# Solving Stochastic Differential Equations with Same Random Numbers

**URL:** <https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189>\
**Category:** Modelling & Simulations\
**Tags:** diffeq, sde\
**Created:** [January 5, 2018, 9:35pm UTC](https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189 "2018-01-05T21:35:58Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![Torkel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/torkel/32/5030_2.png) [@Torkel](https://discourse.julialang.org/u/Torkel)\
**Post date:** [January 5, 2018, 9:35pm UTC](https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189/1 "2018-01-05T21:35:58Z")

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One Line Question: How do I use Julia to solve a system of Stochastic Differential Equations were there are correlated terms of noise (or “the same” noise appears in several places)

Biological Background: Lets say I have two components, X and Y. Now we have a reaction X \<-\> Y. I want to model this using a standard approach. I also want to include the stochastics of the system using the standard Langevin approach. I get (Stochastic) Differential Equations:  
d/dt = r[Y] - q + sqrt(r[Y])\*N1(0,1) - sqrt(q)\*N2(0,1)  
d[Y]/dt = q[Y] - r + sqrt(q)\*N2(0,1) - sqrt(r[Y])\*N1(0,1)  
Here N1 and N2 are normally distributed Noise with mean 0 and std 1. What is important is that the same N1 and N2 appears in both equations (which in this case has the important implication that the sum + [Y] is preserved.

I (think) I understand how to use non-diagonal noise and the stochastic differential equation solver to solve this problem, but then all the 4 instances of noise would all be uncorrelated  
( [http://docs.juliadiffeq.org/latest/tutorials/sde\_example.html](http://docs.juliadiffeq.org/latest/tutorials/sde_example.html) ). However, I would I do to solve the problem were the same noise appears in several places?

(The example above is not the one I am exactly interested in, but a simple example of the Chemical Langevin Equations were I do not know how to solve it in Julia. However, if I understand this example I should be able to do int in a more complicated case as well).

Thanks in advance

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<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:** [January 5, 2018, 11:56pm UTC](https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189/2 "2018-01-05T23:56:37Z")

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Note that I took the liberty of changing the title because correlated noise is different so hopefully it helps people find this in searches better in the future

> [@Torkel](#):
>
> What is important is that the same N1 and N2 appears in both equations

This is what non-diagonal noise is for. Let’s look at the example from the documentation:

[http://docs.juliadiffeq.org/latest/tutorials/sde\_example.html#Example-4:-Systems-of-SDEs-with-Non-Diagonal-Noise-1](http://docs.juliadiffeq.org/latest/tutorials/sde_example.html#Example-4:-Systems-of-SDEs-with-Non-Diagonal-Noise-1)

```julia
f(t,u,du) = du .= 1.01u
function g(t,u,du)
  du[1,1] = 0.3u[1]
  du[1,2] = 0.6u[1]
  du[1,3] = 0.9u[1]
  du[1,4] = 0.12u[2]
  du[2,1] = 1.2u[1]
  du[2,2] = 0.2u[2]
  du[2,3] = 0.3u[2]
  du[2,4] = 1.8u[2]
end
prob = SDEProblem(f,g,ones(2),(0.0,1.0),noise_rate_prototype=zeros(2,4))

```

This is the equation

```julia
du = f(t,u)dt + g(t,u)*dW

```

where W is a vector of 4 noise terms. This means if you look at the matrix multiplication, `g(t,u)*dW`, then that means that the SDEs are:

```julia
du1 = f1(t,u)dt + g11(t,u)*dW1 + g12(t,u)*dW2 + g13(t,u)*dW3 + g14(t,u)*dW4
du2 = f1(t,u)dt + g21(t,u)*dW1 + g22(t,u)*dW2 + g23(t,u)*dW3 + g24(t,u)*dW4

```

As you can see by writing it out, `g[1,1]` and `g[2,1]` are two terms which have the same random number applied to them. Another way to explain this is if I simplify this example a bit:

```julia
f(t,u,du) = du .= 1.01u
function g(t,u,du)
  du[1,1] = 0.3u[1]
  du[2,1] = 1.2u[1]
end
prob = SDEProblem(f,g,ones(2),(0.0,1.0),noise_rate_prototype=zeros(2,1))

```

then by the same logic this is an SDE with a single Brownian motion and both SDEs get the same random number.

> [@Torkel](#):
>
> which in this case has the important implication that the sum + [Y] is preserved.

This blog post shows exactly your same problem (chemical Langivan equations and non-diagonal noise to preserve sums). Hopefully this helps as well.

[http://mbesancon.github.io/post/2017-12-14-diffeq-julia/](http://mbesancon.github.io/post/2017-12-14-diffeq-julia/)

Hopefully together these explain it. But since it the docs didn’t work the first time, please let us know how to improve the documentation so that this is more clear!

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<div class="post-metadata">

**Author:** ![Torkel](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/torkel/32/5030_2.png) [@Torkel](https://discourse.julialang.org/u/Torkel)\
**Post date:** [January 7, 2018, 10:02am UTC](https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189/3 "2018-01-07T10:02:41Z")

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Thanks for your reply, I understand.

Now when I read the documentation I understand it well, but I am unsure why I was not able to at the time. Partially the reason should have been that I was unfamiliar with Diagonal/Non-Diagonal Noise. Maybe when I looked at the two sections the obvious difference I saw was that in one you had several noise terms in each equation and in one not, and missed that in the Non-Diagonal case the various noise terms were the same.

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<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:** [January 7, 2018, 11:06am UTC](https://discourse.julialang.org/t/solving-stochastic-differential-equations-with-same-random-numbers/8189/4 "2018-01-07T11:06:56Z")

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@Torkel since this explanation worked I updated the docs to say something similar. Thanks for reporting this since hopefully this clears it up for more people!