Brownian Motion Type Research Articles

Modelling intermittent anomalous diffusion with switching fractional Brownian motion

The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.

On Financial Distributions Modelling Methods: Application on Regression Models for Time Series

The financial market is a complex system with chaotic behavior that can lead to wild swings within the financial system. This can drive the system into a variety of interesting phenomenon such as phase transitions, bubbles, and crashes, and so on. Of interest in financial modelling is identifying the distribution and the stylized facts of a particular time series, as the distribution and stylized facts can determine if volatility is present, resulting in financial risk and contagion. Regression modelling has been used within this study as a methodology to identify the goodness-of-fit between the original and generated time series model, which serves as a criterion for model selection. Different time series modelling methods that include the common Box–Jenkins ARIMA, ARMA-GARCH type methods, the Geometric Brownian Motion type models and Tsallis entropy based models when data size permits, can use this methodology in model selection. Determining the time series distribution and stylized facts has utility, as the distribution allows for further modelling opportunities such as bivariate regression and copula modelling, apart from the usual forecasting. Determining the distribution and stylized facts also allows for the identification of the parameters that are used within a Geometric Brownian Motion forecasting model. This study has used the Carbon Emissions Futures price between the dates of 1 May 2012 and 1 May 2022, to highlight this application of regression modelling.

Roughness induced rotational slowdown near the colloidal glass transition

HypothesisIn concentrated suspensions, the dynamics of colloids are strongly influenced by the shape and topographical surface characteristics of the particles. As the particles get into close proximity, surface roughness alters the translational and rotational Brownian motions in different ways. Eventually, the rotations will get frustrated due to geometric hindrance from interacting asperities. ExperimentsWe use model raspberry-like colloids to study the effect of roughness on the translational and rotational dynamics. Using Confocal Scanning Laser Microscopy and particle tracking, we simultaneously resolve the two types of Brownian motion and obtain the corresponding Mean Squared Displacements for varying concentrations up to the maximum packing fraction. FindingsRoughness not only lowers the concentration of the translational colloidal glass transition, but also generates a broad concentration range in which the rotational Brownian motion changes signature from high-amplitude diffusive to low-amplitude rattling. This hitherto not reported second glass transition for rough spherical colloids emerges when the particle intersurface distance becomes comparable to the roughness length scale. Our work provides a unifying understanding of the surface characteristics’ effect on the rotational dynamics during glass formation and provides a microscopic foundation for many roughness-related macroscale phenomena in nature and technology.

Identifying heterogeneous diffusion states in the cytoplasm by a hidden Markov model

Diffusion of nanoparticles in the cytoplasm of live cells has frequently been reported to exhibit an anomalous and even heterogeneous character, i.e. particles seem to switch gears during their journey. Here we show by means of a hidden Markov model that individual trajectories of quantum dots in the cytoplasm of living cultured cells feature a dichotomous switching between two distinct mobility states with an overall subdiffusive mode of motion of the fractional Brownian motion (FBM) type. Using the extracted features of experimental trajectories as input for simulations of different variants of a two-state FBM model, we show that the trajectory-intrinsic and the ensemble-wise heterogeneity in the experimental data is mostly due to variations in the (local) transport coefficients, with only minor contributions due to locally varying anomaly exponents. Altogether, our approach shows that diffusion heterogeneities can be faithfully extracted and quantified from fairly short trajectories obtained by single-particle tracking in highly complex media.

Optimal kernel estimation of spot volatility of stochastic differential equations

A unified framework to optimally select the bandwidth and kernel function of spot volatility kernel estimators is put forward. The proposed models include not only classical Brownian motion driven dynamics but also volatility processes that are driven by long-memory fractional Brownian motions or other Gaussian processes. We characterize the leading order terms of the mean squared error, which in turn enables us to determine an explicit formula for the leading term of the optimal bandwidth. Central limit theorems for the estimation error are also obtained. A feasible plug-in type bandwidth selection procedure is then proposed, for which, as a sub-problem, a new estimator of the volatility of volatility is developed. The optimal selection of the kernel function is also investigated. For Brownian Motion type volatilities, the optimal kernel turns out to be an exponential function, while, for fractional Brownian motion type volatilities, easily implementable numerical results to compute the optimal kernels are devised. Simulation studies further confirm the good performance of the proposed methods.

Long-time heat kernel estimates and upper rate functions of Brownian motion type for symmetric jump processes

Let $X$ be a symmetric jump process on $\mathbb{R}^{d}$ such that the corresponding jumping kernel $J(x,y)$ satisfies \[J(x,y)\le\frac{c}{|x-y|^{d+2}\log^{1+\varepsilon}(e+|x-y|)}\] for all $x,y\in\mathbb{R}^{d}$ with $|x-y|\ge1$ and some constants $c,\varepsilon>0$. Under additional mild assumptions on $J(x,y)$ for $|x-y|<1$, we show that $C\sqrt{r\log\log r}$ with some constant $C>0$ is an upper rate function of the process $X$, which enjoys the same form as that for Brownian motions. The approach is based on heat kernel estimates of large time for the process $X$. As a by-product, we also obtain two-sided heat kernel estimates of large time for symmetric jump processes whose jumping kernels are comparable to \[\frac{1}{|x-y|^{d+2+\varepsilon}}\] for all $x,y\in\mathbb{R}^{d}$ with $|x-y|\ge1$ and some constant $\varepsilon>0$.

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

ABSTRACTCentral limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ( and regular ( types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter ) is the so-called 4th moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

Fractal Analysis of Deep Ocean Current Speed Time Series

AbstractFractal properties of deep ocean current speed time series, measured at a single-point mooring on the Madeira Abyssal Plain at 1000- and 3000-m depth, are explored over the range between one week and 5 years, by using the detrended fluctuation analysis and multifractal detrended fluctuation analysis methodologies. The detrended fluctuation analysis reveals the existence of two subranges with different scaling behaviors. Long-range temporal correlations following a power law are found in the time-scale range between approximately 50 days and 5 years, while a Brownian motion–type behavior is observed for shorter time scales. The multifractal analysis approach underlines a multifractal structure whose intensity decreases with depth. The analysis of the shuffled and surrogate versions of the original time series shows that multifractality is mainly due to long-range correlations, although there is a weak nonlinear contribution at 1000-m depth, which is confirmed by the detrended fluctuation analysis of volatility time series.

Fractional Brownian motions: memory, diffusion velocity, and correlation functions**In memoriam Marian Smoluchowski, on the 100th anniversary of the publication of his seminal papers on Brownian motion and diffusion-limited kinetics.

Fractional Brownian motions (FBMs) have been observed recently in the measured trajectories of individual molecules or small particles in the cytoplasm of living cells and in other dense composite systems, among others. Various types of FBMs differ in a number of ways, including the strength, range and type of damping of the memory encoded in their definitions, but share several basic characteristics: distributions, non-ergodic properties, and scaling of the second moment, which makes it difficult to determine which type of Brownian motion (fractional or normal) the measured trajectory belongs to. Here, we show, by introducing FBMs with regulated range and strength of memory, that it is the structure of memory which determines their physical properties, including mean velocity of diffusion; therefore, the course and kinetics of several processes (including coagulation and some chemical reactions). We also show that autocorrelation functions possess characteristic features which enable identification of an observed FBM, and of the type of memory governing its trajectory.

Existence and regularity of the density for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of densities for the solution of a nonlinear heat diffusion with stochastic perturbation of Brownian and fractional Brownian motion type: we use the Malliavin calculus in order to prove that, if the nonlinear term is suitably regular, then the law of the solution has a smooth density with respect to the Lebesgue measure.

Local time of self-affine sets of Brownian motion type – Revisited

In [5], we studied self-affine sets of Brownian motion type. We obtained the image measure induced by the Lebesgue measure on the domain space. If it is absolutely continuous, the density function is called the local time. The local times are obtained by solving so-called jigsaw puzzle. In this paper, we are specially interested in the self-affine set with 3 pieces with the orientations “up, down, up” (see the picture below). [Display omitted]

Threshold-free evaluation of near-surface diffusion and adsorption-dominated motion from single-molecule tracking data of single-stranded DNA through total internal reflection fluorescence microscopy

Total internal reflection fluorescence (TIRF) microscopy enables the single-molecule observation in liquid near substrate surface. However, the evaluation of the diffusion from their individually-tracked positions entails the difficulty in the treatment of molecular adsorption on the substrate. We propose a novel technique to evaluate them, and two types of near-surface Brownian motion were determined for DNA. One is the diffusion near glass surface, and the other is the adsorption-dominated motion, which is also found to be diffusive rather than anchored to the substrate. Our technique does not require the threshold values for the distinction, and even the transition between them can be captured. Objective distinction of Brownian motion with and without adsorption does not require the adsorption-free sample preparation. It is also useful for the characterization of adsorption/desorption kinetics. Our method is not limited to TIRF but applicable to many other systems involving multiple states of Brownian motion.

Microscopic derivation of open quantum Brownian motion: a particular example

The microscopic derivation of a new type of Brownian motion, namely open quantum Brownian motion (OQBM) is presented. The quantum master equation for OQBM is derived for a weakly driven system interacting with a decoherent environment. Examples of the dynamics for initial Gaussian and non-Gaussian distributions are presented. Both examples demonstrate convergence of the OQBM dynamics to Gaussian distributions.

Modeling Generic Mean Reversion Processes with a Symmetrical Binomial Lattice - Applications to Real Options

Abstract In this paper we propose a Symmetrical Binomial Lattice Approach that is equivalent to the well-known and widely utilized Lattice of Cox, Ross & Rubinstein when modeling Geometric Brownian Motion type of processes, but can be utilized for a wide variety of other Markov style stochastic processes, such as Mean Reversion. This is due to the highly intuitive construction in which first the expected value expression of the process is directly used and the variance is modeled in a symmetrical lattice, which is added to the first. We then demonstrate its applicability with several Real Options examples, comparing to the Cox et al model.

Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion

ABSTRACTIn this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion SH with Hurst index H. When , we prove by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on SH. We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index H.

What do market-calibrated stochastic processes indicate about the long-term price of crude oil?

Stochastic process models of commodity prices are important inputs in energy investment evaluation and planning problems. In this paper, we focus on modeling and forecasting the long-term price level, since it is the dominant factor in many such applications. To provide a foundation for our modeling approach we first evaluate the empirical characteristics of crude oil price data from 1990 to 2013 using unit root and variance ratio tests. Statistical evidence from these tests shows only weak support for the applicability of stationary mean-reverting type processes up through 2004, with non-stationary Brownian motion type processes becoming more plausible when the data from 2005 to 2013 is added. We then apply a Kalman filtering method with maximum likelihood approach to estimate the model parameters for both a single-factor Geometric Brownian motion (GBM) process as well as the two-factor Schwartz and Smith (2000) process. The latter process decomposes the spot price into unobservable factors for the long-term equilibrium level and short-term deviation, and it accommodates aspects of both a GBM process and a mean-reverting process. Both empirical and simulated data are analyzed with these models, and we quantify the increases in both the drift rate and volatility of these processes that result from developments in the crude oil markets since the middle of the last decade. We conclude by comparing and contrasting both historical accuracy and forecasts from the parameterized models, and show that when the emphasis is on the long-term expectations, a single factor GBM forecast may be sufficient.

Local time of self-affine sets of Brownian motion type and the jigsaw puzzle problem

Let Ω⊂[0,1]×[0,1] be the solution of the set equation:Ω=⋃i=1k(φIi,1×φJi,τi)(Ω), where for an interval I=[a,b]⊂[0,1] and τ∈{−1,1}, φI,τ:[0,1]→I is the linear map such that φI,1(0)=a, φI,1(1)=b, φI,−1(0)=b, φI,−1(1)=a, and {Ii;i=1,⋯,k} is a partition of [0,1] with |Ji|=|Ii|1/2. Thus, Ω is a graph of a Borel function fΩ almost surely and it is called a self-affine set of Brownian motion type. Let λ be the Lebesgue measure on [0,1] and let μΩ=λ∘fΩ−1. The density ρΩ=dμΩdλ, if it exists, is called the local time of Ω and it has been studied. It is known that dimHΩ=3/2 if ρΩ exists. In the present study, ρΩ is obtained by solving the so-called jigsaw puzzle on {Ji,τi;i=1,⋯,k}, i.e., the problem of decomposing ρΩ into a sum of its self-similar images with the support Ji and the orientation τi for i=1,⋯,k.

On a continuous time stock price model with regime switching, delay, and threshold

Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black–Scholes model for pricing call options.

Stochastic oscillator with random mass: New type of Brownian motion

The model of a stochastic oscillator subject to additive random force, which includes the Brownian motion, is widely used for analysis of different phenomena in physics, chemistry, biology, economics and social science. As a rule, by the appropriate choice of units one assumes that the particle’s mass is equal to unity. However, for the case of an additional multiplicative random force, the situation is more complicated. As we show in this review article, for the cases of random frequency or random damping, the mass cannot be excluded from the equations of motion, and, for example, besides the restriction of the size of Brownian particle, some restrictions exist also of its mass. In addition to these two types of multiplicative forces, we consider the random mass, which describes, among others, the Brownian motion with adhesion. The fluctuations of mass are modeled as a dichotomous noise, and the first two moments of coordinates show non-monotonic dependence on the parameters of oscillator and noise. In the presence of an additional periodic force an oscillator with random mass is characterized by the stochastic resonance phenomenon, when the appearance of noise increases the input signal.

Stochastic Equations and Inclusions with Mean Derivatives and Some Applications

The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing some cost criteria, is proved.