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Spline approximation of a random process with singularity
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2011 (English)In: Journal of Statistical Planning and Inference, ISSN 0378-3758, Vol. 141, no 3, 1333-1342 p.Article in journal (Refereed) Published
Abstract [en]

Let a continuous random process X defined on [0,1] be (m+β)-smooth, 0m, 0<β$\leq$1, in quadratic mean for all t>0 and have an isolated singularity point at t=0. In addition, let X be locally like a m-fold integrated β-fractional Brownian motion for all nonsingular points. We consider approximation of X by piecewise Hermite interpolation splines with n free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic approximation rate n^(m+β) for the whole interval.

Place, publisher, year, edition, pages
Elsevier , 2011. Vol. 141, no 3, 1333-1342 p.
Keyword [en]
Approximation, Random process, Sampling design, Hermite splines
Identifiers
OAI: oai:DiVA.org:umu-41544DiVA: diva2:406794
Available from: 2011-03-28 Created: 2011-03-28 Last updated: 2011-09-02Bibliographically approved
In thesis
1. Numerical analysis for random processes and fields and related design problems
Open this publication in new window or tab >>Numerical analysis for random processes and fields and related design problems
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D).

In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed.

In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided.

In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity.

In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

Place, publisher, year, edition, pages
Umeå: Institutionen för matematik och matematisk statistik, Umeå universitet, 2011. 30 p.
Keyword
stochastic processes, random fields, approximation, numerical integration, Hermite splines, piecewise linear interpolator, local stationarity, point singularity, stratified Monte Carlo quadrature, Asian option, Monte Carlo pricing method, Lévy market models
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-46156 (URN)978-91-7459-249-8 (ISBN)
Public defence
2011-09-29, Samhällsvetarhuset, S213, Umeå Universitet, Umeå, 08:15 (English)
Supervisors
Available from: 2011-09-08 Created: 2011-08-26 Last updated: 2012-11-12Bibliographically approved

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