Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

Backward Stochastic Differential Equations in Finance

Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index

/pdf/the-concentration-of-measure-phenomenon-27qyoor2ef.pdf

The concentration of measure phenomenon

0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov's Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall's Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.

Continuous martingales and Brownian motion

Mesure et intégration

In this chapter, we take up the study of Brownian motion and, more generally, of continuous martingales. We will use the stochastic integration of Chap. IV together with the technique of time changes to be introduced presently.

Representation of Martingales

In previous chapters stochastic differential equations have been mentioned several times in an informal manner. For instance, if M is a continuous local martingale, its exponential e(M) satisfies the equality 

$$\mathcal{E}{(M)_t} = 1 + \int_0^t {\mathcal{E}{{(M)}_s}} d{M_s};$$

this can be stated: e(M) is a solution to the stochastic differential equation 

$${X_t} = 1 + \int_0^t {{X_s}d{M_s},} $$

which may be written in differential form 

$$d{X_t} = {X_t}d{M_t},{X_0} = 1.$$

We have even seen (Exercise (3.10) Chap. IV) that e(M) is the only solution to this equation. Likewise we saw in Sect. 2 Chap. VII, that some Markov processes are solutions of what may be termed stochastic differential equations.

Stochastic Differential Equations

In this section, we will specialize the notions of Sect. 5 Chap. 0 to the Wiener space W d . This space is a Polish space when endowed with the topology of uniform convergence on compact subsets of ℝ+. This topology is associated with the metric 

$$d\left( {\omega ,\omega '} \right) = \sum\limits_1^\infty {{2^{ - n}}} \frac{{{{\sup }_{t \leqslant n}}\left| {\omega \left( t \right) - \omega '\left( t \right)} \right|}}{{1 + {{\sup }_{t \leqslant n}}\left| {\omega \left( t \right) - \omega '\left( t \right)} \right|}}.$$

Daniel Revuz

Papers

Continuous martingales and Brownian motion

Mesure et intégration

Representation of Martingales

Stochastic Differential Equations

Limit Theorems in Distribution