Given a generic SDE for all $t\in\mathbf{Y}\bigcup\mathbf{Z} \subset \mathbf{R}^{+}$

$d\widehat{D}(t)=\Phi((t))dt~+~\Psi(D(t))\otimes d\widehat{\mathcal{B}}(t)$

for some diffusion $\widehat{D}(t)$ defined for all $t\in\mathbf{Y}\bigcup\mathbf{Z}\subset\mathbf{R}^{+}$ and where one can have $\Phi(D(t))=\Psi(D(t))$. Suppose the underlying deterministic ODE is $dD(t)/dt=\Phi(D(t))$ and has a solution, for some initial data, which explodes at some $t=t_{*}$ then $D(t*)=\infty$ for some $t_{*}>t_{\epsilon}$. Then the SDE (1.3) does not blow up for any $t\in\mathbf{Y}\bigcup\mathbf{Z}$ if either $|\mathbf{\mathcal{F}el}(|D_{\epsilon}|,\infty)|=\infty$ or $|\mathbf{\mathcal{F}el}(-\infty,D_{\epsilon}|)|=\infty$

$|\mathbf{\mathcal{F}el}(|D_{\epsilon}|,\infty)| =\bigg|{\displaystyle \int_{|D_{\epsilon}|}^{\infty}}\bigg(\frac{\int_{|D_{\epsilon}|}^{|D(t)|}d\bar{D}(t)[\Psi(\bar{D}(t))]^{-2}\exp[+\Lambda(\bar{D}(t)]} {exp[-\Lambda(D(t))}\bigg)dD(t) \bigg|=\infty$

$|\mathbf{\mathcal{F}el}(-\infty,|D_{\epsilon}|) |= \bigg|{\displaystyle \int_{|D_{\epsilon}|}^{\infty}}\bigg(\frac{\int_{|D_{\epsilon}|}^{|D(t)|}d\bar{D}(t)[\Psi(\bar{D}(t))]^{-2}\exp[+\Lambda(\bar{D}(t)]} {exp[-\Lambda(D(t))}\bigg)dD(t) \bigg|=\infty$

where

$\Lambda(D(t))={\displaystyle \int^{|D(t)|}}2 \Phi[\bar{D}(t)][\Psi(\bar{D}(t))]^{-2} d\bar{D}(t)$

Setting $\Theta[\pm D(t)]=exp[\pm\Lambda[D(t)]$ these can be written even more succintly as

$|\mathbf{\mathcal{F}el}(|D_{\epsilon}|,\infty)| =\bigg|{\displaystyle \int_{|D_{\epsilon}|}^{\infty}}\bigg(\Theta[-D(t)])] \int_{|D_{\epsilon}|}^{|D(t)|}d\bar{D}(t) [\Psi(\bar{D}(t))]^{-2}\Theta[+D(t)] \bigg)dD(t)\bigg|=\infty$

$|\mathbf{\mathcal{F}el}(-\infty,|D_{\epsilon}|) =\bigg|{\displaystyle \int_{|D_{\epsilon}|}^{\infty}}\bigg(\Theta[-D(t)])] \int_{|D_{\epsilon}|}^{|D(t)|}d\bar{D}(t) [\Psi(\bar{D}(t))]^{-2}\Theta[+D(t)] \bigg)dD(t)\bigg|=\infty$

But the SDE explodes if $\mathbf{\mathcal{F}el}(\psi_{\epsilon},\infty)<\infty$ or $\mathbf{\mathcal{F}el}(-\infty,\psi_{\epsilon})<\infty$
\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 $\mathbf{\mathcal{F}el}(D_{o},\infty))$ in terms of a limit and apply the L’Hopital rule. The Feller Tests can be expressed as

$|\mathbf{\mathcal{F}el}(D_{\epsilon},\infty)|=\lim_{D(s)\rightarrow -\infty} \bigg|{\displaystyle \int_{D_{\epsilon}}^{D(t)}}d\bar{D}(t)[\Phi^{-1}(\bar{D}(t)]\bigg|$

$|\mathbf{\mathcal{F}el}(-\infty, D_{\epsilon})|=\lim_{D(s)\rightarrow -\infty}\bigg|{\displaystyle \int_{D_{\epsilon}}^{D(t)}} d\bar{D}(t)[\Phi(\bar{D}(t)]\bigg|$

Hence, all Ito SDEs or pure diffusions of the form

$d\widehat{D}(t)=\Psi(D(t))\otimes d\widehat{\mathcal{B}}(t)$

cannot explode for any  $t \in \mathbf{R}^{+}$ since $\mathbf{\mathcal{F}el}(D_{\epsilon},\infty)|=\infty$ when $\Phi(D(t))=0$

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

$|\mathbf{\mathcal{F}el}(D_{\epsilon},\infty)|={\displaystyle \int_{D_{\epsilon}}^{\infty}} \bigg(\frac{\mathcal{G}(D(t))}{\mathcal{H}(D(t))}\bigg) dD(t)$

$=\left|{\displaystyle \int_{D_{\epsilon}}^{\infty}}\bigg(\frac{\int_{D_{\epsilon}}^{|D(t)|} dD(t)[Y(D(t))]^{-2} exp[+\Lambda(D(t)]}{ exp[+\Lambda(D(t))]}\bigg) dD(t)\right|$

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

$lim_{D(t)\rightarrow\infty}\frac{\mathcal{G}(D(t))}{\mathcal{H}(D(t))}= lim_{D(t)\rightarrow \infty}\frac{\frac{d}{dD(t)}\mathcal{G}(D(t))} {\frac{\delta}{\delta D(t)\mathcal{G}}(D(t))}= lim_{D(t)\rightarrow\infty}\frac{\partial_{D}\mathcal{G}(D(t))} {\partial_{D}\mathcal{H}(D(t))}$

where $\nabla_{D}=d/dD(t)$. Then the convergence properties can be deduced as

$lim_{D(t) \rightarrow \infty} {\displaystyle \int_{D_{\epsilon}}^{|D(t)|}} \bigg( \frac{\mathcal{G} (\bar{D}(t))}{\mathcal{H} (\bar{D}(t))} \bigg)|$

$\sim lim_{D(t)\rightarrow \infty}\bigg|{\displaystyle \int_{D_{\epsilon}}^{D(t)}}\bigg(\frac{\partial_{D}\mathcal{G} (\bar{D}(t))}{\partial_{D}\mathcal{H}(\bar{D}(t))}\bigg)d\bar{D}(t)\bigg |$

so that

$lim_{D(t)\rightarrow \infty}\bigg\lvert{\displaystyle \int_{D_{\epsilon}}^{D(t)}}\bigg(\frac{\partial_{D}\mathcal{G} (\bar{D}(t))}{\partial_{D}\mathcal{H}(\bar{D}(t))}\bigg)d\bar{D}(t)\bigg\rvert$

$=\bigg|{\displaystyle \int_{-\infty}^{D_{\epsilon}}} \bigg(\frac{[\Psi(\bar{D}(t))]^{-2} exp[+\Lambda(\bar{D}(t)]}{\partial_{\bar{D}}{\Lambda (\bar{D}(t))\exp[+\Lambda(\bar{D}(t))]}}\bigg)d\bar{D}(t)\bigg|$

and the exponential terms cancel leaving

$\mathbf{\mathcal{F}el}(D_{\epsilon},\infty)=\lim_{D(t)\rightarrow \infty}\bigg\lvert{\displaystyle \int_{D_{\epsilon}}^{D(t)}}\bigg(\frac{\partial_{D} \Lambda(\bar{D}(t))}{\partial_{D}\mathcal{H}(\bar{D}(t))}\bigg)dD(t)\bigg\rvert$

$=\bigg|{\displaystyle \int_{-\infty}^{D_{\epsilon}}}\bigg([\frac{[\Psi(\bar{D}(t))]^{-2}}{\partial_{D} \Lambda(\bar{D}(t)}\bigg)dD(t)\bigg|$

But the derivative of (2.14) is

$\partial_{D}\Lambda(D(t))=\Phi(D(t))[\Psi(D(t)]^{-2}$

so that

$\mathbf{\mathcal{F}el}(D_{\epsilon},\infty)=\lim_{D(t)\rightarrow \infty}\bigg\lvert{\displaystyle \int_{D_{\epsilon}}^{D(t)}}\bigg(\frac{\partial_{D}\mathcal{G} (\bar{D}(t))}{\partial_{D}\mathcal{H}(\bar{D}(t))}\bigg)d\bar{D}(t)\bigg\rvert=$

$\lim_{D(t)\rightarrow\infty}\bigg|{\displaystyle \int_{D_{\epsilon}}^{\bar{D}(t)}}\frac{d\bar{D}(t)} {\Phi(\bar{D}(t))}\bigg|$

An immediate corollary then is that pure diffusions $d\widehat{D}(t)=\Psi[D(t)]\otimes d\widehat{\mathcal{B}}(t)$ never explode since $\Phi(D(t)]=0$ for these SDEs.