Year: 2025
Type: preprint
Abstract: This paper discusses a special class of nonlinear Hawkes processes, where the rate function is the exponential function. We call these processes loglinear Hawkes processes. Such processes are used with success in computational neuroscience and we provide a mathematical background for them. After obtaining existence from a standard Poisson embedding, we give sufficient conditions for explosion and nonexplosion that cover a large class of memory functions. In order to give sufficient stability conditions, we construct an autoregressive integer-valued time series that gives pathwise upper bounds for the bin counts of a loglinear Hawkes process. From here, a Markov chain argument shows that the regeneration time for a loglinear Hawkes process has finite expectation if the memory function is negative enough in a neighborhood of zero. Stability in total variation follows. Finally, we show that both the loglinear Hawkes process under a large class of initial conditions and its stationary version have finite exponential moments.
Source: arXiv (Cornell University)
Authors: Bielecki, Tomasz R., Jakubowski, Jacek, Kirchner, Matthias, Niewęgłowski, Mariusz
Institutions: University of Warsaw, Pädagogische Hochschule Bern
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W4414945604
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