<p>In recent years, the field of Fractional Brownian motion, Fractional Gaussian noise and long-range dependent processes has gained growing interest. Fractional Brownian motion is of great interest for example in telecommunications, hydrology and the generation of artificial landscapes. In fact, Fractional Brownian motion is a basic continuous process through which we show that it is neither a semimartingale nor a Markov process. In this work, we will focus on the path properties of Fractional Brownian motion and will try to check the absence of the property of a semimartingale. The concept of volatility will be dealt with in this work as a phenomenon in finance. Moreover, some statistical method like R/S analysis will be presented. By using these statistical tools we examine the volatility of shares and we demonstrate empirically that there are in fact shares which exhibit a fractal structure different from that of Brownian motion.</p>

This dissertation presents experimental studies on Brownian motion at fast time scales, as well as our recent developments in Thermal Noise Imaging which uses thermal motions of microscopic particles for spatial imaging. As thermal motions become increasingly important in the studies of soft condensed matters, the study of Brownian motion is not only of fundamental scientific interest but also has practical applications. Optical tweezers with a fast position-sensitive detector provide high spatial and temporal resolution to study Brownian motion at fast time scales. A novel high bandwidth detector was developed with a temporal resolution of 30 ns and a spatial resolution of 1 °A. With this high bandwidth detector, Brownian motion of a single particle confined in an optical trap was observed at the time scale of the ballistic regime. The hydrodynamic memory effect was fully studied with polystyrene particles of different sizes. We found that the mean square displacements of different sized polystyrene particles collapse into one master curve which is determined by the characteristic time scale of the fluid inertia effect. The particle’s inertia effect was shown for particles of the same size but different densities. For the first time the velocity autocorrelation function for a single particle was shown. We found excellent agreement between our experiments and the hydrodynamic theories that take into account the fluid inertia effect. Brownian motion of a colloidal particle can be used to probe three-dimensional nano structures. This so-called thermal noise imaging (TNI) has been very successful in imaging polymer networks with a resolution of 10 nm. However, TNI is not efficient at micrometer scale scanning since a great portion of image acquisition time is wasted on large vacant volume within polymer networks. Therefore, we invented a method to improve the efficiency of large scale scanning by combining traditional point-to-point scanning to explore large vacant space with thermal noise imaging at the proximity of the object. This method increased the efficiency of thermal noise imaging by more than 40 times. This development should promote wider applications of thermal noise imaging in the studies of soft materials and biological systems. / text

Understand the behavior of queueing networks in heavy tra c is very important due to its importance in evaluating the network performance in related applications. However, in many cases, the stationary distributions of such networks are intractable. Based on di usion limits of queueing networks, we can use Re ected Brownian Motion (RBM) processes as reasonable approximations. As such, we are interested in obtaining the stationary distribution of RBM. Unfortunately, these distributions are also in most cases intractable. However, the tail behavior (large deviations) of RBM may give insight into the stationary distribution. Assuming that a large deviations principle holds, we need only solve the corresponding variational problem to obtain the rate function. Our research is mainly focused on how to solve variational problems in the case of rotationally symmetric (RS) data. The contribution of this dissertation primarily consists of three parts. In the rst part we give out the speci c stability condition for the RBM in the octant in the RS vi case. Although the general stability conditions for RBM in the octant has been derived previously, we simplify these conditions for the case we consider. In the second part we prove that there are only two types of possible solutions for the variational problem. In the last part, we provide a simple computational method. Also we give an example under which a spiral path is the optimal solution. / text

This report is mainly based on the paper "Optimal multiple stopping and valuation of swing options" by R. Carmona and N. Touzi (1). Here the authors model and solve optimal stopping problems with more than one exercise time. The existence of optimal stopping times is firstly proved and they then construct the value function of American put options with multiple exercises in the case of the Black-Scholes model, characterizing the exercise boundaries of the perpetual case. Finally, they extend the analysis to the swing contracts with infinitely many exercise rights. In this report, we concentrate on explaining their rigorous mathematical analysis in detail, especially for the valuation of the perpetual American put options with single exercise and two exercise rights, and the characteristics of the exercise boundaries of the multiple stopping case. These results are presented as theorems in Chapter 2 and Chapter 3. / text

Multiple lines of evidence suggest that RNA performed all of the biological functions in the first life forms on earth. These functions included cleavage, ligation, polymerization, recognition, binding, and replication. In order to perform these functions, populations of RNA molecules with unevolved sequences must have been able to fold into compact three dimensional shapes, in unregulated environments, and without the help of proteins. Folding into compact tertiary structures is difficult because of the high charge density of RNA. Consequently, the ranges of temperature, salinity, pH, and pressure that allow RNA to fold into functional shapes is very restricted. We use thermodynamic arguments and Brownian dynamics simulations to compute the range of these environmental parameters that will allow RNA to fold. This is a non-trivial calculation due to the formation of an ion atmosphere around RNA that reduces its electric field. The results can be used to clarify the environments in which the transition to life is possible. Our preliminary calculations suggest that environments with low temperatures ($0-50^\circ C$) and high salt concentrations (greater than 100mM) are the most favorable for unassisted RNA folding and thus the transition to RNA-based life. Applications of our results include determining the environments on early earth where life formed, assesing the habitability of Europa, Titan, and (using modeled parameters) extrasolar planets. / text

This dissertation focuses on the performance analysis of multiclass open queueing networks using semi-martingale reflecting Brownian motion (SRBM) approximation. It consists of four parts. In the first part, we derive a strong approximation for a multiclass feedforward queueing network, where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival; that is, a first-in-first-out (FIFO) discipline is in force, and among groups, jobs are served under a pre-assigned preemptive priority discipline. We obtain an SRBM as the result of strong approximation for the network, through an inductive approach. Based on the strong approximation, some procedures are proposed to approximate the stationary distribution of various performance measures of the queueing network. Our work extends and complements the previous work done on the feedforward queueing network. The numeric examples show that the strong approximation provides a better approximation than that suggested by a straightforward interpretation of the heavy traffic limit theorem. In the second part, we develop a Brownian approximation for a general multiclass queueing network with a set of single-server stations that operate under a combination of FIFO (first-in-first-out) and priority service disciplines and are subject to random breakdowns. Our intention here is to illustrate how to approximate a queueing network by an SRBM, not to justify such approximation. We illustrate through numerical examples in comparison against simulation that the SRBM model, while not always supported by a heavy traffic limit theorem, possesses good accuracy in most cases, even when the systems are moderately loaded. Through analyzing special networks, we also discuss the existence of the SRBM approximation in relation to the stability and the heavy traffic limits of the networks. In most queueing network applications, the stationary distributions of queueing networks are of great interest. It becomes natural to approximate these stationary distributions by the stationary distributions of the approximating SRBMs. Although we are able to characterize the stationary distribution of an SRBM, except in few limited cases, it is extremely difficult to obtain the stationary distribution analytically. In the third part of the dissertation, we propose a numerical algorithm, referred to as BNA/FM (Brownian network analyzer with finite element method), for computing the stationary distribution of an SRBM in a hypercube. SRBM in a hypercube serves as an approximate model of queueing networks with finite buffers. Our BNA/FM algorithm is based on finite element method and an extension of a generic algorithm developed in the previous work. It uses piecewise polynomials to form an approximate subspace of an infinite dimensional functional space. The BNA/FM algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to the BNA/SM (Brownian network analyzer with spectral method) developed in the previous work, where global polynomials are used to form the approximate subspace and they sometime fail to produce meaningful estimates of these stationary probabilities. We also report extensive computational experiences from our implementation that will be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers is presented to illustrate the effectiveness of the Brownian approximation model and our BNA/FM algorithm. In the last part of the dissertation, we extend the BNA/FM algorithm to calculate the stationary distribution of an SRBM in an orthant. This type of SRBM arises as a Brownian approximation model for queueing networks with infinite buffers. We prove the convergence theorems which justify the extension. A three-machine job shop example is presented to illustrate the accuracy of our extended BNA/FM algorithm. In fact, this extended algorithm is also used in the first two parts of this dissertation to analyze the performance of several queueing network examples and it gives fairly good performance estimates in most cases.

No description available.

We evaluated the influence of a landscape dominated by agriculture and an extensive road network on fine-scale movements of black bears (Ursus americanus) in south-central Florida. The objectives of this study were to (1) define landscape functionality including corridor use by the directionality and speed of bear movements, (2) to develop a model reflecting selected habitat characteristics during movements, (3) to identify habitat characteristics selected by bears at road-crossing locations, and (3) to develop and evaluate a predictive model for road-crossing locations based on habitat characteristics. We assessed models using GPS data from 20 adult black bears (9 F, 11 M), including 382 unique road-crossing events by 16 individuals. Directionality of bear movements were influenced by the density of cover and proximity to human infrastructure, and movement speed was influenced by density of cover and proximity to paved roads. We used the Brownian bridge movement model to assess road-crossing behavior. Landscape-level factors like density of cover and density of roads appeared more influential than roadside factors, vegetative or otherwise. Model validation procedures suggested strong predictive ability for the selected road-crossing model. These findings will allow managers to prioritize and implement sound strategies to promote connectivity and reduce road collisions.

This thesis consists of two novel contributions to the computation of first passage time distribution for Brownian motion. First, we extend the known formula for boundary crossing probabilities for Brownian motion to the discontinuous piecewise linear boundary. Second, we derive explicit formula for the first passage time density of Brownian motion crossing piecewise linear boundary. Further, we demonstrate how to approximate the boundary crossing probabilities and density for general nonlinear boundaries. Moreover, we use Monte Carlo simulation method and develop algorithms for the numerical computation. This method allows one to assess the accuracy of the numerical approximation. Our approach can be further extended to compute two-sided boundary crossing probabilities.

We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion / when the initial distribution is Gibbsian.