Let $\varphi$ be the characteristic function of an infinite divisible distribution. It can be expressed in the form $\varphi = e^\psi$ with

$$\psi(\lambda) = i \lambda a - \frac{\sigma^2 \lambda^2}{2} + \int_{\mathbb R} \left(\exp(i \lambda x) - 1 - \frac{i\lambda x}{1+x ^2}\right)\frac{1+x^2}{x^2} \nu(dx)$$

where $(a,\sigma^2,\nu)$ is the Lévy-Khintchine triplet.

What is the intuitive meaning of this triplet?

For example, how can one understand that $\sigma^2=0$ for the compound Poisson distribution?

  • 2
    $\begingroup$ That $\sigma^2=0$ means no Brownian component, as explained in every text on the subject. To begin with, you might want to compute $\psi$ when $X_t=at+\sigma B_t+uN_t$. $\endgroup$ – Did Mar 26 '13 at 6:37
  • $\begingroup$ So? Did you make any progress on this? $\endgroup$ – Did Mar 28 '13 at 10:44

a is drift, sigma σ^2 is variance rate, and ν(x) is "jump measure": the arrival rate of jumps of size x.


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