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Parameter estimation for stochastic differential equation from discrete observations

Suppose we have a time-series $x(t_i)$ at discrete times $t_i$ and we want to estimate the parameters of an underlying SDE corresponding to this time-series:

$$dx_t = f(x_t,\theta)dt + \sigma(x_t,\theta)dB_t$$

I have read that there exists a lot of different numerical methods to approach this problem, but I have not found a comprehensive and comparative survey about this problem.

  1. What is the best method (according to the type of data/model) ?

  2. Is there a Python or Matlab toolbox doing the job ? I have had a look at SDE Toolbox for Matlab but it is not clear to me how to use it, so it would be appreciated to provide an example.

I think it can be useful to have an overview of this topic since it may be interesting for many people. Thank you.

user16215
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