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E-grāmata: Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and Medicine

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Introduction to Continuous-Time Stochastic Processes: Theory, Models, and Applications to Finance, Biology, and MedicinePalielināt
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Expanding on the first edition of An Introduction to Continuous-Time Stochastic Processes, this concisely written book is a rigorous and self-contained introduction to the theory of continuous-time stochastic processes. A balance of theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, industrial applications, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required.

Here is a self-contained introduction to the theory of continuous-time stochastic processes. Balancing theory and application, the book offers examples of real-world modeling from biology, medicine, industry, finance and insurance using stochastic methods.

Recenzijas

From the reviews of the second edition: "The book is useful both for undergraduate and graduate students in stochastics, finances and biology and for all persons who is interested in stochastic calculus and its applications." (Yuliya S. Mishura, Zentralblatt MATH, Vol. 1261, 2013)

Preface to the Second Edition v
Preface to the First Edition vii
Part I Theory of Stochastic Processes
1 Fundamentals of Probability
3(74)
1.1 Probability and Conditional Probability
3(6)
1.2 Random Variables and Distributions
9(8)
1.2.1 Random Vectors
14(3)
1.3 Independence
17(3)
1.4 Expectations
20(10)
1.5 Gaussian Random Vectors
30(3)
1.6 Conditional Expectations
33(8)
1.7 Conditional and Joint Distributions
41(7)
1.8 Convergence of Random Variables
48(9)
1.9 Infinitely Divisible Distributions
57(8)
1.9.1 Examples
64(1)
1.10 Stable Laws
65(6)
1.10.1 Martingales
68(3)
1.11 Exercises and Additions
71(6)
2 Stochastic Processes
77(96)
2.1 Definition
77(8)
2.2 Stopping Times
85(2)
2.3 Canonical Form of a Process
87(1)
2.4 Gaussian Processes
87(1)
2.5 Processes with Independent Increments
88(3)
2.6 Martingales
91(10)
2.7 Markov Processes
101(24)
2.8 Brownian Motion and the Wiener Process
125(15)
2.9 Counting, and Poisson Processes
140(5)
2.10 Marked Point Processes
145(9)
2.10.1 Random Measures
145(1)
2.10.2 Stochastic Intensities
146(8)
2.11 Levy Processes
154(9)
2.12 Exercises and Additions
163(10)
3 The Ito Integral
173(40)
3.1 Definition and Properties
173(12)
3.2 Stochastic Integrals as Martingales
185(5)
3.3 Ito Integrals of Multidimensional Wiener Processes
190(3)
3.4 The Stochastic Differential
193(4)
3.5 Ito's Formula
197(1)
3.6 Martingale Representation Theorem
198(3)
3.7 Multidimensional Stochastic Differentials
201(3)
3.8 The Ito Integral with Respect to Levy Processes
204(3)
3.9 The Ito-Levy Stochastic Differential and the Generalized Ito Formula
207(1)
3.10 Exercises and Additions
208(5)
4 Stochastic Differential Equations
213(64)
4.1 Existence and Uniqueness of Solutions
213(18)
4.2 Markov Property of Solutions
231(6)
4.3 Girsanov Theorem
237(4)
4.4 Kolmogorov Equations
241(10)
4.5 Multidimensional Stochastic Differential Equations
251(3)
4.6 Stability of Stochastic Differential Equations
254(11)
4.7 Ito-Levy Stochastic Differential Equations
265(2)
4.7.1 Markov Property of Solutions of Ito-Levy Stochastic Differential Equations
267(1)
4.8 Exercises and Additions
267(10)
Part II Applications of Stochastic Processes
5 Applications to Finance and Insurance
277(34)
5.1 Arbitrage-Free Markets
278(5)
5.2 The Standard Black-Scholes Model
283(7)
5.3 Models of Interest Rates
290(6)
5.4 Extensions and Alternatives to Black-Scholes
296(5)
5.5 Insurance Risk
301(6)
5.6 Exercises and Additions
307(4)
6 Applications to Biology and Medicine
311(48)
6.1 Population Dynamics: Discrete-in-Space--Continuous-in-Time Models
311(11)
6.2 Population Dynamics: Continuous Approximation of Jump Models
322(4)
6.3 Population Dynamics: Individual-Based Models
326(23)
6.3.1 A Mathematical Detour
329(3)
6.3.2 A "Moderate" Repulsion Model
332(6)
6.3.3 Ant Colonies
338(10)
6.3.4 Price Herding
348(1)
6.4 Neurosciences
349(6)
6.5 Exercises and Additions
355(4)
Measure and Integration
359(18)
A.1 Rings and σ-Algebras
359(2)
A.2 Measurable Functions and Measure
361(3)
A.3 Lebesgue Integration
364(5)
A.4 Lebesgue--Stieltjes Measure and Distributions
369(4)
A.5 Radon Measures
373(1)
A.6 Stochastic Stieltjes Integration
374(3)
Convergence of Probability Measures on Metric Spaces
377(20)
B.1 Metric Spaces
377(11)
B.2 Prohorov's Theorem
388(1)
B.3 Donsker's Theorem
388(9)
Elliptic and Parabolic Equations
397(6)
C.1 Elliptic Equations
397(1)
C.2 The Cauchy Problem and Fundamental Solutions for Parabolic Equations
398(5)
Semigroups of Linear Operators 403(4)
Stability of Ordinary Differential Equations 407(4)
References 411(10)
Nomenclature 421(4)
Index 425
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