Fourier Neural Network Approximation of Transition Densities in Finance

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 $L_2$-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 $L_2$ 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.