Second derivative of measured function is PDF of hidden RV

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second derivative of H_{x,t} is exactly the probability density function of S_t

I’m not sure if I understood this part correctly, but I’d suggest to think of your problem from a generative perspective:

  1. You assume that there’s some timeseries S_t - how would you model just this part? Gaussian Process, Random Walk…
  2. from this timeseries, you can calculate another (plus unknown constant C_1)
  3. from this timeseries, you can then calculate H_{x,t} (again plus unknown constant C_2)
  4. finally create a likelihood function that compares H_{x,t} with data
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