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[X]/dt = r[Y] - q[X] + sqrt(r[Y])*N1(0,1) - sqrt(q[X])*N2(0,1)
d[Y]/dt = q[Y] - r[X] + sqrt(q[X])*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 [X] + [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 ). 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