Can one define a pair of stochastic processes with independent alpha-stable increments and different alpha parameter (index of stability) values (e.g. a Brownian motion and a Cauchy or Levy process) which are correlated?

  • 2
    $\begingroup$ A possible solution consists of constructing the increments using a bivariate copula (e.g. Gaussian) with $\alpha-$stable marginals with the desired parameters. Marginally, they will satisfy the assumptions you mention but the processes will be correlated if $\rho\neq 0$ in the case of a Gaussian copula. $\endgroup$
    – user10525
    Commented Oct 12, 2012 at 13:56
  • $\begingroup$ You should write that up as an answer @Procrastinator. That's what I would do. $\endgroup$
    – John
    Commented Oct 12, 2012 at 14:43
  • $\begingroup$ @Procrastinator Would this work with infinitesimal increments as well? $\endgroup$
    – quant_dev
    Commented Oct 12, 2012 at 14:54
  • $\begingroup$ @quant_dev I am not sure if I follow your question. My point is to use a copula for inducing correlation between the processes. This is, assuming that the increments $(X_t,Y_t)-(X_s,Y_s)$ follow the aforementioned bivariate distribution. Marginally the original assumptions are satisfied. $\endgroup$
    – user10525
    Commented Oct 12, 2012 at 15:11
  • $\begingroup$ @Procrastinator I understand that. My question is, can I specify a copula linking infinitesimal increments $(X_{t+dt},T_{t+dt}) - (X_t,Y_y)$ and obtain a well-defined Levy process? $\endgroup$
    – quant_dev
    Commented Oct 12, 2012 at 15:46


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