Calibrating Multivariate Lévy Processes with Neural Networks

Calibrating a L\'evy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate L\'evy processes and less smooth L\'evy densities, the problem becomes challenging as ECFs decay slowly and have large uncertainty because of limited observations. We solve this problem by approximating the L\'evy density with a parametrized functional form; the characteristic function is then estimated using numerical integration. In our benchmarks, we used deep neural networks and found that they are robust and can capture sharp transitions in the L\'evy density. They perform favorably compared to piecewise linear functions and radial basis functions. The methods and techniques developed here apply to many other problems that involve nonparametric estimation of functions embedded in a system model.