# Coupled stochastic differential equations?

I'm a physics student working on a quantum information project (so please be gentle with me). My work involves stochastic processes and I'm new to the topic, so I'm asking some help about a system of differential equations involving a stochastic term. The system is:

$$\begin{cases} \dot r_1 (t)= -\omega r_2(t) \\ \dot r_2 (t)= \omega r_1(t) - \epsilon B(t) r_3(t) \\ \dot r_3 (t)=\epsilon B(t) r_2(t) \end{cases}$$

where $\omega$ and $\epsilon$ are constants. Now, if $B(t)$ were a function of time, I would call this a system of coupled ODEs. If $B(t)$ is a stochastic process (namely, a Ornstein-Uhlenbeck process), how do I call this? A system of coupled stochastic differential equations?

Can someone point me to some references on numerical solutions (I doubt there are analytical ones) of this kind of SDEs?

• As written, these are random ODEs. Since Brownian motion (and most other processes built from it) is (Holder) continuous you can solve these pathwise the way you are used to. If you wanted to write this with a time derivative of $B$ there, then you would have a coupled family of stochastic differential equations. At this point, it becomes physically important to decide what type of SDE you mean. Commented Apr 8, 2014 at 16:22

Is this the way the system was written in your source or was it something like this for for example the second equation?

$$\mathrm{d}r_2(t) = \omega r_1 \mathrm{d}t - \epsilon \mathrm{d}B(t) r_3(t)$$ Or I guess sometimes equivalently (although less correctly) above could be written as $$\dot{r}_2(t) = \omega r_1(t) - \epsilon \dot{B}(t)r_3(t)$$

The solution of an equation like that would require Ito calculus. The thing you've written down could in principle be integrated with regular calculus actually, because there are no $\mathrm{d}B$ terms. Of course the solution would still be a stochastic process, but the integrals involved are just Riemann Stieltjes.

$B(t)$ is a function of time. It's a collection of random variables parametrized by time. Also usually $B(t)$ just denotes a Brownian motion; are you sure it's supposed to be an OU process? Nothing wrong with that mathematically just strange notation.

Yes you could call this a system of SDEs or coupled SDEs anything like that.

The standard reference for numerics is Kloeden/Platen. If you want something more brief this is good too:

An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

But again you only need to worry about the subtleties of stochastic calculus if in fact you wrote these equations down wrong. If you really wanna solve the system you have above you could just generate samples of $B(t)$ and use a regular numerical ode solver.

• $B(t)$ is a background field that appears in the hamiltonian of the system. In the model $B(t)$ is a realization of an Ornstein-Uhlenbeck process. I will then have to calculate the expectation value of the $r_i$ over the stochastic process. If I could just solve the ODE with generated samples of $B(t)$ it would be great. Commented Apr 8, 2014 at 9:46

my approach would be this . Let $$y(t)$$ represent the Ornstein-Uhlenbeck process. This is defined by some parameters $$\theta$$ and $$\sigma$$ such that we can write $$dy(t)=-\theta y(t)dt+\sigma dW(t)$$ where W(t) is a standard Weiner process.

Adding this system to the the given system will allow you to apply Numerical integration (Euler-Maruyama ) .