Catálogo de publicaciones - libros

Our goal is to analyze the moment uniqueness or non-uniqueness of the one-dimensional distributions of the solution processes of Itô type stochastic differential equations (SDE). We recall some criteria, classical and/or new, and apply them to derive results for the solutions of linear and nonlinear SDEs. Special attention is paid to the Brownian motion, stochastic integrals and geometric Brownian motion. Another possibility is to use the moment convergence theorem (Frećhet-Shohat) for finding explicitly the limit one-dimensional distributions of specific processes. Related moment problems are also outlined with the focus on functional transformations of processes and approximations of the solutions of perturbed SDEs.

In this paper we study Poisson noise with the help of generalized functions. We first discuss the -transform of ()-Poisson functionals. We then consider an extension of the -transform with the help of Sobolev norms. Next we introduce the class of normal functionals and consider the Wick product. We consider the Hida derivatives and the Lévy Laplacian acting on these normal functionals. A relationship between the -transform and Y. Itô’s -transform is given. Finally, we give a stochastic limit characterization of the Lévy Laplacian with the help of two-parameter Poisson processes.

A four-compartment model for the evolution of cancer based on the characteristics of the cells is presented. The model is expanded to account for intrinsic and acquired drug resistance. This model can be explored to see the evolution of drug resistance starting from a single cell. Numerical studies are performed illustrating the palliative nature of chemotherapeutic treatments. Computational results are given for traditional treatment schedules. An alternate schedule for treatments is developed increasing the life expectancy and quality of life for the patient. A framework for the alternate scheduling is presented that addresses life expectancy, quality of life, and risk of metastasis. A key feature of the alternate schedule is that information for a particular patient can be used resulting in a personalized schedule of treatments. Alternate scheduling is compared to traditional treatment scheduling.

This is the first of final two papers in this series which will give complete classification of the finite dimensional estimation algebra of maximal rank (cf. Definition 2 in Sec. 2), a problem proposed by R. Brockett in his invited lecture at the International Congress of Mathematics in 1983. The concept of estimation algebra (Lie algebra) was first introduced by Brockett and Mitter independently. This concept plays a crucial role in the investigation of finite-dimensional nonlinear filters. Since 1990, Yau has launched a program to study Brockett’s problem. He first considered Wong’s anti-symmetric matrix = () = (/ − /), where denotes the drift term in equation (1). He solved the Brockett’s problem when matrix has only constant entries. Yau’s program is to show that matrix must have constant entries for finite dimensional estimation algebra. Recently Chen and Yau studied the structure of quadratic forms in a finite-dimensional estimation algebra. Let be the quadratic rank of the estimation algebra and be the dimension of the state space. They showed that the left upper corner (), 1 ≤ ≤ , of matrix is a matrix with constant coefficients. In this paper, we shall show that the lower right corner (), + 1 ≤ ≤ , of matrix is also a constant matrix.

Based on state space reachable sets we formulate a two-player differential game for stability. The role of one player (the bounded disturbance) is to remove as much of the system’s stability as possible, while the second player (the control) tries to maintain as much of the system’s stability as possible. To obtain explicit computable relationships we limit the control selection to setting basic parameters of the system in their stability range and the disturbance is a bounded scalar function. This stability game provides for an explicit quantification of uncertainty in control systems and the value in the game manifests itself as the -gain of the dynamic input/output disturbance system as a function of the control parameters. It has been discovered recently that the -gain can be expressed as an explicit parametric formula for linear second order games, and design charts here provide quantitative information in third order linear games. A model predictive scheme will be given for the higher order linear and nonlinear stability games.

A model of O-U process with discrete noises is proposed for the price micro-movement, which refers to the transactional price behavior. The model can be viewed as a multivariate point process and framed as a filtering problem with counting process observations. Under this framework, the whole sample paths are observable and are used for parameter estimation. Based on the filtering equation, we construct a consistent recursive algorithm to compute the approximate posterior and the Bayes estimates. Finally, Bayes estimates for a two-month transaction prices of Microsoft are obtained.

This paper is concerned with filtering of a hybrid model with a number of linear systems coupled by a hidden switching process. The most probable trajectory approach is used to derive a finite-dimensional recursive filter. Such scheme is applied to nonlinear systems using a piecewise-linear approximation method. Numerical examples are provided and computational experiments are reported.