Self-exciting process in Finance and Insurance for credit risk and longevity risk modelling in heterogenous portfolios.
Nicole El Karoui, LPMA-UPMC.
Recent regulatory evolution in credit risk management suggests to consider the credit risk of an aggregated portfolio as generated by a family of intercorrelated firms
with defaults propagation. Redemption risk in Life Insurance is very sensitive to contagion effect driven by the level of external variables as inflation and interest rates, but also the behavior of the other insured. rmiii For longevity purposes in an actuarial and demographical context, the individual point of view (Individual based models) allows to take into account specific individual characteristics (socio-economic status, education level, marital status…), and also the age of individuals (or events).
The aim is to take into account population heterogeneity in characteristics or age, impacting the rate of evolution in a way easier to model than the global point of view. Contagion effect, which is well known in seismology or in neuroscience with the spike-and-wave patterns, but also in High Frequency Trading, must be also included.
The simple model of contagion process is the Hakwes process, whose we give a new interpretation in terms of IBM, allowing to develop more complex Markovian model best suited to modeling redemption risk. For age pyramid of human heterogeneous population, we propose an extension of traditional birth and death processes. Using as source of randomness a $sigma$- finite Poisson measure with characteristics augmented by a thinning parameter, the population process is described as the strong solution of a stochastic differential equation, based on complex (birth, move and death) rates processes, depending of age, characteristics and past population, and from environmental factors. This strong representation permits easy comparisons.
As an example, we reproduce by simulation the cohort effect, well-known in UK, where a cohort of people born between $(1927,1040) $ showed an improvement in life-expectation from neighboring cohorts.