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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

$$ d\begin{bmatrix}x_t\\y_t\\z_t\\\widetilde{\sigma_t}\end{bmatrix}=\begin{bmatrix}(10+\widetilde{\sigma_t})(y-x)\\\rho x-y-xz\\xy-\beta z\\-\alpha\widetilde{\sigma_t}\end{bmatrix}dt+\begin{bmatrix}0\\0\\0\\\gamma\end{bmatrix}dW_t $$

  1. 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.

  1. How can I determine if equation above is in Ito form?

Finally, I want to analyze order of convergence through experiments.

  1. Does equation above have exact solution? If not, what should I use as reference solution to calculate error against?
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    $\begingroup$ Your system seems correct, but as is, it is not linear, which quite annoying. However, $\sigma_t$ is decoupled from the other variables and its equation of motion can be solved on its own (use Ito's lemma on the function $f(\sigma_t,t) = e^{\alpha t}\sigma_t$). The remaining system is linear with respect to $x,y,z$. $\endgroup$
    – Abezhiko
    Jan 5 at 16:29
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    $\begingroup$ @Abezhiko Applying Ito's lemma on $f(\sigma_t,t)$, I get $df=\alpha e^{\alpha t}\sigma_tdt+e^{\alpha t}d\sigma_t$. Substituting $-\alpha\sigma_tdt+\gamma dW_t$ for $d\sigma_t$, I get $\gamma e^{\alpha t}dW_t$. I am confused how this implies that $e^{\alpha t}\sigma_t$ is exact solution for $d\sigma_t$. Did I apply Ito's lemma incorrectly? $\endgroup$
    – Midess
    Jan 6 at 12:24
  • $\begingroup$ That's correct, then you find $e^{\alpha t}\sigma_t - e^{\alpha t_0}\sigma_{t_0} = \int_{t_0}^te^{\alpha s}\,\mathrm{d}W_s$ by integration, hence $\sigma_t = e^{-\alpha(t-t_0)}\sigma_{t_0} + e^{-\alpha t}\int_{t_0}^te^{\alpha s}\,\mathrm{d}W_s$ and it cannot be simplified further. Nevertheless, note that $X_t := \int_{t_0}^te^{\alpha s}\,\mathrm{d}W_s$ follows a zero-mean normal law, whose variance is $\mathrm{Var}[X_t] = \int_{t_0}^te^{2\alpha s}\,\mathrm{d}t = \frac{e^{2\alpha t}-e^{2\alpha t_0}}{2\alpha}$ by Itô's isometry. $\endgroup$
    – Abezhiko
    Jan 6 at 12:44
  • $\begingroup$ Oops, I forgot the last part : [...] and thus $\displaystyle\tilde{\sigma}_t\sim\mathcal{N}\left(\tilde{\sigma}_{t_0}e^{-\alpha(t-t_0)},\frac{1-e^{-2\alpha(t-t_0)}}{2\alpha}\right)$. $\endgroup$
    – Abezhiko
    Jan 6 at 14:25
  • $\begingroup$ @Abezhiko I haven't run any experiments yet but from your answers, it seems like $\gamma$ has no effect. Is it true? Also, should I interpret the resulting stochastic differential equation as Ito integral or do I need to convert it from Stratonovich integral to Ito integral in order to apply Euler scheme? $\endgroup$
    – Midess
    Jan 9 at 18:33

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