Deterministic multi-level algorithms for infinite-dimensional integration on RNRN

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E(g(B))E(g(B)), the expectation of a payoff function gg, that depends on a Brownian motion BB. Employing a standard series expansion of BB 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 RNRN. 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.