High dimensional integration of kinks and jumps -- smoothing by\n preintegration

2017/12/04 by Andreas Griewank, Griewank, Andreas, Frances Y. Kuo +5 · 3 citations
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1712.00920

openalex publication_date 2017/12/04 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28

Abstract

We show how simple kinks and jumps of otherwise smooth integrands over\n\ℝ^d can be dealt with by a preliminary integration with respect to a\nsingle well chosen variable. It is assumed that this preintegration, or\nconditional sampling, can be carried out with negligible error, which is the\ncase in particular for option pricing problems. It is proven that under\nappropriate conditions the preintegrated function of d-1 variables belongs to\nappropriate mixed Sobolev spaces, so potentially allowing high efficiency of\nQuasi Monte Carlo and Sparse Grid Methods applied to the preintegrated problem.\nThe efficiency of applying Quasi Monte Carlo to the preintegrated function are\ndemonstrated on a digital Asian option using the Principal Component Analysis\nfactorisation of the covariance matrix.\n

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