Aim
I am trying to write down a system of stochastic differential equations, however, since I lack a background in mathematics I am not sure how to do this.
Let's say I have the following system of ordinary differential equations, a simple Lotka Volterra Model:
$$\left\{ \begin{array}{c} \dot x = \alpha x - \beta x y \\ \dot y = \delta x y - \gamma y \end{array} \right.$$
with initial values for x = 10 and y = 10.
The parameter values for alpha, beta, delta and gamma are 1.1, 0.4, 0.1 and 0.4 respectively (mimicking this example).
In order to rewrite it to a system of stochastic differential equations, I rewrote the ODE system like this:
$$ \left\{ \begin{array}{c} \frac{d x}{d t}=\alpha x -\beta xy \\ \frac{d y}{d t}=\delta xy - \gamma y \\ \frac{d \alpha }{d t}= 0 \\ \end{array} \right. $$
In this way, I am able to let parameter $\alpha$ follow a random walk with $\mu = 0$ and $\sigma = 0.01$ (normal distribution).
Attempt
I thought a system of SDEs has the following format:
$$ \left\{ \begin{array}{c} \ dX(t) = b(X(t))dt + B(X(t))dW(t) \\ \ X(0) = x_0 \end{array} \right. $$
However, I am not sure how to write down this ODE model as an SDE model with random walk.
Question
How do I write down this system of stochastic differential equations mathematically correct?