Stochastic model of a RL circuit

Question

I would like to solve stochastic electric circuits numerically. I already know that I should use Milstein method to solve those systems. However, I'm not sure if the stochastic differential equations are correct, if the modeling is correct. Thus, could you give me a feedback whether or not my reasoning is correct?

Consider a series RL circuit . Mathematically, it can be modeled as:

\begin{equation} \begin{cases} \dot{I}(t) = \frac{V_L(t)}{L} \\ \dot{V_L}(t) = \dot{V}(t) - R \dot{I}(t) \end{cases} \end{equation}

As previously stated, I want it to be a stochastic electric circuit. Thus, all parameters should be random variables. For example, I want them to satisfy:

\begin{equation} \begin{cases} L = L_0 + e^{B_t} \\ R = R_0 + e^{B_t} \\ V(t) = V_0 \sin(120 \pi t) + \sigma B_t \end{cases} \end{equation}

where $B_t$ stands for brownian motion and the other constants are all real positive ones.

With that in mind,

1. Rewrite both equations in differential form:

\begin{equation} \begin{cases} dI = \frac{V_L(t)}{L}dt \\ dV_L = dV - R dI \end{cases} \end{equation}

2. Substitute the parameters:

\begin{equation} \begin{cases} dI = \frac{V_L(t)}{L_0 + e^{B_t}}dt \\ dV_L = 120 \pi V_0 \cos( 120 \pi t) dt + \sigma dB_t - \underbrace{(R_0 + e^{B_t}) dI}_{?} \end{cases} \end{equation}

The part "?" feels weird to me. Because I'm used to solve such kind of systems numerically:

\begin{equation} d \vec{X} = \vec{F}(t, \vec{X}) dt + \vec{G}(t, \vec{X}) d\vec{W} \end{equation}

Surely, I could use substitution. Thus I would arrive at the last kind of equation for one dimension. But I would like to solve systems of SDEs numerically.

Is this reasoning correct? Personally, I do think it is. But this field is quite new to me and sometimes tricky.

Thanks

P.S.: I have just read this book on stochastic calculus. I'm an electrical engineering student.

EDIT

The original problem is: I would like to solve stochastic electrical circuits numerically. That means I have the following things to do:

1. model the electrical circuit using stochastic differential equations. Which I'm yet not sure if the reasoning is correct;

2. find the equations that can entirely describe the dynamics of the circuit;

3. set up the following equation:

\begin{equation} d \vec{X} = \vec{F}(t, \vec{X}) dt + \vec{G}(t, \vec{X}) d\vec{W} \end{equation}

4. use Milstein's method to solve it numerically.

An attempt to answer question 1 has been done right above. There I try to model a series RL circuit.

The answers to question 2 and 3 depends whether or not my attempt to answer question 1 is correct or not. That's why this question focuses on the validity of the stochastic model and its reasoning.

Finally, question 4 is already solved. I know how to implement Milstein's method numerically.

Answer 1

You can uncouple the two SDEs by substituting the first equation into the second one, sucht that $$ \begin{array}{rcl} \mathrm{d}V_L &=&\displaystyle \mathrm{d}V - R\mathrm{d}I = \mathrm{d}V - \frac{R}{L}V_L\mathrm{d}t \\ &=&\displaystyle \left(120\pi V_0\cos(120\pi t) + \frac{R_0+e^{B_t}}{L_0+e^{B_t}}V_L\right)\mathrm{d}t + \sigma\mathrm{d}B_t \end{array} $$ which is a one-dimensional SDE with the standard structure (that you can solve numerically). Once you get it, the other variable is given by $$ I(t) = I(t_0) + \int_{t_0}^t \frac{V_L(s)}{L_0+e^{B_s}} \mathrm{d}s $$