Let $g(t) = E[X_{t}^{2}]$, the starting point is the following set of inequalities which hold for all $t \geq 0$: \begin{align} - \int _{0}^{t} 2g(s)ds + t \leq g(t) \leq \int _{0}^{t} 2g(s)ds + t \end{align}

Then, we need to prove the following:\begin{align} 0 < \text{lim inf}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}\leq \text{lim sup}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}} < \infty \end{align}

An obvious first step is integrating the above inequalities, and Fubini will probably also be useful, since then you can exchange the expectation and integrals. Furtermore, $X_{t}$ is also a submartingale, but I am not sure if that is relevant here.

However, even with the above knowledge I still can not picture where the limsup and liminf in the expression come from.

I would really appreciate any help.

EDIT: As per TheBridges' request, I also give the SDE of $X_{t}$ below, though I am not sure if it is relevant towards answering my question. \begin{align} dX_{t} = |X_{t}|dt + dW_{t}, X_{0} = 0 \end{align}

To derive the first inequalities at the top, I applied Ito to rewrite $X_{t}^2$, and together with the SDE of $X_{t}$ you can fairly easily derive these inequalities. But it is still unclear as to how they should be used to derive the second set of inequalities.

  • $\begingroup$ Where your first two inequalities are coming from ? $\endgroup$ – TheBridge May 20 at 13:01
  • $\begingroup$ The right hand side inequality seems dubious to me with $g(t)=t^2$, I get the condition $-2/3t^3+t<t^2$ equivalent for t>0 to $-2/3t^2+1<t$ which false for small t. $\endgroup$ – TheBridge May 20 at 13:08
  • $\begingroup$ @TheBridge You mean the starting inequalities right? I just applied the Ito formula to $X_{t}^{2}$, and used the original SDE for $X_{t}$ to eventually get those inequalities. If you'd like I can edit those into the question, but sure if it's relevant. $\endgroup$ – Wanderer May 20 at 16:25
  • $\begingroup$ Still not clear to what sde do you apply Itô's lemma to ? I don't see how those inequalities are derived from Itô maybe you could add those details in the question $\endgroup$ – TheBridge May 20 at 20:49
  • $\begingroup$ @TheBridge Ok, I edited the question with the SDE. $\endgroup$ – Wanderer May 20 at 21:15

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