# Jump Process with Drift
**URL:** https://discourse.julialang.org/t/jump-process-with-drift/115166
**Category:** Modelling & Simulations
**Tags:** differentialequation
**Created:** [June 4, 2024, 12:33pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166 "2024-06-04T12:33:55Z")
**Posts on this page:** 5
**Page:** 1
### Author:  [@Mieszko](https://discourse.julialang.org/u/Mieszko)
#### Post date: [June 4, 2024, 12:33pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166/1 "2024-06-04T12:33:55Z")
Hey, I am trying to simulate with the use of JumpProcesses.jl the following stochastic differential equation:
dX\_t = -X\_tdt + dN\_t.
Do you know, if there is any straightforward way of defining the f (drift) function in this case? If you go for the following:
> function f(du, u, p, t)
> du[1] = -u[1]
> end
then as far as I understand your process has the following form:
dX\_t = \exp(-X\_t)dt + dN\_t.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [June 4, 2024, 12:38pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166/2 "2024-06-04T12:38:10Z")
> [@Mieszko](#):
>
> then as far as I understand your process has the following form:
No, that would put your equation in the right form.
---
### Author:  [@Mieszko](https://discourse.julialang.org/u/Mieszko)
#### Post date: [June 5, 2024, 8:59pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166/3 "2024-06-05T20:59:39Z")
Thank you for pointing this.
I am a bit confused since all the paths I get when simulating this process are positive:
For the comparison, using the jumpdiff library from python and simulating the same process, the paths look like this:
Is there any reason, why Julia’s path gives only positive value (some shifting etc.), or is there something else wrong?
I am using the following code for the Julia path:
> > using DifferentialEquations, Plots, JumpProcesses
>
> > timestep = Float64(0.05)
> > t = range(0, timestep, length=100)
>
> function f(du, u, p, t)
> du[1] = -u[1]
> end
>
> rate(u, p, t) = 1.0
>
> function affect!(integrator)
> integrator.u[1] += 1
> nothing
> end
>
> jump = ConstantRateJump(rate, affect!)
>
> prob = ODEProblem(f, [0.0], (10.0, 100.0))
> jump\_prob = JumpProblem(prob, Direct(), jump)
> sol = solve(jump\_prob, Tsit5(), saveat=timestep)
>
> plot(sol)
---
### Author:  [@Dan](https://discourse.julialang.org/u/Dan)
#### Post date: [June 5, 2024, 9:09pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166/4 "2024-06-05T21:09:43Z")
The jump amplitude parameter in Python jumpdiff library is the standard deviation of a Gaussian distributed jump `N(0,sigma^2)` and as such can take both negative and positive values.
In the Julia code:
```julia
function affect!(integrator)
integrator.u[1] += 1
nothing
end
```
the jump is a +1 jump only (at a specified rate). To get the same as the Python effect use:
```julia
function affect!(integrator)
integrator.u[1] += randn()
nothing
end
```
---
### Author:  [@Mieszko](https://discourse.julialang.org/u/Mieszko)
#### Post date: [June 5, 2024, 9:47pm UTC](https://discourse.julialang.org/t/jump-process-with-drift/115166/5 "2024-06-05T21:47:37Z")
Thanks, that makes much more sense now!