This paper introduces FourNet, a novel single-layer feed-forward neural
network (FFNN) method designed to approximate transition densities for which
closed-form expressions of their Fourier transforms, i.e. characteristic
functions, are available. A unique feature of FourNet lies in its use of a
Gaussian activation function, enabling exact Fourier and inverse Fourier
transformations and drawing analogies with the Gaussian mixture model. We
mathematically establish FourNet's capacity to approximate transition densities
in the L2β-sense arbitrarily well with finite number of neurons. The
parameters of FourNet are learned by minimizing a loss function derived from
the known characteristic function and the Fourier transform of the FFNN,
complemented by a strategic sampling approach to enhance training. Through a
rigorous and comprehensive error analysis, we derive informative bounds for the
L2β estimation error and the potential (pointwise) loss of nonnegativity in
the estimated densities. FourNet's accuracy and versatility are demonstrated
through a wide range of dynamics common in quantitative finance, including
L\'{e}vy processes and the Heston stochastic volatility models-including those
augmented with the self-exciting Queue-Hawkes jump process.Comment: 27 pages, 5 figure