Theory of stochastic processes: with applications to financial mathematics and risk theory

This book is a collection of exercises covering all the main topics in the modern theory of stochastic processes and its applications, including finance, actuarial mathematics, queuing theory, and risk theory. The aim of this book isto provide the reader with the theoretical and practical material necessary for deeper understanding of the main topics in the theory of stochastic processes and its related fields. The book is divided into chapters according to the various topics. Each chapter contains problems, hints, solutions, as well as aself-contained theoretical part which gives all the necessary material for solving the problems. References to the literature are also given. The exerciseshave various levels of complexity and vary from simple ones, useful for students studying basic notions and technique, to very advanced ones that reveal some important theoretical facts and constructions. This book is one of the largest collections of problems in the theory of stochastic processes and its applications. The problems in this book can be useful for undergraduate and graduate students, as well as for specialists in the theory of stochastic processes.Contains over 1000 high quality exercises on stochastic processes Presents a modern approach to topics such as sample paths and optimal stopping Ideal for professors who need exercises for exams, and graduate students wishing to learn about stochastic processes INDICE: Preface.- Definition of stochastic process. Cylinder s -algebra, finitedimensional distributions, the Kolmogorov theorem.- Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions.- Trajectories. Modifications. Filtrations.- Continuity. Differentiability. Integrability.- Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures.- Gaussian processes.- Martingales and related processes in discrete and continuous time. Stopping times.- Stationary discrete- and continuous-time processes. Stochastic integral over measure with orthogonal values.- Prediction and interpolation.- Markov chains: Discrete andcontinuous time.- Renewal theory. Queueing theory.- Markov and diffusion processes.- Itˆo stochastic integral. Itˆo formula. Tanaka formula.- Stochastic differential equations.- Optimal stopping of random sequences and processes.- Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems.- Statistics of stochastic processes.- Stochastic processes in financial mathematics (discrete time).- Stochastic processes in financial mathematics (continuous time).- Basic functionals of the risk theory.- Appendix.- Index.

• ISBN: 978-0-387-87861-4
• Editorial: Springer
• Encuadernacion: Cartoné
• Páginas: 375
• Fecha Publicación: 01/12/2009
• Nº Volúmenes: 1
• Idioma: Inglés