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I would like to be able to model an SDE having the form

$$dX_t = \mu(X_t)dt + \sigma(X_t)dW(t) + \int_{\{|c| <1 \}}g(X_t,c)\tilde{N}(dt,dc) + \int_{\{|c| \ge 1 \}}h(X_t,c)N(dt,dc).$$

where $W$ is a standard $1$-dimensional Brownian motion. $N$ is a Poisson process and $\tilde{N}$ is the compensated Poisson process. $N$ and $W$ are independent.

Previously, without the presence of jumps, I had success using the Python package SDEint.

I am looking for recommendation for Python packages or perhaps suggestions outside of Python for accomplishing this.

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    $\begingroup$ What prevents you from having the same kind of success that you had before? The question is just very broad as posted. $\endgroup$ Apr 12, 2022 at 3:32
  • $\begingroup$ @WolfgangBangerth That particular python package doesn't have any methods for the addition of jumps. $\endgroup$ Apr 12, 2022 at 3:37
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    $\begingroup$ You can try DifferentialEquations.jl for jump diffusions. $\endgroup$ Apr 12, 2022 at 12:30
  • $\begingroup$ @ChrisRackauckas That is what I am leaning towards now. I don't any experience with Julia. It says in the documentation I can use it through Python though. $\endgroup$ Apr 12, 2022 at 17:17
  • $\begingroup$ @ChrisRackauckas feel free to give that comment as an answer. I'll accept it as the answer. It seems to be the best option, thanks for the input. $\endgroup$ Apr 12, 2022 at 21:00

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You can try DifferentialEquations.jl for jump diffusions. It has a tutorial on jump diffusion models:

https://diffeq.sciml.ai/stable/tutorials/jump_diffusion/

and more documentation at:

https://diffeq.sciml.ai/stable/types/jump_types/ https://diffeq.sciml.ai/stable/solvers/jump_solve/

The standard jumps are for irregular jumping, i.e. calculating the location of each jump occurrence and adding that to the SDE evaluation. The regular jump methods are for things like tau-leaping where a Poisson process is evaluated at the dt. DiffEq can be used from Python, though I don't think the jump system is tested from Python.

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In terms of simulating a visual representation of a jump diffusion model,matplotlibprovides excellent diagraming capabilities. Here is an article outlining a use case of this precisely.

Additionally, 'jumpdif' is a suitable option.

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