Fourier-cosine Tensor Trains for Density Recovery and Expectation Calculation

Abstract

The Fourier-cosine (COS) method of Fang and Oosterlee (2008) recovers densities and computes expectations semi-analytically from characteristic functions (ch.f.s). In higher dimensions, both expansion synthesis and COS coefficient-tensor construction scale exponentially with dimension. We address this curse of dimensionality with two tensor-train (TT) methods. A straightforward TT decomposition of the COS coefficient tensor gives COS-TT, which removes the exponential cost of online synthesis but not of offline decomposition. Our main contribution, COS-TT-CHF, instead compresses a ch.f. sample tensor and maps it to the original Fourier--cosine coefficient tensor through an alternative COS representation. Apart from the model-dependent cost of one ch.f. evaluation, both offline and online costs scale linearly with dimension, while error grows only algebraically. COS-TT-CHF introduces two additional errors: frequency-domain truncation and discretization errors. Both are controlled by parameter-selection rules from theoretical error analysis. We further propose an approach for expectations of nonseparable functions of additive scalar aggregates by computing the ch.f.s of the aggregates using tensorized COS methods, reducing the original multidimensional problem to a one-dimensional COS calculation. Closed-form formulas are derived for the TT-contraction integrals arising when the aggregate is a weighted sum, as in European basket option pricing. Experiments under geometric Brownian motion (GBM) and variance gamma (VG) show that, with the parameter-selection rules, COS-TT-CHF remains accurate through 150 dimensions for GBM and 100 for VG within a 30-minute offline computation budget on a laptop. As a by-product, our Fourier-cosine TT constructions yield cosine-basis functional TTs (FTTs) under weaker assumptions than in spectral FTT literature.