Stochastic risk estimation due to non-stationary hazards using compound-NHPP

The frequency and the magnitude of the environmental hazards are going to be impacted by the changing climate. The non-stationary hazards can be effectively modelled using a Non-Homogeneous Poisson process (NHPP) with time-dependent rates and damages/losses associated with each hazard event. For risk analysis over a finite time, the compounded version of this process can give the net damages/losses with or without discounting. Panjer’s recursion mainly applicable to the Homogeneous Poisson process (HPP) can be slightly generalized to obtain the distribution of the net losses for an NHPP as well, that can only accommodate the time-dependent rates but there is no way to account for time-dependent losses or discounted losses for that matter. To address this limitation, this paper proposes a method to estimate the mean, variance and other higher-order moments of the compound NHPP under the most generalized condition i.e., both the rate and the losses, being time-dependent. This estimation is achieved by developing simple differential equations from the definition of the process, which can be solved easily using any suitable differential equation solver and the formulations are validated using Monte Carlo Simulations. This analysis can prove to be useful for estimating the risk in non-stationary climatic conditions and help provide a strong economic basis for investment decisions pertaining to climate action.