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A Markov-modulated non-homogeneous Poisson process (MMNPP), whose intensity process is designed to capture both the cyclical and nonrecurring trends, is considered for modelling the total count of cyber incidents. Extending the Expectation-Maximisation (EM) algorithm for the current MMPP literature, we derive the filters and smoothers to support the MMNPP online parameter estimation. A scaling transformation is introduced to address the numerical issue for large data sizes whilst maintaining accuracy. The filter- and smoother-based EM algorithms are then benchmarked to the maximum likelihood-based EM algorithm at the theoretical level. The differences emerge in the E-step of the EM procedure. Both the filtering and smoothing schemes, in conjunction with the change-of-measure technique, avoid the computing complication caused by the hidden regimes. In contrast to the usual EM algorithm, the said two algorithms could be implemented given only the incident counts data without the specific times of jumps. Within the data compiled by the U.S. Department of Health and Human Services, the filter-based algorithm performs better than the algorithm involving smoothers. The benchmarked algorithm may do well in calibration under the presence of extreme incident counts with an extremely low frequency; however, overfitting may occur. For most practical applications involving 2 or 3 regimes, both algorithms are superior when it comes to efficiency, real-time update, and low computational cost. The benchmarked algorithm is better when there are more regimes under relatively closer intensities. Overall, the filter-based algorithm gives better estimation, especially if there is a low-frequency regime and the flexible binning of the data set is an important consideration. [ABSTRACT FROM AUTHOR]

Within the general framework of a multifactor term structure model, the fundamental partial differential equation (PDE) satisfied by a default-free zero-coupon bond price is derived via a martingale-oriented approach. Using this PDE, a result characterizing a model belonging to an exponential affine class is established using only a system of partial derivatives. It is also shown that the solution to the bond price PDE has the conditional expectation representation arising in martingale pricing through the application of a multi-dimensional version of Ito's lemma and a property of the stochastic integral.