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!


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