Nonlinear expectations and stochastic calculus under uncertainty : with robust CLT and G-Brownian motion /: with robust CLT and G-Brownian motion. (2019)

• Record Type: Book
• Further Information: Note: Shige Peng.
• Other Names: Peng, Shige
• Contents: Intro; Preface; Introduction; Contents; Part I Basic Theory of Nonlinear Expectations; 1 Sublinear Expectations and Risk Measures; 1.1 Sublinear Expectations and Sublinear Expectation Spaces; 1.2 Representation of a Sublinear Expectation; 1.3 Distributions, Independence and Product Spaces; 1.4 Completion of Sublinear Expectation Spaces; 1.5 Examples of i.i.d Sequences Under Uncertainty of Probabilities; 1.6 Relation with Coherent Measures of Risk; 1.7 Exercises; 2 Law of Large Numbers and Central Limit Theorem Under Probability Uncertainty 2.1 Some Basic Results of Parabolic Partial Differential Equations2.2 Maximal Distribution and G-Normal Distribution; 2.3 Existence of G-Distributed Random Variables; 2.4 Law of Large Numbers and Central Limit Theorem; 2.5 Exercises; Part II Stochastic Analysis Under G-Expectations; 3 G-Brownian Motion and Itô's Calculus; 3.1 Brownian Motion on a Sublinear Expectation Space; 3.2 Existence of G-Brownian Motion; 3.3 Itô's Integral with Respect to G-Brownian Motion; 3.4 Quadratic Variation Process of G-Brownian Motion; 3.5 Distribution of the Quadratic Variation Process langleB rangle 3.6 Itô's Formula3.7 Brownian Motion Without Symmetric Condition; 3.8 G-Brownian Motion Under (Not Necessarily Sublinear) Nonlinear Expectation; 3.9 Construction of Brownian Motions on a Nonlinear Expectation Space; 3.10 Exercises; 4 G-Martingales and Jensen's Inequality; 4.1 The Notion of G-Martingales; 4.2 Heuristic Explanation of G-Martingale Representation; 4.3 G-Convexity and Jensen's Inequality for G-Expectations; 4.4 Exercises; 5 Stochastic Differential Equations; 5.1 Stochastic Differential Equations; 5.2 Backward Stochastic Differential Equations (BSDE) 5.3 Nonlinear Feynman-Kac Formula5.4 Exercises; 6 Capacity and Quasi-surely Analysis for G-Brownian Paths; 6.1 Integration Theory Associated to Upper Probabilities; 6.1.1 Capacity Associated with P; 6.1.2 Functional Spaces; 6.1.3 Properties of Elements of mathbbLpc; 6.1.4 Kolmogorov's Criterion; 6.2 G-Expectation as an Upper Expectation; 6.2.1 Construction of G-Brownian Motion Through Its Finite Dimensional Distributions; 6.2.2 G-Expectation: A More Explicit Construction; 6.3 The Capacity of G-Brownian Motion; 6.4 Quasi-continuous Processes; 6.5 Exercises Part III Stochastic Calculus for General Situations7 G-Martingale Representation Theorem; 7.1 G-Martingale Representation Theorem; 8 Some Further Results of Itô's Calculus; 8.1 A Generalized Itô's Integral; 8.2 Itô's Integral for Locally Integrable Processes; 8.3 Itô's Formula for General C2 Functions; Appendix A Preliminaries in Functional Analysis; A.1 Completion of Normed Linear Spaces; A.2 The Hahn-Banach Extension Theorem; A.3 Dini's Theorem and Tietze's Extension Theorem; Appendix B Preliminaries in Probability Theory; B.1 Kolmogorov's Extension Theorem; B.2 Kolmogorov's Criterion
• Publisher Details: Berlin, Germany : Springer
• Publication Date: 2019
• Extent: 1 online resource (216 pages)
• Subjects: 519.2/33; Brownian motion processes; Brownian motion processes; Electronic books
• Languages: English
• ISBNs: 9783662599037; 3662599031
• Related ISBNs: 9783662599020
• Notes: Note: Print version record.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.