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On the cumulant transforms for Hawkes processes

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  23 February 2023

Young Lee Open the ORCID record for Young Lee [Opens in a new window]  and
Thorsten Rheinländer
Young Lee*
Affiliation:
Harvard University
Thorsten Rheinländer*
Affiliation:
Technische Universität Wien
*
*Postal address: Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA.
**Postal address: Wiedner Hauptstr. 8/E105-1 1040 Vienna, Austria.
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Abstract

We consider the asset price as the weak solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale whose predictable compensator follows shot-noise and Hawkes processes. In this framework, we discuss the Esscher martingale measure where the conditions for its existence are detailed. This generalizes certain relationships not yet encountered in the literature.

Keywords

Hawkes processstochastic logarithmexponentially special semimartingale

MSC classification

Primary: 60G20: Generalized stochastic processes
Secondary: 60G55: Point processes 60E10: Characteristic functions; other transforms
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 528 - 541
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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