Deterministic multi-level algorithms for infinite-dimensional integration on R N

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E ( g ( B ) ) , the expectation of a payoff function g , that depends on a Brownian motion B . Employing a standard series expansion of B the latter problem is equivalent to the computation of the exp...

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Bibliographic Details
Published in:Journal of Complexity Vol. 27; no. 3; pp. 331 - 351
Main Authors: Niu, Ben, Hickernell, Fred J., Müller-Gronbach, Thomas, Ritter, Klaus
Format: Journal Article
Language:English
Published: Elsevier Inc 01.06.2011
Subjects:
Fixed subspace sampling
Infinite-dimensional quadrature
Low discrepancy points
Multi-level method
Tractability
Variable subspace sampling
Tractability
Fixed subspace sampling
Multi-level method
Infinite-dimensional quadrature
Low discrepancy points
Variable subspace sampling
ISSN:0885-064X, 1090-2708
Online Access:Get full text
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Summary:Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E ( g ( B ) ) , the expectation of a payoff function g , that depends on a Brownian motion B . Employing a standard series expansion of B the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the sequence space R N . The class of integrands studied in this paper is the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2010.08.001