Multivariate rough claim processes: properties and estimation

This article studies a multivariate claim process with stochastic intensities driven by rough mean reverting diffusions. By construction, the dynamic of claim arrivals is induced by a fractional Brownian motion with a Hurst index, H E(0,1/2). Therefore intensities have an infinite quadratic variation and are not semi-martingales. Nevertheless, we show that the moment generating function of the claim process admits a representation in terms of solutions of fractional differential equations. We next propose a procedure to filter the most likely sample path of rough intensities from time-series of claims. To illustrate this work, we estimate one and two dimensional rough models to time-series of cyber-attacks targeting medical and other services in the US from 2014 to 2018.