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Assume that I have some stochastic dynamical model of the form:

$$x_{t+1}=M(x_t)+\epsilon$$

where $M:x\rightarrow x$ is a deterministic function, the subscript $t$ denotes a certain time step, and $\epsilon$ is a realization of some Gaussian noise $\mathcal{N(\mu=0, \sigma)}$ with zero mean and some standard deviation. If I propagate this model in equal time steps $\Delta t$, then I am adding a steady amount of random noise to the forecast over time. That's great.

Now, let's complicate matters. Assume that instead of propagating in steady time steps, I am using some scheme which adjusts the time steps individually. As a consequence, the time between two steps is no longer uniform. To ensure that I am still adding a constant amount of noise over time, I suspect that I would have to adjust the standard deviation $\sigma$ to reflect the length of the individual time step. Is there a general rule to scale additive Gaussian noise for such uneven time steps?

I suspect this might be a relatively common question, but my search on this site and beyond has come up dry. Perhaps I have been looking for the wrong terms. I would appreciate any answers or references to other questions which answered this conundrum.

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The state equation may be seen as the simplest possible discretisation (namely the Euler-Maryuama scheme) of the stochastic differential equation governing "nature". Similar to simulations of Brownian motion, the variance (or covariance matrix) of $\varepsilon$ should scale with $\Delta t$.

Notably, this has the nice consequence that the variance of a sum of errors for many smaller time steps is the same as the variance of the error for the original time step.

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