We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.
A Hawkes process is a simple point process whose intensity depends on its history; the resulting dynamics are generally non-Markovian. We establish a sample-path moderate deviation principle for a nonlinear Hawkes process in the full moderate regime. Since a Poisson cluster representation is unavailable for nonlinear H...
We develop a spectral theory for dynamical large deviations in non-Markov jump processes and non-Markov chains, whose dynamics depends on the past through state- and jump-dependent empirical observables. We demonstrate that a multiscale Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) Ansatz separates fast configurational rel...
This paper develops a time-inhomogeneous Ito diffusion population model subject to multiplicative random catastrophes. Closed-form transient moments of arbitrary integer order are derived. For a homogeneous benchmark, the expected-population exponent depends on E[Y], whereas the almost-sure sample-path exponent depends...
Yousef Al-Zalzalah, B. Al-Eideh· International Journal of Mat...· 0 citations
We study nonlinear stochastic dynamical systems that evolve in a Markov environment with separated fast and slow transition scales. These systems are also subject to impulsive perturbations under a Poisson approximation scheme. The environment is represented on a product state space, and phase aggregation is used to av...
S. Bekešienė, A. Nikitin, A. Prus· Mathematics· 0 citations
This paper develops a random-effects extension of the ergodic basic affine jump-diffusion (BAJD) model for a population of independent trajectories with individual unobserved jump intensities and common structural parameters. Under continuous-time observation, each intensity is estimated by the empirical jump frequency...
We develop a stochastic diffusion model with two component Allee effects via a continu- um limit of a discrete Markov process. The boundary classification proves that zero is exit and infinity is entrance. Using intrinsic ultracontractivity of the associated semigroup, we establish the existence and uniqueness of a qua...
Gui-Jie Lan· International Journal of Bio...· 0 citations
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