Estimating a(x, t) and b(x, t) in the Equation dX(t) = a(x, t) dt + b(x, t) dW(t) + ξdJ(t)

[jaeminSon]

Thanks for sharing nice codes. By the way, I failed to find any method that finds a(x,t) and b(x,t) right away even in your paper (https://www.jstatsoft.org/article/view/v105i04/4421).

Can you provide any hints to do that?

[LRydin]

Hey @jaeminSon, nice to know you are interested in the jumpdiff package. Have you looked at the readme of the github page? Here you find a way to estimate a(x,t) and b(x,t)

import jumpdiff as jd
import matplotlib.plotly as plt

# consider a one-dimensional time series X

edges, moments = jd.moments(timeseries = X)

plt.plot(edges, moments[1,...])  # this is your a(x) function (no time dependence)
plt.plot(edges, moments[2,...])  # this is your b(x) function (no time dependence)

Is this what you are looking for?

[LRydin]

Correction: If you indeed have a jump-diffusion process, to estimate the diffusion coefficient b(x), you need to also estimate the jump parameters. Note that the conditional moments moments, more precisely, moments[2,...] is given by

moments[2,...] = b^2 + λξ

thus you will need to subtract the jump parameter, which you obtain with

# first estimate the jump amplitude
xi_est = jd.jump_amplitude(moments = moments)

# and now estimated the jump rate
lamb_est = jd.jump_rate(moments = moments)