Moments and cumulants in stochastic approximation

Jürgen Dippon

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Moments and cumulants in stochastic approximation

Jürgen Dippon

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Abstract

Moments of the Robbins-Monro process are computed up to 6th order. As a corollary asymptotic expansions of related cumulants are obtained up to 4th order. These can be used to derive a second order formal Edgeworth expansion for the standardized Robbins-Monro process.

Myriam Maumy-Bertrand

Annales De L Isup, 2004

Consider αn,c(•) the compound empirical process based on two independent sequences of i.i.d.r.v. In this paper, we establish an approximation of order n −1/2 log n of the process αn,c(•) by a linear combination of a sequence of Wiener processes and a sequence of Brownian bridges. We obtain also an approximation of √ nαn,c(•) by a linear combination of a two-parameter Wiener process and a Kiefer process.

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A closed-form formula for the conditional moments of the extended CIR process

PHIRAPHAT SUTTHIMAT

Journal of Computational and Applied Mathematics, 2016

In this paper, we derive a simple closed-form formula for the γ th conditional moment of a variance process, based on the extended Cox-Ingersoll-Ross (ECIR) process, for any real number γ. The closed-form formula presented here is a result of successfully finding an exact solution for the partial differential equation, based on the ECIR process. We conduct Monte Carlo simulations to illustrate the accuracy of the current closed-form formula.

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Stochastic Processes

Rolando Rebolledo

International Encyclopedia of Statistical Science, 2011

Abril JC () Asymptotic expansions for time series problems with applications to moving average models. PhD Thesis, The

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Higher-order moments, cumulants and spectral densities of the NGINAR (1) process

Dr. Hassan Bakouch

Statistical Methodology, 2010

The motivation for time series with geometric marginal distributions arises from noting that the Poisson distribution is not always suitable for the modeling and analysis of integer-valued time series. The NGINAR(1) process that has been introduced by Ristić et al. (2009) represents a class of such time series. Joint higher-order (factorial) moments and cumulants with some other related statistical measures of the NGINAR(1) process are constructed. Also, the spectral and bispectral density functions of this process are investigated, including their nonparametric estimators, using the multitapering method. A real data example of the nonparametric multitaper spectral estimates is investigated, with a discussion of the results obtained.

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Some additional remarks on the cumulant expansion for linear stochastic differential equations

Jos B. T. M. Roerdink

Physica A: Statistical Mechanics and its Applications, 1984

We summarize our previous results on cumulant expansions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail, because our earlier results for this case are only valid in special situations. Moreover, a second partially time ordered cumulant is introduced, which is appropriate for the calculation of multi-time averages of the solution process and which differs slightly from the one to be used for single-time averages. By the two supplements just mentioned we complete our earlier investigations on the subject.

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A recursive method for functionals of Poisson processes

Mahmoud Zarepour

2002

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The generalized Wiener process II: Finite systems

Manuel Caceres

Journal of Physics A: Mathematical and General, 1999

A class of Langevin-like equations (non-Markovian processes) are studied in the presence of non-natural boundary conditions. Exact results for all cumulants and the corresponding Kolmogorov hierarchy of distributions are given in terms of our functional approach we previously reported (1997 J. Phys. A: Math. Gen. 30 8427). The generalized Wiener processes-on finite domains-are completely characterized for reflecting and periodic boundary conditions. Some examples are given to show the behaviour of the moments and the probability distributions for different noises. The interplay between the boundary conditions and the structure of the noises is also pointed out.

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Approximations of Stochastic Integrals of the Poisson Process

Gharib M O U S A Gharib

JOURNAL OF ADVANCES IN MATHEMATICS, 2016

In this article the asymptotic behavior of the finite sums of the elements depending on approximations, being linear combinations of approximations by I to and Stratonovich, is investigated. The outcomes of the present work are generalizations of the corresponding theorems from

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A relation between moments of Liu process and Bernoulli numbers

Guanzhong Ma

Fuzzy Optimization and Decision Making, 2020

This paper finds a relation between moments of Liu process and Bernoulli numbers. Firstly, by an exponential generating function of Bernoulli numbers, a useful integral formula is obtained. Secondly, based on this integral formula, the moments of a normal uncertain variable and Liu process are expressed via Bernoulli numbers.

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Polynomial processes and their applications to mathematical Finance

Martin Keller-Ressel

Arxiv preprint arXiv: …, 2008

We introduce a class of Markov stochastic processes called mpolynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains affine processes, Feller processes with quadratic squared diffusion coefficient, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as the classical Black-Scholes, exponential Lévy or affine models are covered by this setting. The applications range from statistical GMM estimation to option pricing. For instance, the efficient and easy computation of moments can successfully be used for variance reduction techniques in Monte Carlo simulations.

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References (10)

K.L. Chung. On a stochastic approximation method. Ann. Math. Statist., 25:463-483, 1954.
J. Dippon. Higher order representations of the Robbins-Monro process. Submitted for publication.
J. Dippon. Asymptotic expansions of the Robbins-Monro process. Submitted for publication.
V. Fabian. Stochastic approximation of minima with improved asymptotic speed. Ann. Math. Statist., 38:191-200, 1967.
P. Hall. The Bootstrap and Edgeworth Expansion. Springer, New York, 1992.
V.V. Petrov. Limit Theorems of Probability Theory. Clarendon Press, Oxford, 1995.
J. Sacks. Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist., 29:373-405, 1958.
L. Schmetterer. Stochastic approximation. In Proc. Fourth Berkeley Symp. Math. Statist. Prob., pages 587-609, Berkeley, 1961. Univ. of California Press.
R.N. Bhattacharya und R.R. Rao. Normal Approximation and Asymptotic Expan- sions. Wiley, New York, 1986.
H. Robbins und S. Monro. A stochastic approximation method. Ann. Math. Statist., 22:400-407, 1951.

Jürgen Dippon

Mathematical Methods of Statistics, 2008

To estimate the root ϑ of an unknown regression function f : R → R, whose function values f (x) at points x can be observed with some random error V only, Robbins and Monro suggested to run the recursion X n+1 = X n − a n Y n with Y n = f (X n) + V n. Under regularity assumptions, the normalized Robbins-Monro process (Z n), given by Z n := (X n+1 − ϑ)/ Var(X n+1), is asymptotically standard normal. In this paper Edgeworth expansions are presented which provide approximations of the distribution function of Z n up to an error of order o(1/ √ n) or even o(1/n). As corollaries asymptotic confidence intervals for the unknown parameter ϑ are obtained with coverage probability errors of order O(1/n). Further results concern Cornish-Fisher expansions of the quantile function of Z n , an Edgeworth correction of the distribution function and a stochastic expansion in terms of a bivariate polynomial in 1/ √ n and a standard normal random variable.

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Higher order representations of the Robbins–Monro process

Jürgen Dippon

Journal of Multivariate Analysis, 2004

For quasi-linear regression functions, the Robbins-Monro process X n is decomposed in a sum of a linear form and a quadratic form both defined in the observation errors. Under regularity conditions, the remainder term is of order Oðn À3=2 Þ with respect to the L p-norm. If a cubic form is added, the remainder term can be improved up to an order of Oðn À2 Þ: As a corollary the expectation of X n is expanded up to an error of order Oðn À2 Þ: This is used to correct the bias of X n up to an error of order Oðn À3=2 log nÞ:

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Asymptotic expansions for moments of functions of stochastic processes and their applications

Jan Hurt

Statistics Amp Decisions, 1986

Let {X(t)> t eT be a stochastic process and g be a real function. Suppose that E|X(t)-0| q+2-» 0 , q being a natural number and θ being a parameter. Under some smoothing conditions on g asymptotic expansions for Ε g(X(t)) and var g(X(t)) are derived with the remainder terms Ο(E|X(t)-6| g+1) and 0(E|x(t)-9| g+2) , respectively. The proofs use the idea of the method of statistical differentials. Function g may depend on t explicitely. A generalization to the multi-dimensional case is also given. The asymptotic formulas are applied to the problem of compa

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Edgeworth expansions for compound Poisson processes and the bootstrap

KESAR SINGH

Annals of the Institute of Statistical Mathematics, 2003

One-term Edgeworth Expansions for the studentized version of compound Poisson processes are developed. For a suitably defined bootstrap in this context, the so called one-term Edgeworth correction by bootstrap is also established. The results are applicable for constructing second-order correct confidence intervals (which make correction for skewness) for the parameter "mean reward per unit time".

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Edgeworth expansions of stochastic trading time

Ann Schepper

2010

Under most local and stochastic volatility models the underlying forward is assumed to be a positive function of a time-changed Brownian motion. It relates nicely the implied volatility smile to the so-called activity rate in the market. Following Young and DeWitt-Morette (1986), we propose to apply the Duru-Kleinert process-cum-time transformation in path integral to formulate the transition density of the forward. The method leads to asymptotic expansions of the transition density around a Gaussian kernel corresponding to the average activity in the market conditional on the forward value. The approximation is numerically illustrated for pricing vanilla options under the CEV model and the popular normal SABR model. The asymptotics can also be used for Monte Carlo simulations or backward integration schemes.

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An elementary introduction to the Wiener process and stochastic integrals

Tamas Szabados

1996

An elementary construction of the Wiener process is discussed, based on a proper sequence of simple symmetric random walks that uniformly converge on bounded intervals, with probability 1. This method is a simplification of F.B. Knight's and P. Révész's. The same sequence is applied to give elementary (Lebesgue-type) definitions of Itô and Stratonovich sense stochastic integrals and to prove the basic Itô formula. The resulting approximating sums converge with probability 1. As a by-product, new elementary proofs are given for some properties of the Wiener process, like the almost sure non-differentiability of the sample-functions. The purpose of using elementary methods almost exclusively is twofold: first, to provide an introduction to these topics for a wide audience; second, to create an approach well-suited for generalization and for attacking otherwise hard problems.

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On a new stopping rule for stochastic approximation

Henry Braun

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1982

A new stopping rule for the Robbins-Monro process, based on an F-statistic criterion is proposed and its asymptotic behavior established. On the basis of evidence obtained through experimental sampling, the procedure seems to work well over a wide variety of situations. A two-stage procedure, coupling the new rule with an earlier one proposed by Sielken [1973] is recommended for practical use.

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Critique of the generalized cumulant expansion method

Ronald Fox

Journal of Mathematical Physics, 1976

The method of ordered cumulants is presented for the solution of multiplicative stochastic processes. The relationship between this method and the irreducible cluster integral method of Mayer, which is used in the theory of the imperfect gas, is elucidated. The cluster property of ordered cumulants is proved. A critique of the literature in this area is presented which exposes some errors in the formulas. Several examples are indicated for the application of the ordered cumulant method.

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Three-term asymptotic expansions for the moments of the random walk with triangular distributed interference of chance

rovshan aliyev

Applied Mathematical Modelling, 2010

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of ergodic distribution of the process X(t) are obtained when the random variable f 1 , which is describing a discrete interference of chance, has a triangular distribution in the interval [s, S] with center (S + s)/2. Based on these results, the asymptotic expansions with three-term are obtained for the first four moments of the ergodic distribution of X(t), as a (S À s)/2 ? 1. Furthermore, the asymptotic expansions for the variance, skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, by using Monte Carlo experiments it is shown that the given approximating formulas provide high accuracy even for small values of parameter a.

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Saddlepoint Approximations in the Case of Intractable Cumulant Generating Functions

John Kolassa

Institute of Mathematical Statistics eBooks, 1991

Saddlepoint Approximations have long been used to approximate densities and distribution functions of random variables with known cumulant generating function defined on an open interval about the origin. This approximation has very desirable asymptotic properties when approximating densities and tail probabilities for sums of random variables, and also often performs remarkably well for small sample sizes, including samples of one. Calculating the saddlepoint approximation requires calculating the Legendre transform of the log of the cumulant generating function. In some cases this cumulant generating function may be unavailable; in other cases the Legendre transform is difficult to calculate analytically. This paper discusses modifications to the saddlepoint approximation necessary when the cumulant generating function is replaced by a similar but more tractable function whose Legendre transform can be given explicitly. Calculations for the logistic distribution are presented to illustrate the case of a known but intractable cumulant generating function, and an example involving an overdispersed binomial model is presented to illustrate the case of an unavailable cumulant generating function. An application to a random effects logistic linear model is discussed. 1. The Problem. Consider the following problem: X lt X^ X3,..., X n ,... are independent and identically distributed random variables. Assume that their common distribution is continuous, with probability density function/^*) and cumulant generating function K(t) = log E[e tX ], defined on some convex set icR. Without loss of generality assume further that E[XH = 0. I wish to approximate the density function/ rt Cs) and distribution function F n (s) of 5= ι analytically, and am faced with three options: a. Convolute the density n times. b. Use the classical Edgeworth series approximation to/ rt (x).

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Moments and cumulants in stochastic approximation

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