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

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

This paper gives an overview of results and developments in the area of pricing and hedging contingent claims in an incomplete market by means of a quadratic criterion. We first present the approach of risk-minimization in the case where the underlying discounted price process X is a local martingale. We then discuss the extension to local risk-minimization when X is a semimartingale and explain the relations to the Follmer-Schweizer decomposition and the minimal martin- gale measure. Finally we study mean-variance hedging, the variance-optimal martingale measure and the connections to closedness properties of spaces of stochastic integrals.

/pdf/a-guided-tour-through-quadratic-hedging-approaches-hb5skmd4af.pdf

A guided tour through quadratic hedging approaches

Using the CRSP (Center for Research in Security Prices) daily stock return data, we revisit the question of whether or not actual stock market prices exhibit long-range dependence. Our study is based on an empirical investigation reported in Teverovsky, Taqqu and Willinger [33] of the modified rescaled adjusted range or R/S statistic that was proposed by Lo [17] as a test for long-range dependence with good robustness properties under “extra” short-range dependence. Our main conclusion is that because the modified R/S statistic shows a strong preference for accepting the null hypothesis of no long-range dependence, irrespective of whether long-range dependence is present in the data or not, Lo's acceptance of the hypothesis for the CRSP data (i.e., no long-range dependence in stock market prices) is less conclusive than is usually regarded in the econometrics literature. In fact, upon further analysis of the data, we find empirical evidence of long-range dependence in stock price returns, but because the corresponding degree of long-range dependence (measured via the Hurst parameter H) is typically very low (i.e., H-values around 0.60), the evidence is not absolutely conclusive.

/pdf/stock-market-prices-and-long-range-dependence-3ei968601d.pdf

Stock market prices and long-range dependence

On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension

Stochastic Analysis and Random Fields.- Detection of Dynamical Systems from Noisy Multivariate Time Series.- A Bakry-Emery Criterion for Self-Interacting Diffusions.- Stationary Solutions for the 2D Stochastic Dissipative Euler Equation.- Volterra Equations Perturbed by a Gaussian Noise.- Dirichlet Forms Methods: An Application to the Propagation of the Error Due to the Euler Scheme.- Individual-Based Probabilistic Models of Adaptive Evolution and Various Scaling Approximations.- A Note on Evolution Systems of Measures for Time-Dependent Stochastic Differential Equations.- Remarks on 3D Stochastic Navier-Stokes Equations.- Slices of a Brownian Sheet: New Results and Open Problems.- An Estimate of the Convergence Rate in Diffusion Approximation of a Particle Motion under Random Forcing.- Long-Time Behaviour for the Brownian Heat Kernel on a Compact Riemannian Manifold and Bismut's Integration-by-Parts Formula.- Probabilistic Deformation of Contact Geometry, Diffusion Processes and Their Quadratures.- Approximation of Stochastic Differential Equations Driven by Fractional Brownian Motion.- Critical Exponents for Semilinear PDEs with Bounded Potentials.- Generalized Ornstein-Uhlenbeck Processes on Separable Banach Spaces.- Approximation of Rough Paths of Fractional Brownian Motion.- A One-Dimensional Analysis of Singularities and Turbulence for the Stochastic Burgers Equation in d Dimensions.- Attractors for Ergodic and Monotone Random Dynamical Systems.- On the Stability of Feynman-Kac Propagators.- Some Applications of the Malliavin Calculus to Sub-Gaussian and Non-Sub-Gaussian Random Fields.- Nonlinear Markovian Problems in Large Dimensions.- Stochastic Methods in Financial Models.- A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints.- Numerical Aspects of Loan Portfolio Optimization.- An Orlicz Spaces Duality for Utility Maximization in Incomplete Markets.- No Free Lunch under Transaction Costs for Continuous Processes.- Robustness of the Hobson-Rogers Model with Respect to the Offset Function.- PDE Approach to Utility Maximization for Market Models with Hidden Markov Factors.- Generalizations of Merton's Mutual Fund Theorem in Infinite-Dimensional Financial Models.

"Seminar on Stochastic Analysis, Random Fields and Applications Iv": "Centro Stefano Franscini, Ascona, May 2002"

Stochastic processes with a multidimensional parameter

Seminar on stochastic analysis, random fields and applications IV

https://hal.archives-ouvertes.fr/hal-00410678/document

Marco Dozzi

Papers

On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension

"Seminar on Stochastic Analysis, Random Fields and Applications Iv": "Centro Stefano Franscini, Ascona, May 2002"

Stochastic processes with a multidimensional parameter

Seminar on stochastic analysis, random fields and applications IV

Finite-Time Blowup and Existence of Global Positive Solutions of a Semi-Linear SPDE