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Topics in stochastic processes 2004 ed. [Pehme köide]

  • Formaat: Paperback / softback, 126 pages, kõrgus x laius: 240x170 mm, 126 p., 1 Paperback / softback
  • Sari: Publications of the Scuola Normale Superiore
  • Ilmumisaeg: 01-Oct-2004
  • Kirjastus: Scuola Normale Superiore
  • ISBN-10: 8876421319
  • ISBN-13: 9788876421310
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Topics in stochastic processes 2004 ed.Suurem pilt
  • Formaat: Paperback / softback, 126 pages, kõrgus x laius: 240x170 mm, 126 p., 1 Paperback / softback
  • Sari: Publications of the Scuola Normale Superiore
  • Ilmumisaeg: 01-Oct-2004
  • Kirjastus: Scuola Normale Superiore
  • ISBN-10: 8876421319
  • ISBN-13: 9788876421310
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9788876421310.html
Märksõnad:
The notes are based on lectures on stochastic processes given at Scuola Normale Superiore in 1999 and 2000. Some new material was added and only selected, less standard results were presented. We did not include several applications to statistical mechanics and mathematical finance, covered in the lectures, as we hope to write part two of the notes devoted to applications of stochastic processes in modelling. The main themes of the notes are constructions of stochastic processes. We present different approaches to the existence question proposed by Kolmogorov, Wiener, Ito and Prohorov. Special attention is also paid to Levy processes. The lectures are basically self-contained and rely only on elementary measure theory and functional analysis. They might be used for more advanced courses on stochastic processes.
Chapter 1 Introduction
1(4)
1.1 Existence questions
2(1)
1.2 Some history
3(2)
Chapter 2 Measure theoretic preliminaries
5(10)
2.1 Measurable spaces
5(1)
2.2 Dynkin's π-λ theorem
6(1)
2.3 Independence
7(1)
2.4 Expectations and conditioning
8(1)
2.5 Gaussian measures
9(1)
2.6 Convergence of measures
9(1)
2.7 Characteristic functions
10(5)
Chapter 3 Kolmogorov's existence theorem
15(10)
3.1 Carathdodory's and Ulam's theorems
15(2)
3.2 Kolmogorov's theorem
17(2)
3.3 Some applications
19(6)
3.3.1 Gaussian processes
19(1)
3.3.2 Wiener processes
20(1)
3.3.3 Markov processes
21(1)
3.3.4 L6vy processes
22(2)
3.3.5 Poisson processes
24(1)
Chapter 4 Doob's regularisation theorem
25(12)
4.1 Martingales and supermartingales
25(4)
4.2 Regularisation theorem
29(8)
4.2.1 An application to Levy processes
30(1)
4.2.2 Proof of the regularisation theorem
31(3)
4.2.3 An application to Markov processes
34(3)
Chapter 5 Wiener's approach
37(10)
5.1 Hilbert space expansions
37(2)
5.2 The Levy-Ciesielski construction of Wiener's process
39(5)
5.2.1 Construction of a Poisson process
41(3)
5.3 The Steinhaus construction
44(3)
Chapter 6 Analytic approach to Levy processes
47(14)
6.1 Infinitely divisible families
47(2)
6.2 The Levy-Khinchin formula
49(6)
6.3 Infinitely divisible families and semigroups
55(6)
6.3.1 Infinitely divisible families on [ 0,+∞)
55(1)
6.3.2 Subordination
56(1)
6.3.3 Semigroups determined by infinitely divisible families
57(1)
6.3.4 Generators of (Pt) on L2(Rd)
58(3)
Chapter 7 Representation of Levy processes
61(8)
7.1 Poissonian random measures
61(3)
7.2 Representation theorem
64(5)
Chapter 8 Stochastic integration and Markov processes
69(12)
8.1 Constructing Markov chains
69(2)
8.2 The Courrege theorem
71(2)
8.3 Diffusion with additive noise
73(2)
8.4 Stochastic integrals
75(6)
8.4.1 Integration with respect to a Wiener process W
75(2)
8.4.2 Integration with respect to a Poissonian measure π
77(1)
8.4.3 Determining equations
78(3)
Chapter 9 Stochastic integration in infinite dimensions
81(894)
9.1 Integration with respect to a Wiener process
81(6)
9.1.1 Wiener process on a Hilbert space
81(4)
9.1.2 Stochastic integration with respect to a Wiener process
85(2)
9.2 Stochastic integration with respect to martingales
87(888)
9.2.1 Introduction
87(1)
9.2.2 Doob-Meyer decomposition
88(3)
9.2.3 Operator valued angle bracket process
91(4)
9.2.4 Final comments
95(880)
Chapter 10 Prohorov's theorem
975
10.1 Motivations
97(1)
10.2 Weak topology
98(2)
10.3 Metrics on M1(E)
100(2)
10.4 Prohorov's theorem
102(3)
Chapter 11 Invariance principle and Kolmogorov's test
105(18)
11.1 Weak convergence in C([ 0,T]; E)
105(5)
11.2 Classical proof of the invariance principle
110(4)
11.3 The Ottaviani inequality
114(1)
11.4 Tightness by the factorisation method
115(3)
11.5 Donsker's theorem
118(1)
11.6 Tightness by Kolmogorov's test
118(5)
Bibliography 123