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Stochastic Calculus with Infinitesimals 2013 ed. [Mīkstie vāki]

  • Formāts: Paperback / softback, 112 pages, height x width: 235x155 mm, weight: 2117 g, XVIII, 112 p., 1 Paperback / softback
  • Sērija : Lecture Notes in Mathematics 2067
  • Izdošanas datums: 07-Nov-2012
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642331483
  • ISBN-13: 9783642331480
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Stochastic Calculus with Infinitesimals 2013 ed.Palielināt
  • Formāts: Paperback / softback, 112 pages, height x width: 235x155 mm, weight: 2117 g, XVIII, 112 p., 1 Paperback / softback
  • Sērija : Lecture Notes in Mathematics 2067
  • Izdošanas datums: 07-Nov-2012
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • ISBN-10: 3642331483
  • ISBN-13: 9783642331480
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783642331480.html
Keywords:
Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial mathematics or theoretical quantum mechanics) and therefore appears in physics and economics curricula as well. However, existing approaches to stochastic analysis either presuppose various concepts from measure theory and functional analysis or lack full mathematical rigour. This short book proposes to solve the dilemma: By adopting E. Nelson's "radically elementary" theory of continuous-time stochastic processes, it is based on a demonstrably consistent use of infinitesimals and thus permits a radically simplified, yet perfectly rigorous approach to stochastic calculus and its fascinating applications, some of which (notably the Black-Scholes theory of option pricing and the Feynman path integral) are also discussed in the book.
1 Infinitesimal Calculus, Consistently and Accessibly
1(6)
1.1 An Accessible Axiom System for Infinitesimal Calculus: Minimal Internal Set Theory
1(3)
1.2 Finer Classification of the Reals: Finite vs. Limited
4(3)
2 Radically Elementary Probability Theory
7(12)
2.1 Random Variables and Stochastic Processes
7(2)
2.2 Integrability and Limitedness
9(4)
2.3 Wiener Walks and Wiener Processes
13(2)
2.4 Distribution of the Wiener Walk
15(1)
2.5 Integrability and Limited Paths of the Wiener Walk
16(3)
3 Radically Elementary Stochastic Integrals
19(16)
3.1 Martingales and I to Integrals
19(4)
3.2 Radically Elementary I to Processes
23(3)
3.3 A Basic Radically Elementary I to Formula
26(2)
3.4 Analytic Excursion: A Radically Elementary Treatment of Geometric I to Processes with Monotone Drift
28(5)
3.5 The Radically Elementary Version of Levy's Characterization of Wiener Processes
33(2)
4 The Radically Elementary Girsanov Theorem and the Diffusion Invariance Principle
35(10)
4.1 Girsanov's Theorem
35(7)
4.2 The Radically Elementary Diffusion Invariance Principle
42(3)
5 Excursion to Financial Economics: A Radically Elementary Approach to the Fundamental Theorems of Asset Pricing
45(10)
6 Excursion to Financial Engineering: Volatility Invariance in the Black-Scholes Model
55(6)
7 A Radically Elementary Theory of I to Diffusions and Associated Partial Differential Equations
61(10)
7.1 I to Diffusions
61(3)
7.2 The Markov Property of I to Diffusions and the Feynman-Kac Formula
64(7)
8 Excursion to Mathematical Physics: A Radically Elementary Definition of Feynman Path Integrals
71(6)
9 A Radically Elementary Theory of Levy Processes
77(16)
9.1 Random Walks and Levy Walks
77(3)
9.2 Integrability of Levy Walks with Limited Increments
80(2)
9.3 Lindstrøm's Characterization of Levy Walks
82(4)
9.4 A Radically Elementary Ito-Doeblin Formula for Levy Walks with Limited-Variation Jump Part
86(4)
9.5 A Brief Look at Levy Finance
90(3)
10 Final Remarks
93(2)
A Excursion to Logic: Some Remarks on the Metamathematics of Minimal Internal Set Theory
95(8)
A.1 An Alternative Road to Minimal Internal Set Theory
95(2)
A.2 A Simple Relative Consistency Proof for a Substantial Subsystem of minIST
97(4)
A.3 Definable Models for (Minimal) Nonstandard Analysis
101(2)
B Robinsonian vs. Minimal Nonstandard Analysis
103(4)
References 107(4)
Index 111