Elements of random walk and diffusion processes [[electronic resource] /] / Oliver C. Ibe

• Record Nr.: UNINA9910139012803321
• Autore: Ibe Oliver C (Oliver Chukwudi), <1947->
• Pubbl/distr/stampa: Hoboken, N.J., : John Wiley & Sons, Inc., 2013
• ISBN: 1-118-61793-2; 1-118-61805-X; 1-118-62985-X
• Descrizione fisica: 1 online resource (278 p.)
• Collana: Wiley series in operations research and management science
• Classificazione: MAT003000
• Disciplina: 519.2/82
• Soggetti: Random walks (Mathematics); Diffusion processes
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Elements of Random Walk and Diffusion Processes; Copyright; Contents; Preface; Acknowledgments; 1 Review of Probability Theory; 1.1 Introduction; 1.2 Random Variables; 1.2.1 Distribution Functions; 1.2.2 Discrete Random Variables; 1.2.3 Continuous Random Variables; 1.2.4 Expectations; 1.2.5 Moments of Random Variables and the Variance; 1.3 Transform Methods; 1.3.1 The Characteristic Function; 1.3.2 Moment-Generating Property of the Characteristic Function; 1.3.3 The s-Transform; 1.3.4 Moment-Generating Property of the s-Transform; 1.3.5 The z-Transform; 1.3.6 Moment-Generating Property of the z-Transform1.4 Covariance and Correlation Coefficient; 1.5 Sums of Independent Random Variables; 1.6 Some Probability Distributions; 1.6.1 The Bernoulli Distribution; 1.6.2 The Binomial Distribution; 1.6.3 The Geometric Distribution; 1.6.4 The Poisson Distribution; 1.6.5 The Exponential Distribution; 1.6.6 Normal Distribution; 1.7 Limit Theorems; 1.7.1 Markov Inequality; 1.7.2 Chebyshev Inequality; 1.7.3 Laws of Large Numbers; 1.7.4 The Central Limit Theorem; Problems; 2 Overview of Stochastic Processes; 2.1 Introduction; 2.2 Classification of Stochastic Processes2.3 Mean and Autocorrelation Function; 2.4 Stationary Processes; 2.4.1 Strict-Sense Stationary Processes; 2.4.2 Wide-Sense Stationary Processes; 2.5 Power Spectral ; Density; 2.6 Counting Processes; 2.7 Independent Increment Processes; 2.8 Stationary Increment Process; 2.9 Poisson Processes; 2.9.1 Compound Poisson Process; 2.10 Markov Processes; 2.10.1 Discrete-Time Markov Chains; 2.10.2 State Transition Probability Matrix; 2.10.3 The k-Step State Transition Probability; 2.10.4 State Transition Diagrams; 2.10.5 Classification of States; 2.10.6 Limiting-State Probabilities2.10.7 Doubly Stochastic Matrix; 2.10.8 Continuous-Time Markov Chains; 2.10.9 Birth and Death Processes; 2.11 Gaussian Processes; 2.12 Martingales; 2.12.1 Stopping Times; Problems; 3 One-Dimensional Random Walk; 3.1 Introduction; 3.2 Occupancy Probability; 3.3 Random Walk as a Markov Chain; 3.4 Symmetric Random Walk as a Martingale; 3.5 Random Walk with Barriers; 3.6 Mean-Square Displacement; 3.7 Gambler's Ruin; 3.7.1 Ruin Probability; 3.7.2 Alternative Derivation of Ruin Probability; 3.7.3 Duration of a Game; 3.8 Random Walk with Stay; 3.9 First Return to the Origin3.10 First Passage Times for Symmetric Random Walk; 3.10.1 First Passage Time via the Generating Function; 3.10.2 First Passage Time via the Reflection Principle; 3.10.3 Hitting Time and the Reflection Principle; 3.11 The Ballot Problem and the Reflection Principle; 3.11.1 The Conditional Probability Method; 3.12 Returns to the Origin and the Arc-Sine Law; 3.13 Maximum of a Random Walk; 3.14 Two Symmetric Random Walkers; 3.15 Random Walk on a Graph; 3.15.1 Proximity Measures; 3.15.2 Directed Graphs; 3.15.3 Random Walk on an Undirected Graph; 3.15.4 Random Walk on a Weighted Graph
• Sommario/riassunto: "Featuring an introduction to stochastic calculus, this book uniquely blends diffusion equations and random walk theory and provides an interdisciplinary approach by including numerous practical examples and exercises with real-world applications in operations research, economics, engineering, and physics. It covers standard methods and applications of Brownian motion and discusses Levy motion; addresses fractional calculus; introduces percolation theory and its relationship to diffusion processes; and more"--