My question is how to I simulate sample paths from a Cauchy process? I know this can be done using two Brownian motions, but I am trying to do it from the basics.

It's known that if we have a Levy process $\{X_t\}$ with Levy triplet $(0,0,\nu)$ then the measure $\nu$ determines the jump intensity such that if $\lambda:=\nu(\mathbb{R})<\infty$ then the probability measure $\nu_1=\nu/\lambda$ determines the distribution of jump sizes; furthermore $$X_t=\sum_{j=1}^{N_t}V_j ,$$ where $\{N_t\}$ is a Poisson process with intensity $\lambda$ and the $V_j$ are iid random variables that are distributed according to $\nu_1$.

However, in the case of a Cauchy process we have the triplet $(0,0,W)$, where $$W(dx)=\frac{dx}{\pi x^2}.$$ And here $W(\mathbb{R})$ is not finite. Is there a way to side-step this?

Thanks in advance.


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