Continuous Optimization for Data Science (1版)

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The text is divided into three main parts: unconstrained optimization, constrained optimization, and linear programming. The first part addresses unconstrained optimization in single-variable and multivariable functions, introducing key algorithms such as steepest descent, Newton, and quasi-Newton methods. The second part focuses on constrained optimization, starting with linear equality constraints and extending to more general cases, including inequality constraints. It details optimality conditions, sensitivity analysis, and relevant algorithms for solving these problems. The third part covers linear programming, presenting the formulation of LP problems, the simplex algorithm, and sensitivity analysis. Throughout, the text provides numerous applications to data science, such as linear regression, maximum likelihood estimation, expectation-maximization algorithms, support vector machines, and linear neural networks.

Continuous Parameter Markov Processes and Stochastic Differential Equations

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This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples. After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô’s fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.

Arbitrage Theory in Continuous Time (4版)

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ARBITRAGE THEORY IN CONTINUOUS TIME 4E 2021 (H) ISBN： 9780198851615 類別： 財務金融Financial Management 出版社： OXFORD UNIVERSITY PRESS 作者： BJORK 年份： 2020 裝訂別： 精裝 頁數： 592頁 Table of Contents 1:Introduction I. Discrete Time Models 2:The Binomial Model 3:A More General One period Model II. Stochastic Calculus 4:Stochastic Integrals 5:Stochastic Differential Equations III. Arbitrage Theory 6:Portfolio Dynamics 7:Arbitrage Pricing 8:Completeness and Hedging 9:A Primer on Incomplete Markets 10:Parity Relations and Delta Hedging 11:The Martingale Approach to Arbitrage Theory 12:The Mathematics of the Martingale Approach 13:Black-Scholes from a Martingale Point of View 14:Multidimensional Models: Martingale Approach 15:Change of Numeraire 16:Dividends 17:Forward and Futures Contracts 18:Currency Derivatives 19:Bonds and Interest Rates 20:Short Rate Models 21:Martingale Models for the Short Rate 22:Forward Rate Models 23:LIBOR Market Models 24:Potentials and Positive Interest IV. Optimal Control and Investment Theory 25:Stochastic Optimal Control 26:Optimal Consumption and Investment 27:The Martingale Approach to Optimal Investment 28:Optimal Stopping Theory and American Options V. Incomplete Markets 29:Incomplete Markets 30:The Esscher Transform and the Minimal Martingale Measure 31:Minimizing f-divergence 32:Portfolio Optimization in Incomplete Markets 33:Utility Indifference Pricing and Other Topics 34:Good Deal Bounds VI. Dynamic Equilibrium Theory 35:Equilibrium Theory: A Simple Production Model 36:The Cox-Ingersoll-Ross Factor Model 37:The Cox-Ingersoll-Ross Interest Rate Model 38:Endowment Equilibrium: Unit Net Supply