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Stochastic Integration with Jumps [Hardback]

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(University of Texas, Austin)
  • Formāts: Hardback, 516 pages, height x width x depth: 242x164x34 mm, weight: 897 g
  • Sērija : Encyclopedia of Mathematics and its Applications
  • Izdošanas datums: 13-May-2002
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521811295
  • ISBN-13: 9780521811293
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  • Cena: 205,57 €
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Stochastic Integration with JumpsPalielināt
  • Formāts: Hardback, 516 pages, height x width x depth: 242x164x34 mm, weight: 897 g
  • Sērija : Encyclopedia of Mathematics and its Applications
  • Izdošanas datums: 13-May-2002
  • Izdevniecība: Cambridge University Press
  • ISBN-10: 0521811295
  • ISBN-13: 9780521811293
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9780521811293.html
Keywords:
Bichteler (mathematics, U. of Texas at Austin) aims to present the mathematical underpinning of stochastic analysis. Wiener process is treated for economics students and driving terms with jumps are covered to give mathematics students the background to connect with the literature and discrete time martingales. This leads to the most general Lebesgue-Stieltjes integral. Bichteler identifies the useful Lebesgue-Stieltjes distribution functions among all functions on the line and looks at criteria for process to be useful as "random distribution functions." Integration theory is demonstrated to be useful for finding these criteria. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Stochastic processes with jumps and random measures are gaining importance as drivers in applications like financial mathematics and signal processing. This book develops stochastic integration theory for both integrators (semimartingales) and random measures from a common point of view. Using some novel predictable controlling devices, the author furnishes the theory of stochastic differential equations driven by them, as well as their stability and numerical approximation theories. Highlights feature DCT and Egoroff's Theorem, as well as comprehensive analogs to results from ordinary integration theory, for instance, previsible envelopes and an algorithm computing stochastic integrals of c`agl`ad integrands pathwise.

The complete theory of stochastic differential equations driven by jumps, their stability, and numerical approximation theories.

Recenzijas

Review of the hardback: 'The material in the book is presented well: it is detailed, motivation is stressed throughout and the text is written with an enjoyable pinch of dry humour.' Evelyn Buckwar, Zentralblatt MATH Review of the hardback: 'The highlights of the monograph are: Girsanov-Meyer theory on shifted martingales, which covers both the Wiener and Poisson setting; a Doob-Meyer decomposition statement providing really deep information that the objects that can go through the Daniell-like construction of the stochastic. This is an excellent and informative monograph for a general mathematical audience.' EMS

Papildus informācija

The complete theory of stochastic differential equations driven by jumps, their stability, and numerical approximation theories.
Preface xi
Introduction
1(42)
Motivation: Stochastic Differential Equations
1(8)
The Obstacle
4(1)
Ito's Way Out of the Quandary
5(1)
Summary: The Task Ahead
6(3)
Wiener Process
9(11)
Existence of Wiener Process
11(3)
Uniqueness of Wiener Measure
14(3)
Non-Differentiability of the Wiener Path
17(1)
Supplements and Additional Exercises
18(2)
The General Model
20(23)
Filtrations on Measurable Spaces
21(1)
The Base Space
22(1)
Processes
23(4)
Stopping Times and Stochastic Intervals
27(2)
Some Examples of Stopping Times
29(3)
Probabilities
32(1)
The Sizes of Random Variables
33(1)
Two Notions of Equality for Processes
34(2)
The Natural Conditions
36(7)
Integrators and Martingales
43(44)
Step Functions and Lebesgue-Stieltjes Integrators on the Line
43(3)
The Elementary Stochastic Integral
46(7)
Elementary Stochastic Integrands
46(1)
The Elementary Stochastic Integral
47(1)
The Elementary Integral and Stopping Times
47(2)
Lp-Integrators
49(2)
Local Properties
51(2)
The Semivariations
53(5)
The Size of an Integrator
54(2)
Vectors of Integrators
56(1)
The Natural Conditions
56(2)
Path Regularity of Integrators
58(9)
Right-Continuity and Left Limits
58(3)
Boundedness of the Paths
61(1)
Redefinition of Integrators
62(1)
The Maximal Inequality
63(1)
Law and Canonical Representation
64(3)
Processes of Finite Variation
67(4)
Decomposition into Continuous and Jump Parts
69(1)
The Change-of-Variable Formula
70(1)
Martingales
71(16)
Submartingales and Supermartingales
73(1)
Regularity of the Paths: Right-Continuity and Left Limits
74(2)
Boundedness of the Paths
76(1)
Doob's Optional Stopping Theorem
77(1)
Martingales Are Integrators
78(2)
Martingales in Lp 80
80(7)
Extension of the Integral
87(100)
Daniell's Extension Procedure on the Line
87(1)
The Daniell Mean
88(6)
A Temporary Assumption
89(1)
Properties of the Daniell Mean
90(4)
The Integration Theory of a Mean
94(12)
Negligible Functions and Sets
95(2)
Processes Finite for the Mean and Defined Almost Everywhere
97(2)
Integrable Processes and the Stochastic Integral
99(2)
Permanence Properties of Integrable Functions
101(1)
Permanence Under Algebraic and Order Operations
101(1)
Permanence Under Pointwise Limits of Sequences
102(2)
Integrable Sets
104(2)
Countable Additivity in p-Mean
106(4)
The Integration Theory of Vectors of Integrators
109(1)
Measurability
110(5)
Permanence Under Limits of Sequences
111(1)
Permanence Under Algebraic and Order Operations
112(1)
The Integrability Criterion
113(1)
Measurable Sets
114(1)
Predictable and Previsible Processes
115(8)
Predictable Processes
115(3)
Previsible Processes
118(1)
Predictable Stopping Times
118(4)
Accessible Stopping Times
122(1)
Special Properties of Daniell's Mean
123(7)
Maximality
123(1)
Continuity Along Increasing Sequences
124(1)
Predictable Envelopes
125(3)
Regularity
128(1)
Stability Under Change of Measure
129(1)
The Indefinite Integral
130(15)
The Indefinite Integral
132(3)
Integration Theory of the Indefinite Integral
135(2)
A General Integrability Criterion
137(1)
Approximation of the Integral via Partitions
138(2)
Pathwise Computation of the Indefinite Integral
140(4)
Integrators of Finite Variation
144(1)
Functions of Integrators
145(12)
Square Bracket and Square Function of an Integrator
148(2)
The Square Bracket of Two Integrators
150(3)
The Square Bracket of an Indefinite Integral
153(2)
Application: The Jump of an Indefinite Integral
155(2)
Ito's Formula
157(14)
The Doleans-Dade Exponential
159(2)
Additional Exercises
161(1)
Girsanov Theorems
162(6)
The Stratonovich Integral
168(3)
Random Measures
171(16)
σ-Additivity
174(1)
Law and Canonical Representation
175(2)
Example: Wiener Random Measure
177(3)
Example: The Jump Measure of an Integrator
180(3)
Strict Random Measures and Point Processes
183(1)
Example: Poisson Point Processes
184(1)
The Girsanov Theorem for Poisson Point Processes
185(2)
Control of Integral and Integrator
187(84)
Change of Measure --- Factorization
187(22)
A Simple Case
187(4)
The Main Factorization Theorem
191(4)
Proof for p > 0
195(10)
Proof for p = 0
205(4)
Martingale Inequalities
209(12)
Fefferman's Inequality
209(4)
The Burkholder-Davis-Gundy Inequalities
213(3)
The Hardy Mean
216(2)
Martingale Reprsentation on Wiener Space
218(1)
Additional Exercises
219(2)
The Doob-Meyer Decomposition
221(11)
Doleans-Dade Measures and Processes
222(3)
Necessity, Uniqueness, and Existence
225(2)
The Inequalities
227(1)
The Previsible Square Function
228(3)
The Doob-Meyer Decomposition of a Random Measure
231(1)
Semimartingales
232(6)
Integrators Are Semimartingales
233(1)
Various Decompositions of an Integrator
234(4)
Previsible Control of Integrators
238(15)
Controlling a Single Integrator
239(7)
Previsible Control of Vectors of Integrators
246(5)
Previsible Control of Random Measures
251(2)
Levy Processes
253(18)
The Levy-Khintchine Formula
257(4)
The Martingale Representation Theorem
261(4)
Canonical Components of a Levy Process
265(2)
Construction of Levy Processes
267(1)
Feller Semigroup and Generator
268(3)
Stochastic Differential Equations
271(92)
Introduction
271(11)
First Assumptions on the Data and Definition of Solution
272(1)
Example: The Ordinary Differential Equation (ODE)
273(5)
ODE: Flows and Actions
278(2)
ODE: Approximation
280(2)
Existence and Uniqueness of the Solution
282(16)
The Picard Norms
283(2)
Lipschitz Conditions
285(4)
Existence and Uniqueness of the Solution
289(4)
Stability
293(3)
Differential Equations Driven by Random Measures
296(1)
The Classical SDE
297(1)
Stability: Differentiability in Parameters
298(12)
The Derivative of the Solution
301(2)
Pathwise Differentiability
303(2)
Higher Order Derivatives
305(5)
Pathwise Computation of the Solution
310(20)
The Case of Markovian Coupling Coefficients
311(3)
The Case of Endogenous Coupling Coefficients
314(2)
The Universal Solution
316(1)
A Non-Adaptive Scheme
317(3)
The Stratonovich Equation
320(1)
Higher Order Approximation: Obstructions
321(5)
Higher Order Approximation: Results
326(4)
Weak Solutions
330(13)
The Size of the Solution
332(1)
Existence of Weak Solutions
333(4)
Uniqueness
337(6)
Stochastic Flows
343(8)
Stochastic Flows with a Continuous Driver
343(3)
Drivers with Small Jumps
346(1)
Markovian Stochastic Flows
347(2)
Markovian Stochastic Flows Driven by a Levy Process
349(2)
Semigroups, Markov Processes, and PDE
351(12)
Stochastic Representation of Feller Semigroups
351(12)
Appendix A Complements to Topology and Measure Theory 363(107)
A.1 Notations and Conventions
363(3)
A.2 Topological Miscellanea
366(25)
The Theorem of Stone-Weierstraß
366(7)
Topologies, Filters, Uniformities
373(3)
Semi-continuity
376(1)
Separable Metric Spaces
377(2)
Topological Vector Spaces
379(3)
The Minimax Theorem, Lemmas of Gronwall and Kolmogoroff
382(6)
Differentiation
388(3)
A.3 Measure and Integration
391(30)
σ-Algebras
391(1)
Sequential Closure
391(3)
Measures and Integrals
394(4)
Order-Continuous and Tight Elementary Integrals
398(3)
Projective Systems of Measures
401(1)
Products of Elementary Integrals
402(2)
Infinite Products of Elementary Integrals
404(1)
Images, Law, and Distribution
405(1)
The Vector Lattice of All Measures
406(1)
Conditional Expectation
407(1)
Numerical and σ-Finite Measures
408(1)
Characteristic Functions
409(4)
Convolution
413(1)
Liftings, Disintegration of Measures
414(5)
Gaussian and Poisson Random Variables
419(2)
A.4 Weak Convergence of Measures
421(11)
Uniform Tightness
425(1)
Application: Donsker's Theorem
426(6)
A.5 Analytic Sets and Capacity
432(8)
Applications to Stochastic Analysis
436(4)
Supplements and Additional Exercises
440(1)
A.6 Suslin Spaces and Tightness of Measures
440(3)
Polish and Suslin Spaces
440(3)
A.7 The Skorohod Topology
443(5)
A.8 The Lp-Spaces
448(15)
Marcinkiewicz Interpolation
453(2)
Khintchine's Inequalities
455(3)
Stable Type
458(5)
A.9 Semigroups of Operators
463(7)
Resolvent and Generator
463(2)
Feller Semigroups
465(2)
The Natural Extension of a Feller Semigroup
467(3)
Appendix B Answers to Selected Problems 470(7)
References 477(6)
Index of Notations 483(6)
Index 489