Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.