Advances in Applied Probability

A dynamic contagion process

Angelos Dassios and Hongbiao Zhao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.

Article information

Adv. in Appl. Probab., Volume 43, Number 3 (2011), 814-846.

First available in Project Euclid: 23 September 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 60G55: Point processes 60G44: Martingales with continuous parameter 91G40: Credit risk

Dynamic contagion process Cox process with shot noise intensity piecewise-deterministic Markov process cluster point process self-exciting point process Hawkes process


Dassios, Angelos; Zhao, Hongbiao. A dynamic contagion process. Adv. in Appl. Probab. 43 (2011), no. 3, 814--846. doi:10.1239/aap/1316792671.

Export citation


  • Azizpour, S. and Giesecke, K. (2008). Self-exciting corporate defaults: contagion vs. frailty. Working paper, Stanford University.
  • Bordenave, C. and Torrisi, G. L. (2007). Large deviations of Poisson cluster processes. Stoch. Models 23, 593–625.
  • Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.
  • Brémaud, P. and Massoulié, L. (2002). Power spectra of general shot noises and Hawkes processes with a random excitation. Adv. Appl. Prob. 34, 205–222.
  • Chavez-Demoulin, V., Davison, A. C. and McNeil, A. J. (2005). Estimating value-at-risk: a point process approach. Quant. Finance 5, 227–234.
  • Costa, O. L. V. (1990). Stationary distributions for piecewise-deterministic Markov processes. J. Appl. Prob. 27, 60–73.
  • Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.
  • Dassios, A. and Jang, J.-W. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stoch. 7, 73–95.
  • Dassios, A. and Jang, H.-W. (2005). Kalman-Bucy filtering for linear systems driven by the Cox process with shot noise intensity and its application to the pricing of reinsurance contracts. J. Appl. Prob. 42, 93–107.
  • Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.
  • Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.
  • Duffie, D. and Gârleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analysts J. 57, 41–59.
  • Duffie, D. and Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Rev. Financial Studies 12, 687–720.
  • Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.
  • Errais, E., Giesecke, K. and Goldberg, L. (2009). Affine point processes and portfolio credit risk. Working paper, Stanford University.
  • Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.
  • Hawkes, A. G. and Oakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.
  • Jarrow, R. A. and Yu, F. (2001). Counterparty risk and the pricing of defaultable securities. J. Finance 56, 1765–1799.
  • Jarrow, R. A., Lando, D. and Turnbull, S. M. (1997). A Markov model for the term structure of credit risk spreads. Rev. Financial Stud. 10, 481–523.
  • Lando, D. (1998). On Cox processes and credit risky securities. Derivatives Res. 2, 99–120.
  • Longstaff, F. A. and Rajan, A. (2008). An empirical analysis of the pricing of collateralized debt obligations. J. Finance 63, 529–563.
  • Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Process. Appl. 75, 1–30.
  • Oakes, D. (1975). The Markovian self-exciting process. J. Appl. Prob. 12, 69–77.
  • Stabile, G. and Torrisi, G. L. (2010). Risk processes with non-stationary Hawkes claims arrivals. Methodology Comput. Appl. Prob. 12, 415–429.