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

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option. This article presents a new approach to modeling term structures of bonds and other contingent claims that are subject to default risk. As in previous “reduced-form” models, we treat default as an unpredictable event governed by a hazard-rate process. 1 Our approach is distinguished by the parameterization of losses at default in terms of the fractional reduction in market value that occurs at default. Specifically, we fix some contingent claim that, in the event of no default, pays X at time T . We take as given an arbitrage-free setting in which all securities are priced in terms of some short-rate process r and equivalent martingale measure Q [see Harrison and Kreps (1979) and Harrison and Pliska (1981)]. Under this “risk-neutral” probability measure, we letht denote the hazard rate for default at time t and let Lt denote the expected fractional loss in market value if default were to occur at time t , conditional

Modeling Term Structures of Defaultable Bonds

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

The idea of coherent risk measures has been introduced by Artzner, Delbaen, Eber and Heath [1] We think of a special class of coherent risk measures and give a characterization of it Let (Ω, ℱ, P) be a probability space We denote L ∞(Ω, ℱ, P) by L ∞ Following [1], we give the following definition

On law invariant coherent risk measures

We study some mathematical models on default risk. First, we study a “standard model” which is an abstract setting widely used in parctice. Then we study how the hazard rates changes, if we change a basic probability measure. We show that the usual assumptions on hazard rates hold in a standard model, but do not hold in general if we change a basic measure. Finally we study a filtering model.

https://link.springer.com/content/pdf/10.1007%2F978-4-431-65895-5_5.pdf

A Remark on default risk models

Publisher Summary This chapter discusses the application of Malliavin's calculus to various problems in stochastic analysis and the theory of partial differential equations. The chapter examines the regularity estimates on the distribution of functionals to which Malliavin's procedure is applicable. It is shown that solutions of Ito stochastic integral equations are smooth functions in the sense of Malliavin's calculus. The distribution of the solution to a general Ito equation has the same regularity properties as that of a classical diffusion just so long as the coefficients of the white noise are non-degenerate. Any other method, of deducing this result and believing that it is a good example to illustrate the power of Malliavin's calculus, is not known.

Applications of the Malliavin calculus, Part III

Asymptotics of the spectral gap with applications to the theory of simulated annealing

: Second order subelliptic operators have been the subject of a considerable amount of research in recent years. Starting with the paper Rothschild and Stein, in which the sharp form of Hormander's famous subellipticity theorem is proved, and continuing through the work of Fefferman and Phong and Sanchez-Calle, it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration. The main purpose of this article is to obtain bounds, from above and below, on p(t,x,y), t epsilon (1, infinity), in terms of standard heat kernels.

/pdf/long-time-estimates-for-the-heat-kernel-associated-with-a-15ez5cgodn.pdf

Shigeo Kusuoka

Papers

On law invariant coherent risk measures

A Remark on default risk models

Applications of the Malliavin calculus, Part III

Asymptotics of the spectral gap with applications to the theory of simulated annealing

Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator