Stochastic control for diffusions with self-exciting jumps: An overview

Alain Bensoussan; Benoît Chevalier-Roignant

doi:10.3934/mcrf.2024038

Article Contents

2024, Volume 14, Issue 4: 1452-1476. Doi: 10.3934/mcrf.2024038

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Stochastic control for diffusions with self-exciting jumps: An overview

Alain Bensoussan^1, and
Benoît Chevalier-Roignant^2, ,

1.
International Center for Decision and Risk Analysis, Naveen Jindal School of Management, University of Texas at Dallas, United States of America
2.
emlyon business school, France

^*Corresponding author: Benoît Chevalier-Roignant
^*Corresponding author: Benoît Chevalier-Roignant

Received: April 2024

Revised: July 2024

Early access: August 13, 2024

Published online: August 13, 2024

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Abstract

We provide an overview of continuous-time processes subject to jumps that do not originate from a compound Poisson process, but from a compound Hawkes process. Such stochastic processes allow for a clustering of self-exciting jumps, a phenomenon for which empirical evidence is strong. Our presentation, which omits certain technical details, focuses on the main ideas to facilitate applications to stochastic control theory. Among other things, we identify the appropriate infinitesimal generators for a set of problems involving various (possibly degenerate) cases of diffusions with self-exciting jumps. Compared to higher-dimensional diffusions, we note a degeneracy of the second-order infinitesimal generator. We derive a Feynman-Kac Theorem for a dynamic system driven by such a jump diffusion and also discuss a problem of continuous control of such a system and provide a verification theorem establishing a link between the value function and a novel type of Hamilton-Jacobi-Bellman (HJB) equation.

Mathematics Subject Classification: Primary: 49K20, 93E20.

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References

[1]	Y. Aït-Sahalia and T. R. Hurd, Portfolio choice in markets with contagion, Journal of Financial Econometrics, 14 (2015). doi: 10.1093/jjfinec/nbv024.
[2]	E. Bacry, I. Mastromatteo and J.-F. Muzy, Hawkes processes in finance, Market Microstructure and Liquidity, 1 (2015). doi: 10.1142/S2382626615500057.
[3]	A. Bensoussan and J.-L. Lions, Impulse Control and Quasi-Variational Inequalities, North Holland, 1984.
[4]	B. Bian, X. Chen and X. Zeng, Optimal portfolio choice in a jump-diffusion model with self-exciting, Journal of Mathematical Finance, 09 (2019). doi: 10.4236/jmf.2019.93020.
[5]	R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004.
[6]	L. Cui, A. Hawkes and H. Yi, An elementary derivation of moments of Hawkes processes, Advances in Applied Probability, 52 (2020), 102-137. doi: 10.1017/apr.2019.53.
[7]	A. Dassios and H. Zhao, A dynamic contagion process, Advanced Applied Probability, 43 (2011), 814-846. doi: 10.1239/aap/1316792671.
[8]	D. Duffie, J. Pan and K. Singleton, Transform analysis and asset pricing for affine jump‐diffusions, Econometrica, 68 (2000), 1343-1376. doi: 10.1111/1468-0262.00164.
[9]	E. B. Dynkin, Markov Processes, Springer Berlin Heidelberg, 1965. doi: 10.1007/978-3-662-00031-1.
[10]	E. Errais, K. Giesecke and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM Journal on Financial Mathematics, 1 (2010), 642-665. doi: 10.1137/090771272.
[11]	D. Hainaut, Impact of volatility clustering on equity indexed annuities, Insurance: Mathematics and Economics, 71 (2016), 367-381. doi: 10.1016/j.insmatheco.2016.10.009.
[12]	A. G. Hawkes, Point spectra of some mutually exciting point processes, Journal of the Royal Statistical Society: Series B, 33 (1971), 438-443. doi: 10.1111/j.2517-6161.1971.tb01530.x.
[13]	A. G. Hawkes, Hawkes processes and their applications to finance: A review, Quantitative Finance, 18 (2018), 193-198. doi: 10.1080/14697688.2017.1403131.
[14]	S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101. doi: 10.1287/mnsc.48.8.1086.166.
[15]	R. C. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.
[16]	P. Protter, Stochastic Integration and Differential Equations, Springer Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-10061-5.