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I need a set of stochastic processes $x_i(t)$ with the following characteristics:

  • At each time $t$, the jump of each variable can be just $+s$ or $-s$;
  • The processes have to be mean reverting, so that in the limit $t\rightarrow \infty $ the variance of the precess does not diverge;
  • For each time $t$, $\sum_i x_i(t) = \sum_i x_i(0) = const$, or in other words, the sum over all the increments $ \sum_i \Delta x_i(t)$ has to be equal to zero.

My idea was to use something like:

$x_i(t+1) = x_i(t) + s_i(t)$

and $s_i(t)$ is equal to $+s$ with probability $p_i(t)$, and to $-s$ with probability $(1-p_i(t))$, where:

$p_i(t) = \frac{1}{2} + \frac{\alpha}{2} \left[ \frac{x_i(t-1)-m}{\mid x_i(t-1)-m\mid_{max_i} } \right] $

where the $max_i$ is the maximum value taken over all the stochastic processes, and $m$ is the mean value around which the processes should oscillate. I can then simulate the processes and impose manually the last of the three conditions.

Now my question is, how can I find analytic distribution of the random variable $x_i(t)$ ? I see from the simulation that as far as $\alpha$ is higher than zero, the variance does not diverge anymore as $t \rightarrow \infty$. Is there a way to see it analytically? Is there anything similar in literature already developed?

Thanks a lot for your help. Sam

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1 Answer 1

The construction suggested in the post will not satisfy the desired condition on $\sum\limits_ix_i(t)$.

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