Improved Euler-Maruyama Scheme for the Calibration of Deterioration Models

Engineering systems are subject to deterioration processes that reduce their ability to sustain the demands for which they were originally designed. These processes can be classified as gradual processes if they continuously affect the system over time (e.g., corrosion) and shock processes if they affect the system at specific, instantaneous moments (e.g., the occurrence of disastrous events like earthquakes and floods). Recent literature integrates the effect of gradual and shock deterioration into a unified formulation that uses Stochastic Differential Equations (SDEs) to model the evolution of the system’s state variables (i.e., the physical properties that affect the capacity and demand of the system). A semimartingale driving noise is used in the SDE to account for the presence of shocks. These models need to be calibrated based on available readings of the value of the state variables, which can be collected with Non-Destructive Testing (NDT) and Structural Health Monitoring (SHM). Most methods for SDE calibration, such as the Euler-Maruyama scheme, allow to calibrate SDE models based on sparse realizations of the resulting process. However, these procedures only work with continuous processes and cannot be applied to processes driven by semimartingale driving noise (i.e., with jumps). In this paper, we propose an improvement over the classical Euler-Maruyama scheme for SDE calibration that allows to incorporate the presence of discontinuities due to shocks. We detail how to account for the possible presence of a shock between two readings of the state variables and formulate the likelihood of the unknown parameters conditioned on such readings. We also obtain closed-form solutions for the likelihood in a few selected cases. Numerical examples are provided to showcase the performance of the proposed methodology.