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

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Stochastic equations in infinite dimensions

The flow of homogeneous fluids through porous media: analogies with other physical problems

In this paper, as an application of the theoretical result in [1], we exhibit a family of payoffs $\widehat\varphi _h (x) indexed by a measure h, for the American price of which an almost closed formula holds, which are very close to the Put payoff in the following sense: they are continuous, match with (K-x)+ outside of the range ]K*,K[ (where K* is the perpetual Put strike), are analytic inside with the right derivative (-1) at both ends. Moreover a numerical procedure to select the best h in some sense yields excellent results. The first interest is to yield the corresponding sub- and super- very sharp hedging strategies, since there is a hedge ratio associated to our price. Also since the error is smoothed by the probability of crossing the free boundary, the corresponding errors regarding the price is much smaller, thus yielding good approximations of the Black-Scho- les American Put price.

https://hal.inria.fr/inria-00072805/document

Yet Another Approximation of the American Put

Stochastic volatility models can be seen as a particular family of two-dimensional stochastic differential equations (SDE) in which the volatility process follows an autonomous one-dimensional SDE. We take advantage of this structure to propose an efficient discretization scheme with order two of weak convergence. We prove that the order two holds for the asset price and not only for the log-asset as usually found in the literature. Numerical experiments confirm our theoretical result and we show the superiority of our scheme compared to the Euler scheme, with or without Romberg extrapolation.

Efficient Second-order Weak Scheme for Stochastic Volatility Models

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). 
In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.

Stability of the Weak Martingale Optimal Transport Problem

where U is a general function defined on some Wasserstein space of probability measures on R and (Xi)i≥1 is a given sequence of R -valued random vectors. In the mathematical statistics literature, Von Mises [1] [2] (see also Chapter 6 of [9]) was the first to develop an approach for deriving the asymptotic distribution theory of functionals U( 1 N ∑N i=1 δXi) (called “statistical functions”) under the assumption of independent and identically distributed real-valued random variables (Xi)i≥1. Through the use of the Gâteaux differential, he introduced a Taylor expansion of U( 1 N ∑N i=1 δXi) around U(μ) where μ is the common distribution of the random variables. He proved that if the linear term is the first nonvanishing term in the Taylor expansion of the functional U(.) at μ, the limit distribution is normal (under the usual restrictions corresponding to the central limit theorem). One of the main difficulty was to prove that the remainder in the Taylor expansion goes to zero. In dimension d = 1, Boos and Serfling [10] assumed the existence of a differential for U(.) at μ in a sense stronger than the Gâteaux differential. They assumed the existence of d dǫ |ǫ=0+U (μ+ ǫ (ν − μ)) = dU (μ, ν − μ) at μ that is linear in ν − μ (for ν any probability measure on the real line ) and such that

Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting

We present in this article a stochastic algorithm applied to the integration of the General Dynamics Equation (GDE) for aerosols. This algorithm is validated by comparison with an analytical solution of a coagulation-condensation-evaporation model.

Benjamin Jourdain

Papers

Yet Another Approximation of the American Put

Efficient Second-order Weak Scheme for Stochastic Volatility Models

Stability of the Weak Martingale Optimal Transport Problem

Central limit theorem over non-linear functionals of empirical measures: beyond the iid setting

Modeling aerosol dynamics: A stochastic algorithm