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The effect of adding a jump at time t=5. The difference between the two processes is the Malliavin derivative.
A realisation of a nonlinear self-inhibiting Hawkes process by thinning from an underlying Poisson measure.
Convergence to the continuous-time Hawkes process with a Gaussian noise, as the time-step goes to zero. Time step h=0.002.
Strong (trajectorial) approximation of the continuous time Hawkes intensity. Time step h=0.2.
Time step h=0.02.
For the periodic Poisson autoregression, if the kernel vanishes exponentially fast then the process 'converges' to the cyclo-stationary regime exponentially fast, both almost surely and in Lp.
The approximation of the kernel
ϕ(t)=(1-t)/(1+t^{2.5}) by a sum of two and three exponentials.
A path of a state dependent Hawkes jump-diffusion with kernel ϕ (blue) as well as the paths resulting from thre exponential approximations.
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