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Numerical analysis for random processes and fields and related design problems
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
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 [en]
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: urn:nbn:se:umu:diva-46156ISBN: 978-91-7459-249-8 (print)OAI: oai:DiVA.org:umu-46156DiVA: diva2:437283
Public defence
2011-09-29, Samhällsvetarhuset, S213, Umeå Universitet, Umeå, 08:15 (English)
Opponent
Supervisors
Available from: 2011-09-08 Created: 2011-08-26 Last updated: 2012-11-12Bibliographically approved
List of papers
1. Spline approximation of a random process with singularity
Open this publication in new window or tab >>Spline approximation of a random process with singularity
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<β1, 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
Keyword
Approximation, Random process, Sampling design, Hermite splines
Identifiers
urn:nbn:se:umu:diva-41544 (URN)10.1016/j.jspi.2010.10.006 (DOI)
Available from: 2011-03-28 Created: 2011-03-28 Last updated: 2011-09-02Bibliographically approved
2. Multivariate piecewise linear interpolation of a random field
Open this publication in new window or tab >>Multivariate piecewise linear interpolation of a random field
2011 (English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider a multivariate piecewise linear interpolation of a continuous random field on a-dimensional cube. The approximation performance is measured by the integrated mean square error. Multivariate piecewise linear interpolator is defined by N field observations on a locations grid (or design). We investigate the class of locally stationary random fields whose local behavior is like a fractional Brownian field in mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large N. Moreover, for certain classes of continuous and continuously differentiable fields we provide the upper bound for the approximation accuracy in the uniform mean square norm.

Keyword
approximation, random field, sampling design, multivariate piecewise linear interpolator
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-60995 (URN)
Available from: 2012-11-06 Created: 2012-11-06 Last updated: 2012-11-12Bibliographically approved
3. Stratified Monte Carlo quadrature for continuous random fields
Open this publication in new window or tab >>Stratified Monte Carlo quadrature for continuous random fields
2015 (English)In: Methodology and Computing in Applied Probability, ISSN 1387-5841, E-ISSN 1573-7713, Vol. 17, no 1, 59-72 p.Article in journal (Refereed) Published
Abstract [en]

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H¨older class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

Place, publisher, year, edition, pages
New York: Springer Science+Business Media B.V., 2015
Keyword
numerical integration, random field, sampling design, stratified sampling, Monte Carlo methods
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-60994 (URN)10.1007/s11009-013-9347-6 (DOI)000349406400005 ()
Available from: 2012-11-06 Created: 2012-11-06 Last updated: 2017-12-07Bibliographically approved
4. On the error of the Monte Carlo pricing method for Asian option
Open this publication in new window or tab >>On the error of the Monte Carlo pricing method for Asian option
2008 (English)In: Journal of Numerical and Applied Mathematics, ISSN 0868-6912, Vol. 96, no 1, 1-10 p.Article in journal (Refereed) Published
Abstract [en]

We consider a Monte Carlo method to price a continuous arithmetic Asian option with a given precision. Piecewise constant approximation and plain simulation are used for a wide class of models based on L\'{e}vy processes. We give bounds of the possible discretization and simulation errors. The sufficient numbers of discretization points and simulations to obtain requested accuracy are derived. To demonstrate the general approach, the Black-Scholes model is studied in more detail. We undertake the case of continuous averaging and starting time zero,  but the obtained results can be applied to the discrete case  and generalized for any time before an execution date. Some numerical experiments and comparison to the PDE based method are also presented.

Keyword
Asian option, Levy processes, Monte Carlo
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-30785 (URN)
Available from: 2010-01-15 Created: 2010-01-15 Last updated: 2011-09-02Bibliographically approved

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Abramowicz, Konrad

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