Compensated Jump process

[Max_Beez]

I have been familiarizing myself with DifferentialEquations.jl

I was trying to find an example of an SDE simulation such that the SDE is a jump-diffusion such that the jumps are of the compensated (Poisson) form.

Is there an already built-in method I should be using or perhaps someone has an example I can look at?

Thank you.

[ChrisRackauckas]

@Max_Beez

I was trying to find an example of an SDE simulation such that the SDE is a jump-diffusion such that the jumps are of the compensated (Poisson) form.

Is there an already built-in method I should be using or perhaps someone has an example I can look at?

I don’t think so. It’s worth an issue.

[isaacsas]

Is your jump rate time homogeneous?

[Max_Beez]

Yes they are.

[Albert_Zevelev]

Is a “Compensated Poisson Process” the same thing as a “Compound Poisson Process”?

[Max_Beez]

No, they are not the same thing. A compensated Poisson process is a Poisson process which is made into a Martingale by subtracting the mean from the process.

[isaacsas]

If the jump process has rate \lambda, couldn’t you just add a -\lambda into the drift term for the same jump-diffusion but using an uncompensated Poisson process?

Edit: Assuming unit jumps…

[Max_Beez]

yeah that is what I was thinking is the best solution at present. More variability would be nice but I am sure I can figure out how to get what I want of it.

[rveltz]

@Max_Beez

More variability

what do you mean?

[Max_Beez]

Jumps that aren’t necessarily unit jumps, that is jumps with variable size that can be dependent upon other factors.

[ChrisRackauckas]

@isaacsas

If the jump process has rate \lambda λ\lambda , couldn’t you just add a -\lambda −λ-\lambda into the drift term for the same jump-diffusion but using an uncompensated Poisson process?

yeah, the user would have to do that right now. It’s not hard, but it’s not automated.