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First Course in Stochastic Calculus [Mīkstie vāki]

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  • Formāts: Paperback / softback, 270 pages, height x width: 254x178 mm, weight: 511 g
  • Sērija : Pure and Applied Undergraduate Texts
  • Izdošanas datums: 30-Mar-2022
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470464888
  • ISBN-13: 9781470464882
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  • Mīkstie vāki
  • Cena: 97,63 €
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First Course in Stochastic CalculusPalielināt
  • Formāts: Paperback / softback, 270 pages, height x width: 254x178 mm, weight: 511 g
  • Sērija : Pure and Applied Undergraduate Texts
  • Izdošanas datums: 30-Mar-2022
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470464888
  • ISBN-13: 9781470464882
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470464882.html
Keywords:
A First Course in Stochastic Calculus is a complete guide for advanced undergraduate students to take the next step in exploring probability theory and for master's students in mathematical finance who would like to build an intuitive and theoretical understanding of stochastic processes. This book is also an essential tool for finance professionals who wish to sharpen their knowledge and intuition about stochastic calculus.

Louis-Pierre Arguin offers an exceptionally clear introduction to Brownian motion and to random processes governed by the principles of stochastic calculus. The beauty and power of the subject are made accessible to readers with a basic knowledge of probability, linear algebra, and multivariable calculus. This is achieved by emphasizing numerical experiments using elementary Python coding to build intuition and adhering to a rigorous geometric point of view on the space of random variables. This unique approach is used to elucidate the properties of Gaussian processes, martingales, and diffusions. One of the book's highlights is a detailed and self-contained account of stochastic calculus applications to option pricing in finance.

Recenzijas

Louis-Pierre Arguin's masterly introduction to stochastic calculus seduces the reader with its quietly conversational style; even rigorous proofs seem natural and easy. Full of insights and intuition, reinforced with many examples, numerical projects, and exercises, this book by a prize-winning mathematician and great teacher fully lives up to the author's reputation. I give it my strongest possible recommendation."" Jim Gatheral, Baruch College

""I happen to be of a different persuasion, about how stochastic processes should be taught to undergraduate and MA students. But I have long been thinking to go against my own grain at some point and try to teach the subject at this leveltogether with its applications to financein one semester. Louis-Pierre Arguin's excellent and artfully designed text will give me the ideal vehicle to do so."" Ioannis Karatzas, Columbia University, New York

Foreword ix
Preface xi
Chapter 1 Basic Notions of Probability
1(22)
1.1 Probability Space
1(3)
1.2 Random Variables and Their Distributions
4(7)
1.3 Expectation
11(5)
1.4 Inequalities
16(1)
1.5 Numerical Projects and Exercices
17(2)
Exercises
19(2)
1.6 Historical and Bibliographical Notes
21(2)
Chapter 2 Gaussian Processes
23(28)
2.1 Random Vectors
23(5)
2.2 Gaussian Vectors
28(6)
2.3 Gaussian Processes
34(6)
2.4 A Geometric Point of View
40(5)
2.5 Numerical Projects and Exercises
45(1)
Exercises
46(3)
2.6 Historical and Bibliographical Notes
49(2)
Chapter 3 Properties of Brownian Motion
51(16)
3.1 Properties of the Distribution
51(3)
3.2 Properties of the Paths
54(6)
3.3 A Word on the Construction of Brownian Motion
60(1)
3.4 A Point of Comparison: The Poisson Process
61(1)
3.5 Numerical Projects and Exercises
62(2)
Exercises
64(2)
3.6 Historical and Bibliographical Notes
66(1)
Chapter 4 Martingales
67(32)
4.1 Elementary Conditional Expectation
67(3)
4.2 Conditional Expectation as a Projection
70(10)
4.3 Martingales
80(4)
4.4 Computations with Martingales
84(5)
4.5 Reflection Principle for Brownian Motion
89(2)
4.6 Numerical Projects and Exercises
91(2)
Exercises
93(4)
4.7 Historical and Bibliographical Notes
97(2)
Chapter 5 Ito Calculus
99(36)
5.1 Preliminaries
99(1)
5.2 Martingale Transform
100(4)
5.3 The Ito Integral
104(10)
5.4 Ito's Formula
114(9)
5.5 Gambler's Ruin for Brownian Motion with Drift
123(2)
5.6 Tanaka's Formula
125(2)
5.7 Numerical Projects and Exercises
127(1)
Exercises
128(4)
5.8 Historical and Bibliographical Notes
132(3)
Chapter 6 Multivariate Ito Calculus
135(18)
6.1 Multidimensional Brownian Motion
135(2)
6.2 Ito's Formula
137(7)
6.3 Recurrence and Transience of Brownian Motion
144(2)
6.4 Dynkin's Formula and the Dirichlet Problem
146(2)
6.5 Numerical Projects and Exercises
148(1)
Exercises
149(2)
6.6 Historical and Bibliographical Notes
151(2)
Chapter 7 Ito Processes and Stochastic Differential Equations
153(24)
7.1 Definition and Examples
153(3)
7.2 Ito's Formula
156(6)
7.3 Multivariate Extension
162(1)
7.4 Numerical Simulations of SDEs
163(2)
7.5 Existence and Uniqueness of Solutions of SDEs
165(5)
7.6 Martingale Representation and Levy's Characterization
170(1)
7.7 Numerical Projects and Exercises
171(1)
Exercises
172(3)
7.8 Historical and Bibliographical Notes
175(2)
Chapter 8 The Markov Property
177(22)
8.1 The Markov Property for Diffusions
177(3)
8.2 The Strong Markov Property
180(3)
8.3 Kolmogorov's Equations
183(9)
8.4 The Feynman-Kac Formula
192(2)
8.5 Numerical Projects and Exercises
194(1)
Exercises
195(3)
8.6 Historical and Bibliographical Notes
198(1)
Chapter 9 Change of Probability
199(20)
9.1 Change of Probability for a Random Variable
199(3)
9.2 The Cameron-Martin Theorem
202(7)
9.3 Extensions of the Cameron-Martin Theorem
209(4)
9.4 Numerical Projects and Exercises
213(1)
Exercises
214(3)
9.5 Historical and Bibliographical Notes
217(2)
Chapter 10 Applications to Mathematical Finance
219(46)
10.1 Market Models
220(1)
10.2 Derivatives
221(4)
10.3 No Arbitrage and Replication
225(2)
10.4 The Black-Scholes Model
227(5)
10.5 The Greeks
232(4)
10.6 Risk-Neutral Pricing
236(9)
10.7 Exotic Options
245(1)
10.8 Interest Rate Models
246(5)
10.9 Stochastic Volatility Models
251(4)
10.10 Numerical Projects and Exercises
255(2)
Exercises
257(6)
10.11 Historical and Bibliographical Notes
263(2)
Bibliography 265(4)
Index 269
Louis-Pierre Arguin, Baruch College, City University of New York, NY, and Graduate Center, City University of New York, NY.