A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

Recent advancements in massively parallel biosensing and sequencing technologies have revolutionized the field of molecular biology and paved the way to novel and exciting innovations in medicine, biology, and environmental monitoring. Among them, biosensor arrays (e.g., DNA and protein microarrays) have gained a lot of attention. DNA microarrays are parallel affinity biosensors that can detect the presence and quantify the amounts of nucleic acid molecules of interest. They rely on chemical attraction between target nucleic acid sequences and their Watson-Crick complements that serve as probes and capture the targets. The molecular binding between the probes and targets is a stochastic process and hence the number of captured targets at any time is a random variable. Detection in conventional DNA microarrays is based on a single measurement taken in the steady state of the binding process. Recently developed real-time DNA microarrays, on the other hand, acquire multiple temporal measurements which allow more precise characterization of the reaction and enable faster detection based on the early dynamics of the binding process. In this thesis, I study target estimation and limits of performance of real time affinity biosensors. Target estimation is mapped to the problem of estimating parameters of discretely observed nonlinear diffusion processes. Performance of the estimators is characterized analytically via Cramer-Rao lower bound on the mean-square error. The proposed algorithms are verified on both simulated and experimental data, demonstrating significant gains over state-of-the-art techniques. In addition to biosensor arrays, in this thesis I present studies of the signal processing aspects of next-generation sequencing systems. Novel sequencing technologies will provide significant improvements in many aspects of human condition, ultimately leading towards the understanding, diagnosis, treatment and prevention of diseases. Reliable decision-making in such downstream applications is predicated upon accurate base-calling, i.e., identification of the order of nucleotides from noisy sequencing data. Base-calling error rates are nonuniform and typically deteriorate with the length of the reads. I have studied performance limits of base-calling, characterizing it by means of an upper bound on the error rates. Moreover, in the context of shotgun sequencing, I analyzed how accuracy of an assembled sequence depends on coverage, i.e., on the average number of times each base in a target sequence is represented in different reads. These analytical results are verified using experimental data. Among many downstream applications of high-throughput biosensing and sequencing technologies, reconstruction of gene regulatory networks is of particular importance. In this thesis, I consider the gene network inference problem and propose a probabilistic graphical approach for solving it. Specifically, I develop graphical models and design message passing algorithms which are then verified using experimental data provided by the Dialogue for Reverse Engineering Assessment and Methods (DREAM) initiative. / text

In a 2005 article in Science [45], Park et al. measured in vacuum the friction between a coated atomic-force-microscope tip and the clean two-fold surface of an AlNiCo quasicrystal. Because the two-fold surface is periodic in one direction and aperiodic (with a quasiperiodicity related to the Fibonacci sequence) in the perpendicular direction, frictional anisotropy is not unexpected; however, the magnitude of that anisotropy in the Park experiment, a factor of eight, is unprecedented. By eliminating chemistry as a variable, the experiment also demonstrated that the low friction of quasicrystals must be tied in some way to their quasiperiodicity. Through various models, we investigate generic geometric mechanisms that might give rise to this anisotropy.

We consider a general stochastic differential delay equation (SDDE) with multiplicative colored noise. We study the limit as the time delays and the correlation times of the noises go to zero at the same rate. First, we derive the limiting equation for the equation obtained by Taylor expanding the SDDE to first order in the time delays. The limiting equation contains a noise-induced drift term that depends on the ratios of the time delays to the correlation times of the noises. We prove that, under appropriate assumptions, the solution of the equation obtained by the Taylor expansion converges to the solution of this limiting equation in probability with respect to the sup norm over compact time intervals. Next, we derive the limiting equation for the SDDE and prove a similar convergence result regarding convergence of the solution of the SDDE to the solution of this limiting equation. We see that the limiting equation corresponding to the equation obtained by the Taylor expansion is an approximation of the limiting equation corresponding to the SDDE. Finally, we study the effects of time delay on a particular model of active Brownian motion.

Model-based control and process optimization technologies are becoming more commonly used by chemical engineers. These algorithms rely on fundamental or empirical models that are frequently described by systems of differential equations with unknown parameters. It is, therefore, very important for modellers of chemical engineering processes to have access to reliable and efficient tools for parameter estimation in dynamic models. The purpose of this thesis is to develop an efficient and easy-to-use parameter estimation algorithm that can address difficulties that frequently arise when estimating parameters in nonlinear continuous-time dynamic models of industrial processes. The proposed algorithm has desirable numerical stability properties that stem from using piece-wise polynomial discretization schemes to transform the model differential equations into a set of algebraic equations. Consequently, parameters can be estimated by solving a nonlinear programming problem without requiring repeated numerical integration of the differential equations. Possible modelling discrepancies and process disturbances are accounted for in the proposed algorithm, and estimates of the process disturbance intensities can be obtained along with estimates of model parameters and states. Theoretical approximate confidence interval expressions for the parameters are developed. Through a practical two-phase nylon reactor example, as well as several simulation studies using stirred tank reactors, it is shown that the proposed parameter estimation algorithm can address difficulties such as: different types of measured responses with different levels of measurement noise, measurements taken at irregularly-spaced sampling times, unknown initial conditions for some state variables, unmeasured state variables, and unknown disturbances that enter the process and influence its future behaviour. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2008-06-20 16:34:44.586

We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions.

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research. Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution. Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method. In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula. Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

The fact that loosely bonded DNA bases could tolerate large structural fluctuations, form a dissipative environment for a charge traveling through the DNA. Nonlinear stochastic nature of structural fluctuations facilitates rich charge dynamics in DNA. We study the complex charge dynamics by solving a nonlinear, stochastic, coupled system of differential equations. Charge transfer between donor and acceptor in DNA occurs via different mechanisms depending on the distance between donor and acceptor. It changes from tunneling regime to a polaron assisted hopping regime depending on the donor-acceptor separation. Also we found that charge transport strongly depends on the feasibility of polaron formation. Hence it has complex dependence on temperature and charge-vibrations coupling strength. Mismatched base pairs, such as different conformations of the G・A mispair, cause only minor structural changes in the host DNA molecule, thereby making mispair recognition an arduous task. Electron transport in DNA that depends strongly on the hopping transfer integrals between the nearest base pairs, which in turn are affected by the presence of a mispair, might be an attractive approach in this regard. I report here on our investigations, via the I –V characteristics, of the effect of a mispair on the electrical properties of homogeneous and generic DNA molecules. The I –V characteristics of DNA were studied numerically within the double-stranded tight-binding model. The parameters of the tight-binding model, such as the transfer integrals and on-site energies, are determined from first-principles calculations. The changes in electrical current through the DNA chain due to the presence of a mispair depend on the conformation of the G・A mispair and are appreciable for DNA consisting of up to 90 base pairs. For homogeneous DNA sequences the current through DNA is suppressed and the strongest suppression is realized for the G(anti)・A(syn) conformation of the G・A mispair. For inhomogeneous (generic) DNA molecules, the mispair result can be either suppression or an enhancement of the current, depending on the type of mispairs and actual DNA sequence.

These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.

By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model is formulated and it leads to a system of multi-time scale stochastic differential equations. The classical Picard-Lindel\"{o}f successive approximations scheme is expended to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this generates to a problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations. To illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are exhibited. Without loss in generality, the modeling and analysis of three time-scale fractional stochastic differential equations is followed by the development of the numerical algorithm for multi-time scale dynamic equations. The development of numerical algorithm is based on the idea if numerical integration in the context of the notion of multi-time scale integration. The multi-time scale approach is applied to explore the study of higher order stochastic differential equations (HOSDE) is presented. This study utilizes the variation of constant parameter technique to develop a method for finding closed form solution processes of classes of HOSDE. Then then probability distribution of the solution processes in the context of the second order equations is investigated.