Abstract
We have seen in Chap. 4 that self-excited processes offer a natural way to introduce contagion between shocks in financial markets. In this approach, the occurrence of a shock depends on previous ones. In the most common specification, the intensity of jumps, that is akin to the instantaneous probability of a shock, increases as soon as a jump is observed. The influence of this jump on the intensity next decays with time according to a memory function, also called a kernel. When this memory kernel is exponential as in the SEJD, the pair jump intensity is a bivariate Markov process. In this case, the moment generating function (mgf) admits an analytical solution, found by Itô calculus. If the memory kernel is not exponential, we lose most of the analytical tractability offered by stochastic calculus. This chapter fills a gap in the literature by providing a closed form expression of the moment generating function (mgf) of non-Markov self-exciting jump processes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The q l(⋅) are computed with the relation \(\boldsymbol {q}_{l}(t-\Delta ,s)=i\,b_{l}\mathrm {e}^{i\,b_{l}\,\Delta }\boldsymbol {q}_{\lambda }(t,s)\,\Delta +\mathrm {e}^{i\,b_{l}\,\Delta }\boldsymbol {q}_{l}(t,s)\) coming from Eq. (5.20) instead of using the related ODEs.
References
Barsotti, F., Milhaud, X., Salhi, Y.: Lapse risk in life insurance: correlation and contagion effects among policyholders’ behaviors. Insurance Math. Econ. 71, 317–331 (2016)
Bessy-Roland, Y., Boumezoued, A., Hillairet, C.: Multivariate Hawkes process for cyber insurance. Ann. Actuarial Sci. 15(1), 14–39 (2020)
Dassios, A., Jang, J.: Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stochast. 7(1), 73–95 (2003)
Embrechts, P., Liniger, T., Lu, L.: Multivariate Hawkes processes: an application to financial data. J. Appl. Probab. 48(A), 367–378 (2011)
Hainaut, D.: Moment generating function of non-Markov self-excited claims processes. Insurance Math. Econ. 101, 406–424 (2021)
Jang, J., Dassios, A.: A bivariate shot noise self-exciting process for insurance. Insurance Math. Econ. 53(3), 524–532 (2013)
Jang, J., Dassios, A., Hongbiao, Z.: Moments of renewal shot-noise processes and their applications. Scand. Actuarial J. 8, 727–752 (2018)
Muzy, J.-F., Delattre, S., Hoffmann, M., Bacry, E.: Some limit theorems for Hawkes processes and application to financial statistics. Stoch. Process. Appl. 123(7), 2475–2499 (2013)
Stabile, G., Torrisi, G.L.: Risk processes with non-stationary Hawkes claims arrivals. Methodol. Comput. Appl. Probab. 12(3), 415–429 (2010)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hainaut, D. (2022). Non-Markov Models for Contagion and Spillover. In: Continuous Time Processes for Finance. Bocconi & Springer Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-031-06361-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-031-06361-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-06360-2
Online ISBN: 978-3-031-06361-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)