# Stability of parameters in sde

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?

• It means that we use Stratonovich noise. Does that change the argument? – thaumoctopus Dec 15 '18 at 13:14
• I'm not familiar with that, sorry. – Tki Deneb Dec 15 '18 at 14:13
• @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? – Sayantan Dec 28 '18 at 13:16