Volume 12, Issue 2

In this paper we introduce reproducing kernel Hilbert spaces based on Taylor series. The unit ball of this space contains functions which are infinite at the boundary. We investigate multivariate integration in such spaces and show how functions in such spaces can be integrated with order O ( N −τ ) for τ > 0 arbitrarily large, in spite of the unboundedness of the functions at the boundary. Further we prove that the Taylor space contains functions with infinite variance and hence the function space contains functions for which a simple Monte Carlo algorithm converges with probability one but convergence could be arbitrarily slow.

Given a system of m equations F(x) = 0 (where m is large and x is an unknown m -vector), we seek to apply sequential Monte Carlo [SMC] methods to find solutions efficiently. This paper follows up on a previous paper by the same author, in which consideration was limited to linear systems of the form Ax = a (where, again, m is large, A is a known ( m × m ) matrix, a is a known m -vector, and x is an unknown m -vector). It was shown there that effective techniques could reduce computation times dramatically (speed-up factors of 550 to 26,000 were obtained in sample calculations). The methods presented here rely on the use of Newtonian linearization, combined with the SMC methods previously described. Incidentally, the optimization of these SMC methods is discussed here and should clarify the parametrization of the SMC techniques so as to yield the highest efficiency.

Convergence, consistency, stability and pathwise positivity of balanced Milstein methods for numerical integration of ordinary stochastic differential equations (SDEs) are discussed. This family of numerical methods represents a class of highly efficient linear-implicit schemes which generate mean square converging numerical approximations with qualitative improvements and global rate 1. 0 of mean square convergence, compared to commonly known numerical methods for SDEs with Lipschitzian coefficients.

In this paper, we develop an importance sampling method with the help of flexible control on the Lévy measure in the density transformation. The method has significant efficacy even on evaluating random variables with complex path-dependent structures. Numerical examples are presented to illustrate convergence acceleration through variance reduction with a view towards financial derivatives pricing.