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From lecture notes on SDE's.

Consider the Stratonovich equation $dX_t=rX_tdt+\sigma X_t\circ dB_t$. It has initial condition $X_0=x$. What are the conditions for the parameters $(\sigma,r)$, for $\mathbb{E}(X_t)$ and $\mathbb{E}(X_t^2)$ to stay bounded as $t\rightarrow \infty$?

I thought about using the Dynkin formula, but I think this has more to do with stochastic stability? I am not sure which theorem to use, at there does not seem to be some general approach?

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  • $\begingroup$ It means that we use Stratonovich noise. Does that change the argument? $\endgroup$ – thaumoctopus Dec 15 '18 at 13:14
  • $\begingroup$ I'm not familiar with that, sorry. $\endgroup$ – Tki Deneb Dec 15 '18 at 14:13
  • $\begingroup$ @thaumoctopus Could you please give some context behind this question? Second, if this were a standard Ito SDE, is there a way to characterize the parameters $\sigma$ and $r$ so that the first and second moments remain bounded? $\endgroup$ – Sayantan Dec 28 '18 at 13:16

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