Book contents
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Basic notions
- 2 Brownian motion
- 3 Martingales
- 4 Markov properties of Brownian motion
- 5 The Poisson process
- 6 Construction of Brownian motion
- 7 Path properties of Brownian motion
- 8 The continuity of paths
- 9 Continuous semimartingales
- 10 Stochastic integrals
- 11 Itô's formula
- 12 Some applications of Itô's formula
- 13 The Girsanov theorem
- 14 Local times
- 15 Skorokhod embedding
- 16 The general theory of processes
- 17 Processes with jumps
- 18 Poisson point processes
- 19 Framework for Markov processes
- 20 Markov properties
- 21 Applications of the Markov properties
- 22 Transformations of Markov processes
- 23 Optimal stopping
- 24 Stochastic differential equations
- 25 Weak solutions of SDEs
- 26 The Ray–Knight theorems
- 27 Brownian excursions
- 28 Financial mathematics
- 29 Filtering
- 30 Convergence of probability measures
- 31 Skorokhod representation
- 32 The space C[0, 1]
- 33 Gaussian processes
- 34 The space D[0, 1]
- 35 Applications of weak convergence
- 36 Semigroups
- 37 Infinitesimal generators
- 38 Dirichlet forms
- 39 Markov processes and SDEs
- 40 Solving partial differential equations
- 41 One-dimensional diffusions
- 42 Lévy processes
- Appendices
- A Basic probability
- B Some results from analysis
- C Regular conditional probabilities
- D Kolmogorov extension theorem
- References
- Index
D - Kolmogorov extension theorem
from Appendices
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Frequently used notation
- 1 Basic notions
- 2 Brownian motion
- 3 Martingales
- 4 Markov properties of Brownian motion
- 5 The Poisson process
- 6 Construction of Brownian motion
- 7 Path properties of Brownian motion
- 8 The continuity of paths
- 9 Continuous semimartingales
- 10 Stochastic integrals
- 11 Itô's formula
- 12 Some applications of Itô's formula
- 13 The Girsanov theorem
- 14 Local times
- 15 Skorokhod embedding
- 16 The general theory of processes
- 17 Processes with jumps
- 18 Poisson point processes
- 19 Framework for Markov processes
- 20 Markov properties
- 21 Applications of the Markov properties
- 22 Transformations of Markov processes
- 23 Optimal stopping
- 24 Stochastic differential equations
- 25 Weak solutions of SDEs
- 26 The Ray–Knight theorems
- 27 Brownian excursions
- 28 Financial mathematics
- 29 Filtering
- 30 Convergence of probability measures
- 31 Skorokhod representation
- 32 The space C[0, 1]
- 33 Gaussian processes
- 34 The space D[0, 1]
- 35 Applications of weak convergence
- 36 Semigroups
- 37 Infinitesimal generators
- 38 Dirichlet forms
- 39 Markov processes and SDEs
- 40 Solving partial differential equations
- 41 One-dimensional diffusions
- 42 Lévy processes
- Appendices
- A Basic probability
- B Some results from analysis
- C Regular conditional probabilities
- D Kolmogorov extension theorem
- References
- Index
Summary
Suppose S is a metric space.We use Sℕ for the product space S×S×… furnished with the product topology. We may view Sℕ as the set of sequences (x1, x2, …) of elements of S. We use the σ-field on Sℕ generated by the cylindrical sets. Given an element x = (x1, x2, …) of Sℕ, we define πn(x) = (x1, …, xn) ∈ Sn.
We suppose we have a Radon probability measure μn defined on Sn for each n. (Being a Radon measure means that we can approximate μn(A) from below by compact sets; see
Folland (1999) for details.) The μn are consistent if μn+1(A × S) = μn(A) whenever A is a Borel subset of Sn. The Kolmogorov extension theorem is the following.
Theorem D.1Suppose for each n we have a probability measure μnon Sn. Suppose the μn's are consistent. Then there exists a probability measure μ Sℕ such that μ(A × Sℕ) = μn(A) for all A ⊂ Sn.
Proof Define μ on cylindrical sets by μ(A × Sℕ) = μn(A) if A ⊂ Sn. By the consistency assumption, μ is well defined. By the Carathéodory extension theorem, we can extend μ to the σ-field generated by the cylindrical sets provided we show that whenever An are cylindrical sets decreasing to ∅, then μ(An) → 0.
Suppose An are cylindrical sets decreasing to ∅ but μ(An) does not tend to 0; by taking a subsequence we may assume without loss of generality that there exists ε > 0 such that μ(An) ≥ ε for all n. We will obtain a contradiction.
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- Stochastic Processes , pp. 382 - 384Publisher: Cambridge University PressPrint publication year: 2011