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.