Bernt Øksendal

Bio: Bernt Øksendal is an academic researcher from University of Oslo. The author has contributed to research in topics: Stochastic differential equation & Stochastic control. The author has an hindex of 46, co-authored 279 publications receiving 18699 citations. Previous affiliations of Bernt Øksendal include French Institute for Research in Computer Science and Automation & Norwegian School of Economics.

Stochastic differential equations : an introduction with applications

Abstract: Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.

Stochastic Differential Equations

Abstract: We now return to the possible solutions X t (ω) of the stochastic differential equation (5.1) where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X t satisfies the stochastic integral equation or in differential form (5.2) .

Applied Stochastic Control of Jump Diffusions

Abstract: Preface.- Stochastic Calculus with Levy Processes.- Financial Markets Modelled by Jump Diffusions.- Optimal Stopping of Jump Diffusions.- Backward Stochastic Differential Equations and Risk Measures.- Stochastic Control of Jump Diffusions.- Stochastic Differential Games.- Combined Optimal Stopping and Stochastic Control of Jump Diffusions.- Viscosity Solutions.- Solutions of Selected Exercises.- References.- Notation and Symbols.

Stochastic Calculus for Fractional Brownian Motion and Applications

Abstract: Fractional Brownian motion.- Intrinsic properties of the fractional Brownian motion.- Stochastic calculus.- Wiener and divergence-type integrals for fractional Brownian motion.- Fractional Wick Ito Skorohod (fWIS) integrals for fBm of Hurst index H >1/2.- WickIto Skorohod (WIS) integrals for fractional Brownian motion.- Pathwise integrals for fractional Brownian motion.- A useful summary.- Applications of stochastic calculus.- Fractional Brownian motion in finance.- Stochastic partial differential equations driven by fractional Brownian fields.- Stochastic optimal control and applications.- Local time for fractional Brownian motion.

Fractional white noise calculus and applications to finance

Abstract: The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Ito type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Ito type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Ito fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Ito fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

Gaussian Processes for Machine Learning

Abstract: A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Controlled Markov processes and viscosity solutions

Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

A Survey of Recent Results in Networked Control Systems

Abstract: Networked control systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. We review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packet-rates, sampling, network delay, and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.