Classical Brownian Motion Research Articles

Deterministic Brownian motion generated from differential delay equations

This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation. Our results show that solutions of the deterministic equation with randomly selected initial conditions display a Gaussian-like density for long time, but the densities are supported on an interval of finite measure. Using these chaotic solutions as velocities, we are able to produce Brownian-like motions, which show statistical properties akin to those of a classical Brownian motion over both short and long time scales. Several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation. Numerical studies suggest that these conjectures could be "universal" for similar types of "chaotic" dynamics, but we have been unable to prove this.

Quantum stochastic processes: a case study

We present a detailed study of a simple quantum stochastic process, the quantum phasespace Brownian motion, which we obtain as the Markovian limit of a simple model of anopen quantum system. This physical description of the process allows us to specify and toconstruct the dilation of the quantum dynamical maps, including conditional quantumexpectations. The quantum phase space Brownian motion possesses many propertiessimilar to those of the classical Brownian motion, notably its increments are independentand identically distributed. Possible applications to dissipative phenomena in the quantumHall effect are suggested.

Random attractors for a class of stochastic partial differential equations driven by general additive noise

The existence of random attractors for a large class of stochastic partial differential equations (SPDE) driven by general additive noise is established. The main results are applied to various types of SPDE, as e.g. stochastic reaction–diffusion equations, the stochastic p-Laplace equation and stochastic porous media equations. Besides classical Brownian motion, we also include space-time fractional Brownian motion and space-time Lévy noise as admissible random perturbations. Moreover, cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained.

THE COMPLEXITY OF BROWNIAN PROCESSES RUN WITH NONLINEAR CLOCKS

Anomalous diffusion occurs in many branches of physics. Examples include diffusion in confined nanofilms, Richardson turbulence in the atmosphere, near-surface ocean currents, fracture flow in porous formations and vortex arrays in rotating flows. Classically, anomalous diffusion is characterized by a power law exponent related to the mean-square displacement of a particle or squared separation of pairs of particles: 〈|X(t)|2〉 ~tγ. The exponent γ is often thought to relate to the fractal dimension of the underlying process. If γ > 1 the flow is super-diffusive, if it equals 1 it is classical, otherwise it is sub-diffusive. In this work we illustrate how time-changed Brownian position processes can be employed to model sub-, super-, and classical diffusion, while time-changed Brownian velocity processes can be used to model super-diffusion alone. Specific examples presented include transport in turbulent fluids and renormalized transport in porous media. Intuitively, a time-changed Brownian process is a classical Brownian motion running with a nonlinear clock (Bm-nlc). The major difference between classical and Bm-nlc is that the time-changed case has nonstationary increments. An important novelty of this approach is that, unlike fractional Brownian motion, the fractal dimension of the process (space filling character) driving anomalous diffusion as modeled by Bm-nlc positions or velocities does not change with the scaling exponent, γ.

Brownian motion on Lie groups and open quantum systems

We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

Stock loan valuation under a regime-switching model with mean-reverting and finite maturity

Stock loans are business contracts between borrowers and lenders in which the borrower uses shares of stock as collateral for the loan. Since the value of the collateral is subject to wide and frequent price swings, valuing such a transaction behaves more like an option pricing problem than a debt valuation problem. This paper will list, prove, and analyze formulas for stock loan valuation with finite horizon under various stock models, including classical geometric Brownian motion, mean-reverting, and two-state regime-switching with both mean-reverting and geometric Brownian motion states. Numerical examples are reported to illustrate the results.

Extension and Application of Itô's Formula Under G-Framework

In continuing his study of the intrinsically nonlinear expectation and conditional expectation under the so-called G-framework, Peng introduced a nonlinear Itô calculus; here, the G refers to the generator of a nonlinear heat equation. There, he derived the corresponding Itô formula for C 2-functions with bounded Lipschiz derivatives. This restrictive class of functions limits its applicatory value to stochastic finances and cannot be applied to study the powers of the G-Brownian motion. We extend the Itô formula to a slightly more general class of functions (C 2-functions with uniformly continuous derivatives). This enables us to compute the G-expectations of the even powers of the G-Brownian motion. The G-expectation of odd powers behave differently; in particular, we show that the G-expectation of the cube of the G-Brownian motion is positive, which is qualitatively different from the classical Brownian motion case. We remark that we are not able to get a formula for the G-expectation of the general odd powers of the G-Brownian motion.

Quantum Brownian motion and generalizations of Hermite polynomials

We introduce new families of orthogonal polynomials H D , n , motivated by the non-equilibrium evolution of a quantum Brownian particle (qBp). The H D , n ’s generalize non-trivially the standard Hermite polynomials, employed for classical Brownian motion. We treat several models (labelled by D ) for a non-equilibrium qBp, by means of the Wigner function W , in the presence of a “heat bath” at thermal equilibrium, with and without ab initio friction. For long times (for a suitable class of initial conditions), the non-equilibrium Wigner function W should approach, in some sense, the (time-independent) equilibrium Wigner function W e q , D , which describes the thermal equilibrium of the qBp with the “heat bath” and plays a central role. W e q , D is chosen to be the weight function which orthogonalizes the H D , n ’s. New results on W e q , D and on the H D , n ’s are reported. We justify the key role of the H D , n ’s as follows. Using the H D , n ’s, moments W e q , D , n and W n are introduced for W e q , D and W , respectively. At equilibrium, all moments W e q , D , n except the lowest one ( W e q , D , 0 ) vanish identically. Off-equilibrium, one expects that, for long times (for suitable initial conditions): (i) all non-equilibrium moments W n (except the lowest moment W 0 ), will approach zero, while (ii) the lowest non-equilibrium moment W 0 will tend to W e q , D , 0 ( ≠ 0 ) . To complete the justification, we outline how the approximate long-time non-equilibrium theories determined by W 0 for the different models ( D ) yield Smoluchowski equations and irreversible evolutions of the qBp towards thermal equilibrium.

Modelling and Pricing Temperature Derivatives Using Wavelet Networks and Wavelet Analysis

Weather derivatives are financial instruments that can be used by organizations or individuals as part of a risk management strategy to reduce risk associated with adverse or unexpected weather conditions. Just as traditional contingent claims, whose payoffs depend upon the price of some fundamental, a weather derivative has an underlying measure such as: rainfall, temperature, humidity or snowfall. In this thesis the problem of pricing weather futures written on various temperature indices, as well as weather options on weather futures is addressed. In order to accurately price weather derivatives based on temperature indices, first, a model that describes the evolution of the daily average temperature was developed. This thesis provides a concise and rigorous treatment of the stochastic modelling of weather market. The Ornstein-Uhlenbeck process is described as the basic modelling tool for daily average temperature dynamics, while the innovations are driven by a Brownian motion. We emphasize in the accurate estimation of the seasonal component in the mean and variance using wavelet analysis. A modelling approach that efficiently extracts all the seasonalities from the temperature was developed. In addition we use wavelet networks in order to examine the time dependence of the speed of the mean reversion parameter of the process, α. We estimate nonparametrically with a wavelet network a model of the temperature process and then compute the derivative of the network output with respect to the network input, in order to obtain a series of daily values for α. To our knowledge, this is the first time that this has been done, and it gives us a much better insight into the temperature dynamics and temperature derivative pricing. Our results indicate strong time dependence in the daily values of α , and no seasonal patterns. This is important, since in all relevant studies performed thus far, α was assumed to be constant. Our analysis is based on seven cities that weather derivatives are actively traded on the Chicago Mercantile Exchange. Comparing our method with alternative approaches, our results indicate that our model significantly outperforms them both in-sample and out-of-sample. Furthermore, the residuals of the wavelet neural network provide a better fit to the normal distribution when compared with the residuals of the classic linear models used in the context of temperature modelling. Our model captured efficiently and successfully all the seasonal components of the temperature and completely removed the autocorrelation in the residuals. Finally, our results indicate greater accuracy of our model in forecasting the temperature indices in contrast to alternative models. In order to obtain a better understanding of the distributions of the residuals we expanded our analysis by fitting additional distributions besides the classical Brownian motion. More precisely, a Levy family distribution was fitted to the residuals. Our results indicate that a hyperbolic distribution provides a better fit. Finally, we provide the pricing equations for temperature futures and options on futures written on the most common temperature indices, when κ is time dependent, under the assumption of both a Brownian motion and a Levy process. Our results are very promising and suggest that the proposed method significantly outperforms other methods previously proposed in literature

Theory of Long‐Range Diffusion of Proteins on a Spherical Biological Membrane: Application to Protein Cluster Formation and Actin‐Comet Tail Growth

Breaking of symmetry is often required in biology in order to produce a specific function. In this work we address the problem of protein diffusion over a spherical vesicle surface towards one pole of the vesicle in order to produce ultimately an active protein cluster performing a specific biological function. Such a process is, for example, prerequisite for the assembling of proteins which then cooperatively catalyze the polymerization of actin monomers to sustain the growth of actin tails as occurs in natural vesicles such as those contained in Xenopus eggs. By this process such vesicles may propel themselves within the cell by the principle of action-reaction. In this work the physicochemical treatment of diffusion of large biomolecules within a cellular membrane is extended to encompass the case when proteins may be transiently poised by corral-like structures partitioning the membrane as has been recently documented in the literature. In such case the exchange of proteins between adjacent corrals occurs by energy-gated transitions instead of classical Brownian motion, yet the present analysis shows that long-range movements of the biomolecules may still be described by a classical diffusion law though the diffusion coefficient has then a different physical meaning. Such a model explains why otherwise classical diffusion of proteins may give rise to too small diffusion coefficients compared to predictions based on the protein dimension. This model is implemented to examine the rate of proteins clustering at one pole of a spherical vesicle and its outcome is discussed in relevance to the mechanism of actin comet tails growth.

Chaotic dynamics of super‐diffusion revisited

[1] Super-diffusive mixing in geophysics occurs in atmospheric turbulence, near surface currents in the oceans, and fracture flow in the subsurface to name a few examples. Models of super-diffusion have been around since L. F. Richardson's pioneering work in the 1920's. Here we construct a family of super-diffusive stochastic processes Xα(t), 0 < α, with independent, nonstationary increments, but a priori defined mean-square displacement given by tα+1. The case α = 2 corresponds to Richardson super-diffusion. The family of processes is a nonstationary extension of Brownian motion and hence completely characterized by its first two moments. The Fokker-Planck equation for Xα(t) is classical diffusive with time dependent diffusion coefficient given by tα/2. In contrast to what is found in fractional Brownian motion, where the fractal dimension depends on the Hurst exponent, here the fractal dimension and hence complexity of the process is the same for all exponents α as that of classical Brownian motion. An analytical expression is developed for the finite-size Lyapunov exponent and numerical examples presented.

Finite difference approximations for a fractional advection diffusion problem

The use of the conventional advection diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative diffusion models have been proposed. Fractional space derivatives are used to model anomalous diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a diffusion or dispersion model, it leads to enhanced diffusion, also called superdiffusion. We consider a one-dimensional advection–diffusion model, where the usual second-order derivative gives place to a fractional derivative of order α , with 1 < α ⩽ 2 . We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection–diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test problem.

Brownian motion, quantum corrections and a generalization of the Hermite polynomials

The nonequilibrium evolution of a Brownian particle, in the presence of a “heat bath” at thermal equilibrium (without imposing any friction mechanism from the outset), is considered. Using a suitable family of orthogonal polynomials, moments of the nonequilibrium probability distribution for the Brownian particle are introduced, which fulfill a recurrence relation. We review the case of classical Brownian motion, in which the orthogonal polynomials are the Hermite ones and the recurrence relation is a three-term one. After having performed a long-time approximation in the recurrence relation, the approximate nonequilibrium theory yields irreversible evolution of the Brownian particle towards thermal equilibrium with the “heat bath”. For quantum Brownian motion, which is the main subject of the present work, we restrict ourselves to include the first quantum correction: this leads us to introduce a new family of orthogonal polynomials which generalize the Hermite ones. Some general properties of the new family are established. The recurrence relation for the new moments of the nonequilibrium distribution, including the first quantum correction, turns out to be also a three-term one, which justifies the new family of polynomials. A long-time approximation on the new three-term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order. The standard Smoluchowski equations for the lowest order moments are recovered consistently, both classically and quantum-mechanically.

Comparison results for stochastic volatility models via coupling

The aim of this paper is to investigate the properties of stochastic volatility models, and to discuss to what extent, and with regard to which models, properties of the classical exponential Brownian motion model carry over to a stochastic volatility setting. The properties of the classical model of interest include the fact that the discounted stock price is positive for all t but converges to zero almost surely, the fact that it is a martingale but not a uniformly integrable martingale, and the fact that European option prices (with convex payoff functions) are convex in the initial stock price and increasing in volatility. We explain why these properties are significant economically, and give examples of stochastic volatility models where these properties continue to hold, and other examples where they fail.

Implicit finite difference approximation for time fractional diffusion equations

Time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model. In this paper we develop an implicit unconditionally stable numerical method to solve the one-dimensional linear time fractional diffusion equation, formulated with Caputo’s fractional derivative, on a finite slab. Several numerical examples of interest are also included.

On fake Brownian motions

Recently Albin constructed an example of a continuous martingale different from the classical Brownian motion but with the same marginal distributions, thus improving on the result of Hamza and Klebaner. We present a simpler solution to this problem.

Hyperbolic and fractional hyperbolic Brownian motion

We examine the hyperbolic, planar Brownian motion and its time-fractional version. The analogy between the hyperbolic Brownian motion and Brownian motion on the sphere is also analysed. We examine in detail the connection between the equations governing the distributions in the Cartesian and hyperbolic coordinates. We discuss the time-fractional generalization of hyperbolic Brownian motion and give a representation of it as composition of classical hyperbolic Brownian motion with a reflecting Brownian motion on the line.

Irreversible Investment with Cox-Ingersoll-Ross Type Mean Reversion

We solve a Dixit and Pindyck type irreversible investment problem in continuous time under the assumption that the project value follows a Cox-Ingersoll-Ross process. We indicate how the solution qualitatively differs from the two classical cases geometric Brownian motion and geometric mean reversion. Furthermore we discuss analytical properties of the Cox-Ingersoll-Ross process and demonstrate potential advantageous of this process as a model of the project value with regards to the classical ones.

Radiation model for nanoparticle: extension of classical Brownian motion concepts

Emissive power per unit area of a blackbody has been modeled as a function of frequency using quantum electrodynamics, semi-classical and classical approaches in the available literature. Present work extends the classical lumped-parameter systems model of Brownian motion of nanoparticle to abstract an emissive power per unit area model for nanoparticle radiating at temperature greater than absolute zero. The analytical model developed in present work has been based on synergism of local deformation leading to local motion of nanoparticle due to photon impacts. The work suggests the hypothesis of a free parameter f′ characterizing the damping coefficient of resistive forces to local motion of nanoparticle and the manipulation of which is possible to realize desired emissivity from nanoparticles. The model is validated with the well established Planck’s radiation law.

Stochastic excitation and delayed oscillation of a micro-oscillator

Stochastic processes in the form of the classical Brownian motion or internal phonon fluctuations may be studied by the use of a microcantilever. We present certain dynamic aspects of the behavior of the noise and the delayed fluctuations exhibited by a microcantilever. In particular, we present an analytic solution describing the transient response of the oscillator and discuss how self-excitation may determine the presence of small delays and, conversely, how delay-induced excitation may provide sensory information. As an application of the analysis, we present dynamic measurement of surface displacement with a preliminary sensitivity of 2 Hz/m in the chosen frequency window, which may effectively be extended to the nanometer regime.