For stochastic differential equations, why do we care if the process is $L^2$ bounded?

I have been studying Stochastic Differential Equations, and one theorem relates to the existence of a solution to the SDE:

$$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t$$

with $X_0 = x_0$ and $0 \leq t \leq T$.

The theorem states that if the SDE satisfies a space-variable Lipschitz condition and a spatial growth condition, then there would exist a continuous adapted solution $X_t$ of the differential equation above that is uniformly bounded in $L^2(dP)$, ie.:

$$\sup_{0 \leq t \leq T}E(X_t^2) < \infty$$

What perplexes me is why the last part about $L^2$ boundedness is useful. The only thing I can think of is that I could know that the variance is finite, but I am not sure why that in and of itself is useful. Does anyone have any input? Thanks!