Given a generic SDE for all

for some diffusion defined for all and where one can have . Suppose the underlying deterministic ODE is and has a solution, for some initial data, which explodes at some then for some . Then the SDE (1.3) does not blow up for any if either or

where

Setting these can be written even more succintly as

But the SDE explodes if or

\end{thm}

Rigorous technical proofs of Feller’s test for explosion of SDEs relies on a detailed analysis of exit times from an interval and the application of the Feynman-Kac formula. (Pinsky, Klebaner etc.) Rather than attempting to compute (-) explicitly for a SDE, it is much simpler to reduce down the expression for in terms of a limit and apply the L’Hopital rule. The Feller Tests can be expressed as

Hence, all Ito SDEs or pure diffusions of the form

cannot explode for any since when

Concentrating on equation (-) only, it is useful to write (-) in the form

Applying the L’Hopital rule to the integrand in (-)

where . Then the convergence properties can be deduced as

so that

and the exponential terms cancel leaving

But the derivative of (2.14) is

so that

An immediate corollary then is that pure diffusions never explode since for these SDEs.

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