I am trying to figure out how exactly to determine the rate of convergence for a stochastic dynamical system.

I am given a stochastic equation, i.e., a quadratic SDE, and I am interested to see how to determine the velocity of convergence near a stationary point for which I already know it does not converge exponentially. I can't seem to find a proper definition or even a proper way to determine this.

What my professor has given me riddles me a little. He suggested that I take the SDE, which is given in terms of $\dot{p}$ and take a look at $\frac{d}{dt}\|p - p^*\|^2$ where $p^*$ is a statement to $\dot{p} = H(p)=0$, which makes it an equilibrium or stationary point (but with negative eigenvalues, so it does not attract). On the other hand the internet suggests rather imprecisely that one should regard $$ \| \dot{p}-p^*\|$$ rather than taking the derivative outside the norm. This produces different results.

Although I am quite aware that the differences may well be insignificant, but here is basically where i am stuck. How do I determine a proper convergence rate? I mean I only know that despite having negative eigenvalues my dynamical system converges (with a certain probability, but right now the stochastical perspective is of less interest)

I am thankful for any help

We consider the one dimensional Riccati equation $$\dot{p}= -2ap + r_1 - p^2 \frac{b^2}{r_2},$$ where $b, r_2,r_1$ are positive. I have already considered the case in which a is positive and that leaves a good solution which is exponentially stable and converges at at least exponential rate. But now the case for which a is negative seems tricky, or rather I am not sure what I am to compute here to ascertain a convergence rate.

The stochastic contribution lies in $a$. for now i am considering a very simple case, namely that $a$ is constant. later i want to consider the case where $a$ is the derivative of a differientible function $A$ where it is $A(X_t)$ with a stochastic process $X_t$. $a$ will then be bounded. The last step will be to see the multi dimensional case. My aim is to see whether or not I can get a strong solution as in the paper(see comments) by allowing negative $a$. To research here i need to know what is a decent way to compute the convergence rate.

Also referring to the comment, i have tried to solve the equation by substitution but i am having problems resubstituting. This and the fact that the multidimensional case do no posess known general solutions leave me opting to get a convergence rate here.

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    $\begingroup$ For reference, please give the full SDE. Is there a link to the internet source? $\endgroup$ – LutzL Jun 12 '18 at 12:58
  • $\begingroup$ Basically it's a Riccati equation: $\endgroup$ – Danny Jun 12 '18 at 15:14
  • $\begingroup$ In what sense is that a stochastic DE? I can see no noise term. Is this equation scalar? Then you can solve it via separation of variables. $\endgroup$ – LutzL Jun 12 '18 at 15:49
  • $\begingroup$ i have already solved this. the more general setting with the stochastic term is found in this paper arxiv.org/abs/1606.08251 $\endgroup$ – Danny Jun 12 '18 at 15:53

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