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

Convergence of moderately interacting particle systems to a diffusion–convection equation

In this paper, we summarize the results about the strong convergence rate of
the Ninomiya-Victoir scheme and the stable convergence in law of its normalized
error that we obtained in previous papers. We then recall the properties of the
multilevel Monte Carlo estimators involving this scheme that we introduced and
studied before. Last, we are interested in the error introduced by discretizing
the ordinary differential equations involved in the Ninomiya-Victoir scheme. We
prove that this error converges with strong order 2 when an explicit
Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to
the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order
5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the
same order of strong convergence. Moreover, the properties of our multilevel
Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.

/pdf/ninomiya-victoir-scheme-multilevel-monte-carlo-estimators-2vdnhfk2sp.pdf

Ninomiya-Victoir scheme : Multilevel Monte-Carlo estimators and discretization of the involved Ordinary Differential Equations

In the present paper, we first deal with the discretization of stochastic dierential equa- tions. We elaborate on the analysis of the weak error of the Euler scheme by Talay and Tubaro (31) to contruct schemes with quicker weak rate of convergence for SDEs corresponding to an infinitesimal generator with smooth coecients. We also extend this analysis to the case of a discontinuous drift coecient. In a second part, we present two applications of stochastic gradient algorithms in finance.

/pdf/on-two-numerical-problems-in-applied-probability-17pfzn0esg.pdf

On two numerical problems in applied probability : discretization of Stochastic Differential Equations and optimization of an expectation depending on a parameter

We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Rockner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.

Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise

In a previous work, we proved strong convergence with order 1 of the Ninomiya-Victoir scheme $X^{\rm NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N(X−X^{\rm NV})$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually 1 when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.

Benjamin Jourdain

Papers

Convergence of moderately interacting particle systems to a diffusion–convection equation

Ninomiya-Victoir scheme : Multilevel Monte-Carlo estimators and discretization of the involved Ordinary Differential Equations

On two numerical problems in applied probability : discretization of Stochastic Differential Equations and optimization of an expectation depending on a parameter

Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise

Asymptotic error distribution for the Ninomiya-Victoir scheme in the commutative case