# Compensated Jump process

**URL:** <https://discourse.julialang.org/t/compensated-jump-process/81038>\
**Category:** Modelling & Simulations\
**Tags:** sde, modelling\
**Created:** [May 13, 2022, 6:19pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038 "2022-05-13T18:19:10Z")\
**Posts on this page:** 11\
**Page:** 1

<div class="post-metadata">

**Author:** ![Max\_Beez](https://avatars.discourse-cdn.com/v4/letter/m/7bcc69/32.png) [@Max\_Beez](https://discourse.julialang.org/u/Max_Beez)\
**Post date:** [May 13, 2022, 6:19pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/1 "2022-05-13T18:19:10Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 13, 2022, 6:20pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/2 "2022-05-13T18:20:31Z")

</div>

> [@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.

---

<div class="post-metadata">

**Author:** ![isaacsas](https://avatars.discourse-cdn.com/v4/letter/i/f6c823/32.png) [@isaacsas](https://discourse.julialang.org/u/isaacsas)\
**Post date:** [May 13, 2022, 8:42pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/3 "2022-05-13T20:42:41Z")

</div>

Is your jump rate time homogeneous?

---

<div class="post-metadata">

**Author:** ![Max\_Beez](https://avatars.discourse-cdn.com/v4/letter/m/7bcc69/32.png) [@Max\_Beez](https://discourse.julialang.org/u/Max_Beez)\
**Post date:** [May 13, 2022, 9:13pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/4 "2022-05-13T21:13:21Z")

</div>

Yes they are.

---

<div class="post-metadata">

**Author:** ![Albert\_Zevelev](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/albert_zevelev/32/11844_2.png) [@Albert\_Zevelev](https://discourse.julialang.org/u/Albert_Zevelev)\
**Post date:** [May 13, 2022, 10:37pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/5 "2022-05-13T22:37:01Z")

</div>

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

---

<div class="post-metadata">

**Author:** ![Max\_Beez](https://avatars.discourse-cdn.com/v4/letter/m/7bcc69/32.png) [@Max\_Beez](https://discourse.julialang.org/u/Max_Beez)\
**Post date:** [May 13, 2022, 10:57pm UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/6 "2022-05-13T22:57:52Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![isaacsas](https://avatars.discourse-cdn.com/v4/letter/i/f6c823/32.png) [@isaacsas](https://discourse.julialang.org/u/isaacsas)\
**Post date:** [May 14, 2022, 2:42am UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/7 "2022-05-14T02:42:21Z")

</div>

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…

---

<div class="post-metadata">

**Author:** ![Max\_Beez](https://avatars.discourse-cdn.com/v4/letter/m/7bcc69/32.png) [@Max\_Beez](https://discourse.julialang.org/u/Max_Beez)\
**Post date:** [May 14, 2022, 3:19am UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/8 "2022-05-14T03:19:16Z")

</div>

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.

---

<div class="post-metadata">

**Author:** ![rveltz](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rveltz/32/2707_2.png) [@rveltz](https://discourse.julialang.org/u/rveltz)\
**Post date:** [May 14, 2022, 7:00am UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/9 "2022-05-14T07:00:39Z")

</div>

> [@Max\_Beez](#):
>
> More variability

what do you mean?

---

<div class="post-metadata">

**Author:** ![Max\_Beez](https://avatars.discourse-cdn.com/v4/letter/m/7bcc69/32.png) [@Max\_Beez](https://discourse.julialang.org/u/Max_Beez)\
**Post date:** [May 14, 2022, 7:40am UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/10 "2022-05-14T07:40:25Z")

</div>

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

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 14, 2022, 8:43am UTC](https://discourse.julialang.org/t/compensated-jump-process/81038/11 "2022-05-14T08:43:40Z")

</div>

> [@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.