Consider Lorenz model
$$ \begin{align*} \frac{dx}{dt}&=\sigma(y-x)\\ \frac{dy}{dt}&=\rho x-y-xz\\ \frac{dz}{dt}&=xy-\beta z \end{align*} $$
with $\sigma=10$, $\rho=28$ and $\beta=\frac{8}{3}$.
I want to embed this model in stochastic environment by adding exponentially colored noise process to parameter $\sigma$:
$$ d\widetilde{\sigma_t}=-\alpha\widetilde{\sigma_t}dt+\gamma dW_t\text{, }\sigma_t=10+\widetilde{\sigma_t} $$
where $\alpha$ and $\gamma$ are values I will do experiments with.
I came up with
$$ d\begin{bmatrix}x_t\\y_t\\z_t\\\widetilde{\sigma_t}\end{bmatrix}=\begin{bmatrix}(10+\widetilde{\sigma_t})(y-x)\\\rho x-y-xz\\xy-\beta z\\-\alpha\widetilde{\sigma_t}\end{bmatrix}dt+\begin{bmatrix}0\\0\\0\\\gamma\end{bmatrix}dW_t $$
- Is stochastic differential equation above correct?
I want to apply Euler scheme to approximate given model numerically. I know that Euler scheme requires equation to be in Ito form.
- How can I determine if equation above is in Ito form?
Finally, I want to analyze order of convergence through experiments.
- Does equation above have exact solution? If not, what should I use as reference solution to calculate error against?