed that the uncertain weight matrices belong to the given interval matrices. Some sufficient conditions for the stochastic exponential robust stability of the system are established by applying vector Lyapunov function method and M-matrix theory. The obtained results involving the effect of reaction–diffusion improve the existing conditions. Finally, two examples with numerical simulations are given to illustrate the obtained results.     

Response of Duffing system with delayed feedback control under combined harmonic and real noise excitations

This paper presents a procedure for predicting the response of Duffing system with delayed feedback bang–bang control under combined harmonic and real noise excitations by using the stochastic averaging method. First, the time-delayed feedback bang–bang control force is expressed approximately in terms of the system state variables without time delay. Then the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations. It is shown that the time delay in feedback control can deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

ESTIMATING A STOCHASTIC MODEL OF POPULATION DYNAMICS WITH AN APPLICATION TO KANGAROOS

In this study, we derive stochastic models of population dynamics and devise a new method of estimating the models. The models allow growth and harvest to be nonlinear functions of stochastic processes and the error terms to be nonlinear and heteroskedastic. Ordinary least-squares estimates would be biased and inefficient and generalized least-squares estimates cannot be calculated. Therefore, we implement nonlinear maximum likelihood methods to find unbiased and efficient estimates of parameters. The method is applied to the population dynamics of kangaroos in South Australia. Aerial survey data of kangaroo numbers are combined with harvest, effort and rainfall data to estimate the growth and harvest functions and the variances of the stochastic processes which drive the system. Results suggest that growth and harvest should be modeled as functions of stochastic processes and that observations on kangaroo numbers are critical for estimating population dynamics. The results also indicate that the estimation method works well and is a viable alternative to ARIMA and GARCH models, particularly for small data sets.     

An optimal data assimilation method and its application to the numerical simulation of the ocean dynamics

An original data assimilation (DA) scheme with a general dynamics model is considered. It is shown that this scheme can be approximated by the stochastic diffusion process. The sufficient conditions to provide this approximation are formulated. Based on this algorithm a new DA method is developed. The method combines variational and statistical approaches commonly used in DA theory and minimizes the variance of the trajectory of a diffusion process in conjunction with a dynamics numerical model. In this sense the method is optimal in contrast to other DA approaches. The proposed scheme takes the model dynamics into account and in this way it differs from the well-known Kalman filter. Furthermore, the derived DA method can be applied to a very wide field of dynamical systems, for example, gas dynamics, fluid dynamics and other disciplines. However, the current study deals with oceanography and DA in oceanography specifically. Then the method is applied to the HYbrid Coordinate Ocean Model and assimilates satellite sea level anomaly data from the Archiving, Validating and Interpolating Satellite Oceanography Data over the Atlantic Ocean to correct the model state. Several numerical experiments have been performed. The experiments show that the method substantially changes the synoptic and mesoscale structure of ocean dynamics. Also, the distribution of the obtained result is estimated through the solution of the Fokker–Planck–Kolmogorov equation.     

Switching diffusion logistic models involving singularly perturbed Markov chains: Weak convergence and stochastic permanence

Focusing on stochastic dynamics involve continuous states as well as discrete events, this article investigates stochastic logistic model with regime switching modulated by a singular Markov chain involving a small parameter. This Markov chain undergoes weak and strong interactions, where the small parameter is used to reflect rapid rate of regime switching among each state class. Two-time-scale formulation is used to reduce the complexity. We obtain weak convergence of the underlying system so that the limit has much simpler structure. Then we utilize the structure of limit system as a bridge, to invest stochastic permanence of original system driving by a singular Markov chain with a large number of states. Sufficient conditions for stochastic permanence are obtained. A couple of examples and numerical simulations are given to illustrate our results.     

Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.     

Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein–Uhlenbeck process with Lévy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein–Uhlenbeck process, where in particular we must restrict to operator-valued Lévy processes with “non-decreasing paths”. It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein–Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.     

Stochastic Retarded Evolution Equations: Green Operators,Convolutions, and Solutions

Abstract

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.     

Bogoliubov averaging principle of stochastic reaction–diffusion equation

The purpose of this paper is to establish Bogoliubov averaging principle of stochastic reaction–diffusion equation with a stochastic process and a small parameter. The solutions to stochastic reaction–diffusion equation can be approximated by solutions to averaged stochastic reaction–diffusion equation in the sense of convergence in probability and in distribution. Namely, we establish a weak law of large numbers for the solution of stochastic reaction–diffusion equation.