# Product of two stochastic processes in DifferentialEquations.jl
**URL:** https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391
**Category:** Numerics
**Tags:** diffeq, rode
**Created:** [August 14, 2021, 12:47pm UTC](https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391 "2021-08-14T12:47:41Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@mastrof](https://discourse.julialang.org/u/mastrof)
#### Post date: [August 14, 2021, 12:47pm UTC](https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391/1 "2021-08-14T12:47:42Z")
Hello, I have recently started to play out with DifferentialEquations.jl; I’m gradually reading through the docs but there’s _a lot_ of stuff in there which will keep me busy for a while, so I’m still at a beginner level.
I would like to understand what would be the preferred way of modeling the following problem:
I have a Langevin equation
\begin{cases} \dot{x} = v \\ \dot{v} = -Av + B\xi\_1(t)\xi\_2(t) \end{cases}
where \xi\_1 is Gaussian white noise and \xi\_2 is another stochastic process, switching between values 0 and 1 according to a Poisson process with distinct (but constant) forward and backward rates (i.e., \lambda\_0 is the rate for the 0\rightarrow1 transition, while \lambda\_1 is the rate for the 1\rightarrow0 transition).
Fundamentally, the system can switch randomly from a “diffusive” mode (when \xi\_2=1) to a “non-diffusive” mode (when \xi\_2=0).
If \xi\_2 is not involved the representation is straightforward, I would use the following code (assume appropriate initial conditions `u0`, integration interval `tspan` and parameters `p` are given)
```julia
function f!(du, u, p, t)
A, B = p
du[1] = u[2]
du[2] = -A*u[2]
end # function
function g!(du, u, p, t)
A, B = p
du[1] = 0.0
du[2] = B
end # function
prob = SDEProblem(f!, g!, u0, tspan, p)
sol = solve(prob)
```
At this point, I had some trouble understanding how to “correctly” couple \xi\_2 _only_ with the stochastic term of this SDEProblem.
The easiest solution I conceived is to introduce an additional control parameter `s` (and two further parameters included in `p` that represent the forward and backward rates for \xi\_2)which can be switched between values 0 and 1 via a `ConstantRateJump` (constant rate should be fine since the time dependence is embedded in `s` and there is no explicit dependence on `u`) then I would have
```julia
function f!(du, u, p, t)
A, B, s, λ0, λ1 = p
du[1] = u[2]
du[2] = -A*u[2]
end # function
function g!(du, u, p, t)
A, B, s, λ0, λ1 = p
du[1] = 0.0
du[2] = B*s
end # function
u0 = [0.0, 1.0]
tspan = (0.0, 10.0)
p = [1.0, 1.0, 1.0, 0.02, 0.1]
prob = SDEProblem(f!, g!, u0, tspan, p)
function rate(u,p,t)
A, B, s, λ0, λ1 = p
(1-s)*λ0 + s*λ1
end # function
affect!(integrator) = (integrator.p[3] = 1.0 - integrator.p[3])
jump = ConstantRateJump(rate, affect!)
jump_prob = JumpProblem(prob, Direct(), jump)
sol = solve(jump_prob)
```
This approach seems to work properly (both using ConstantRateJump and VariableRateJump, so I assume ConstantRate is good enough; does this mean the `rate` function is re-evaluated at every step?).
The question however is, is this the “best” approach for the problem?
Is there another way in which I could explicitly include the \xi\_2(t) term (which in general could also be different from a Poisson process) in the equation and solve it like a `SDEProblem` rather than a `JumpProblem`?
Also, if this approach is adopted, then is it possible to obtain the timeseries for the parameter `s` throughout the integration?
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [August 14, 2021, 8:56pm UTC](https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391/2 "2021-08-14T20:56:48Z")
The problem is that this is not a stochastic differential equation. It’s a random ordinary differential equation, so it’ll need to use the tools for that.
---
### Author:  [@mastrof](https://discourse.julialang.org/u/mastrof)
#### Post date: [August 17, 2021, 12:31pm UTC](https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391/3 "2021-08-17T12:31:26Z")
Ok thanks, I initially excluded RODEs because I would get strange results…
It took me quite a while to realize that while SDEProblem takes the Wiener process as “noise”, to obtain the same result on a RODEProblem I would have to provide the “differentiated” process (i.e., a properly normalized Gaussian white noise).
I have been seriously thinking this was some kind of bug.
I’m not really sure of the details behind this, and whether it’s a choice or necessity; however, if what I understood is correct, maybe this should be properly pointed out at least in the RODE tutorial?
I could propose a little PR myself for this, but I want to be sure I’m not misunderstanding.
– I also _just_ noticed this related issue [https://github.com/SciML/DiffEqDocs.jl/issues/211](https://github.com/SciML/DiffEqDocs.jl/issues/211)
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [August 26, 2021, 4:27am UTC](https://discourse.julialang.org/t/product-of-two-stochastic-processes-in-differentialequations-jl/66391/4 "2021-08-26T04:27:32Z")
> [@mastrof](#):
>
> I’m not really sure of the details behind this, and whether it’s a choice or necessity; however, if what I understood is correct, maybe this should be properly pointed out at least in the RODE tutorial?
> I could propose a little PR myself for this, but I want to be sure I’m not misunderstanding.
Yeah, it probably should be. But the real issue is that the RODE tools are still underdeveloped.