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This process is a simple random walk with constant volatility and Student innovations (see the process for constant volatility and normal innovations).
For the present analysis, the innovations with a normal distribution gives results that are mostly “trivial”, as the Gaussian is the fixed point of the central limit theorem. A Student distribution for the increments produces more interesting results, as it shows the convergence to the Gaussian for the returns at large δtr due to the central limit theorem. Therefore, this model allows to witness the pure aggregation case, without dynamic and/or memory effect on the volatility.
More complex models always reproduce better the empirical data with a Student distribution for ϵ.
The parameters for the simulations are:
The simulation time corresponds to 200 years with a time increment δt = 3 minutes.