$\bf{Proposition:Existence ~and ~strong ~solutions}$ Let $\mathbf{Y}=[t_{\epsilon},t_{*}]$, and $t_{*}=t_{\epsilon}+\epsilon$ is the comoving proper time for the density function blowup. And $\mathbf{Y}\bigcup\mathbf{Z}\subset\mathbf{R}^{+}$ The original nonlinear ODE defined over $\mathbf{C}=[0,t_{*}]$ is given by (-) so that $dD(t)=\Psi[D(t)]dt=k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}dt$. In the stochastic extension, the SDE for the density function diffusion for all $t\ge t_{\epsilon}$ is $d\widehat{D}(t)=\Psi[D(t)\circ d\widehat{\mathcal{B}}(t)$ with initial data $D(t_{\epsilon})=D_{\epsilon}$ with $\widehat{\mathcal{B}}(t_{\epsilon})=0$. Then for the coefficient $\Psi[D(t)]$ the following hold

(1) For all $t\ge s\ge t_{\epsilon}$, there is a $C>0$ such that the Lipschitz condition

$|\Psi(D(t))-\Psi(D(s))|\le C|D(t)-D(s)|$

holds for all $t\ge t_{\epsilon}$ and $D(t)\in[D_{\epsilon},\infty)$

(2) There is a $C>0$ such that the linear growth condition of the coefficient holds for all $D(t)\ in [D_{\epsilon},\infty)$ and $t\in\mathbf{Y}=[t_{\epsilon},t_{*}]$, namely

$|\Psi[D(t)|\le C(1+|D(t)]$

There exists  a strong solution $\widehat{D}(t)$ for at least all $t \in \mathbf{Y} \bigcup \mathbf{Z}$ such that for some $H>0$

$\mathbf{E} \lbrace sup_{t_{\epsilon}}$ <=$\infty |D(t)|^{2}\rbrace <= H[1+\mathbf{E}\lbrace|D(t_{\epsilon})|^{2}\rbrace$

To prove (1), let $s=t_{\epsilon}$ then $D(s)=D(t_{\epsilon})=D_{\epsilon} \ge 0$  then

$|\Psi(D(t))-\Psi(D_{\epsilon})| <= C|D(t) - D_{\epsilon}|$

which is

$|k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}|-|D_{\epsilon}^{2}(D_{\epsilon}-1)^{1/2}~<=~ C|D(t)-D_{\epsilon}|$ or

$|k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}|~<=~ C|D(t)|-|D_{\epsilon}|+|D_{\epsilon}^{2}(D_{\epsilon}-1)^{1/2}$

giving

$1~<=~C|k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}|^{-1}-|D_{\epsilon}||k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}|^{-1}+|D_{\epsilon}^{2}(D_{\epsilon}-1)^{1/2}|k^{1/2}D(t)(D(t)-1)^{1/2}|^{-1}$ For $D(t)$ finite but $D(t)> D_{\epsilon}\gg D(0)$

$1~<=~|k^{1/2}D(t)(D(t)-1)^{1/2}|^{-1}$ which holds for all finite density functions $D(t)>1$ since $D_{\epsilon}>1$. To prove the linear growth condition (2). Let $t \ge t_{\epsilon}$ then

$k^{1/2}(D(t))^{2}(D(t)-1)^{1/2}\le C|1+|D(t)||$

which is

$k^{1/2}(1+D(t))^{-1}\le C(D(t))^{-2}(D(t)-1)^{-1/2}$

To prove (3), The second moment over $\mathbf{Y}\bigcup\mathbf{Z}$ is

$\mathbf{E} (\widehat{D}(t))^{2}$=$\mathbf{E} \bigg((D_{\epsilon}$+$k^{1/2}{\displaystyle \int_{t_{\epsilon}}^{t_{*}}}$  $(D(s))^{2} (D(s)-1)^{1/2} \otimes d \widehat{\mathcal{B}} (s) \bigg)^{2}$=

$D_{\epsilon}^{2} + 2\mathbf{E} D_{\epsilon} k^{1/2} {\displaystyle \int_{t_{\epsilon}}^{t_{*}}}(D(s))^{2} (D(s)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(s)$+

$k\bigg (\mathbf{E}{\displaystyle \int_{t_{\epsilon}}^{t_{*}}}(D(s))^{2} (D(s)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(s))^{2}$

Using the result of (-) this reduces to

$\mathbf{E}\widehat{D}(t))^{2}=D_{\epsilon}^{2}+k \mathbf{E} \bigg({\displaystyle \int_{t_{\epsilon}}^{t_{*}}} (D(s))^{2}(D(s)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(s)\bigg)^{2}$

Using the Ito isometry and the Ito calculas whereby $d\widehat{\mathcal{B}}(t)\circ d\widehat{\mathcal{B}}(t)=dt$, this gives

$\mathbf{E}[\sup_{t From the growth condition for the coefficient $\exists~C>0$ for any $D(t)$ such that

$(D(t))^{2}(D(t)-1)^{1/2}\le C(1+\| D(t)\|)$

$(D(t))^{4}(D(t)-1)^{1/2}\le \bar{C}(1+\| D(t)\|)^{2}$

or $C>D^{2}(D-1)^{1/2}(D+1)^{-1}$ using (67) and $\bar{C}>D^{4}(D-1)(D+1)^{-2}$.

We then have:

$\mathbf{E}\lbrace\lbrace\sup_{t

$<= \mathbf{E}[|D_{\epsilon}|^{2}]+\mathbf{E}{\displaystyle \int_{t_{\epsilon}}^{T}}$ $Q(1+| D(v)|^{2})dv$

<=$\mathbf{E}[|D_{\epsilon}|^{2}]+Q(T-t_{\epsilon})+Q\mathbf{E}{\displaystyle \int_{t_{\epsilon}}^{T}} | D(v)|^{2})dv$

Recall the Gronwall inequality. Let

$(C, f,\beta)\mathbf{R}^{+}\rightarrow\mathbf{R}^{+}$

be continuous functions where $\dot{C}(t)$ exists then if

$f(t) <= C(t)+{\displaystyle \int_{t_{0}}^{t} }f(v) \beta(v)dv$

and $C(t)=C$ then

$f(t)\le C exp\bigg({\displaystyle \int_{t_{0}}^{t}}\beta(v)dv\bigg)$

If $f(t)=\mathbf{E}(...)$ then

$\mathbf{E}[...] <= C exp \bigg({\displaystyle \int_{t_{0}}^{t} }\beta(v)dv \bigg)$

Then

$\mathbf{E}\lbrace\lbrace sup_{t  Given $Q>0,\exists \bar{Q}>0$

such that

$\mathbf{E}\lbrace\lbrace\sup_{t

so (-) is established. Taking $T\rightarrow\infty$ then $\mathbf{E}\lbrace\lbrace sup_{t<\infty} [(\widehat{D}(t))^{2}\rbrace \rbrace <= \infty$.

A stronger result is established in the following proposition. It will be useful to note the following classical results from functional analysis. Let $\gamma$ and $\delta$ be conjugate exponents such that $\gamma^{-1}+\delta^{-1}=1$ with $\gamma,\delta\in\mathbb{Z}$. Let $(f,g):\mathbf{R}^{+}\rightarrow\mathbf{R}^{+}$

${\displaystyle \int_{0}^{t}}|f(v)g(v)|^{\gamma}dv\le \bigg({\displaystyle \int }|f(v)|^{\gamma}\bigg)^{1/\gamma}\bigg({\displaystyle \int} |f(v)|^{\delta}\bigg)^{1/\delta}$

$\bigg({\displaystyle \int_{0}^{t}} \bigg(f(v)+g(v)\bigg)^{\gamma}\bigg)^{\frac{1}{\gamma}}dv$ <=$\bigg({\displaystyle \int }|f(v)|^{\gamma}\bigg)^{1/\gamma}+\bigg({\displaystyle \int} |f(v)|^{\gamma}\bigg)^{1/\gamma}$

The following result gives further support to (-) in that $\mathbf{E}\lbrace |\widehat{D}(t)|\rbrace<\infty$ and $\mathbf{E}\lbrace |\widehat{D}(t)|^{2}\rbrace<\infty$ for all $t>t_{\epsilon}$.

$\bf{Theorem: Finite~ diffusion~expectations~and ~ upper~bounds}$: Let $t\in\mathbf{Y}\bigcup\mathbf{Z}$ or $t>t_{\epsilon}$ and
$d\widehat{D}(t)=\Psi[D(t)]\otimes d\widehat{\mathcal{B}}(t)$ with $D_{\epsilon}\gg 1$. If the coefficient $\Psi[D(t)]$ has linear growth then $\Psi[D] <= Q(1+|D|)$ for some $Q>0$ then
$\widehat{D}(t)$ does not blowup for all $t\in\mathbf{Y}\bigcup\mathbf{Z}$ or $t>t_{\epsilon}$. The stochastic expectations are bounded from above such that for any $s\le T$ $\exists (q,Q)>0$ such that

$\mathbf{E}\lbrace sup_{s <= T}|\widehat{D}(s)|\rbrace <= (q|D_{\epsilon}|+Q(T-t_{\epsilon})^{1/2})exp(QT)$

$\mathbf{E}\lbrace sup_{s\le T}|\widehat{D}(s)|^{2}\rbrace\le (q|D_{\epsilon}|^{2}+Q(T-t_{\epsilon}) exp(QT)$

and to all orders $p\in\mathbb{Z}$

$\mathbf{E}\lbrace \sup_{s <= T}|\widehat{D}(s)|^{p}\rbrace <= (q_{p}|D_{\epsilon}|^{p/2}+Q(T-t_{\epsilon})^{p/2})\exp(QT)$

Then in the limit as $T\rightarrow\infty$

$\mathbf{M}(t) = \mathbf{E}\lbrace sup_{t\le \infty}|\widehat{D}(t)|\rbrace <\infty,$

$\mathbf{V}(t) = \mathbf{E}\lbrace sup_{t\le \infty}|\widehat{D}(t)|^{2}\rbrace <\infty,$

$\mathbf{V}_{p}=\mathbf{E}\lbrace sup_{t\le \infty}|\widehat{D}(t)|^{p}\rbrace <\infty$ If $\widehat{D}(t)$ is a solution of the SDE for all $t\in\mathbf{Y}\bigcup\mathbf{Z}$ then $\widehat{D}(t)=D_{\epsilon}+k^{1/2}{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(v)$. We consider the general case for all $P>0$ and then consider the specific cases for $p=1$ and $p=2$ for the mean and variance. There is a constant $q_{p}$ such that

$|\widehat{D}(t)|^{p} <= q_{p}|D_{\epsilon}|^{p}+q_{p} k^{1/2}\bigg|{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(v)\bigg|^{p}$

The expectation is then

$\mathbf{E}( sup_{s\le t}|\widehat{D}(s)|^{p})\le q_{p}|D_{\epsilon}|^{p}$+

$q_{p} k^{1/2}\mathbf{E}\bigg|{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(v)\bigg|^{p}$

Applying the Buckholder-Gundy_Davis inequality, $\exists$ $Q_{p}$ such that

$\mathbf{E}(sup_{s\le t}|\widehat{D}(s)|^{p} <= q_{p}|D_{\epsilon}|^{p}q_{p} Q_{P}$ * $\mathbf{E}\bigg|\bigg\langle{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(v)\bigg |{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\otimes d\widehat{\mathcal{B}}(v)\bigg\rangle\bigg|^{p/2}$

$<= q_{p}|D_{\epsilon}|^{p}+q_{p}Q_{p}\bigg |{\displaystyle \int_{t_{\epsilon}}^{t}}(D(v))^{2}(D(v)-1)^{1/2}\bigg|^{p/2}$

From the conditions $\Psi[D]=D^{2}(D-1)^{1/2} <= \bar{C}(1+|C|)$ and $latex |\Psi[D]|^{2}=D^{4}(D-1) <= C(1+|C|)^{2}$  we have

\$\mathbf{E}( sup_{s\le t}|\widehat{D}(s)|^{p}\le q_{p}|D_{\epsilon}|^{p}+q_{p} Q_{P}\mathbf{E}\bigg|{\displaystyle \int_{t_{\epsilon}}^{t}}C(1+|D(v)|)^{2}dv\bigg|^{p/2}$

From the inequalities (-)

${\displaystyle \int_{t_{\epsilon}}^{t}}C(1+|D(v)|)^{2}dv <= C(t-t_{\epsilon})^{p/2}+C{\displaystyle \int_{t_{\epsilon}}^{t}}|D(v)|^{2}dv$

giving

$\mathbf{E}( sup_{s\le t}|\widehat{D}(s)|^{p} <= q_{p}|D_{\epsilon}|^{p}+C(t-t)_{\epsilon}^{p/2} +C {\displaystyle \int_{t_{\epsilon}}^{t}}\mathbf{E}|D(v)|^{2}dv$

From the Gronwall inequality it follows that

$\mathbf{E}( sup_{t\le T}|\widehat{D}(t)|^{p} <=(q_{p}|D_{\epsilon}|^{p}+C(T-t)_{\epsilon}^{p/2}) exp(CT)$

The cases for $p=1,2$ follow. In the limit as $T\rightarrow\infty$

$\mathbf{E}(|\widehat{D}(t)|^{p})= lim_{T\rightarrow\infty}\mathbf{E}(sup_{t <= T}|\widehat{D}(t)|^{p}<= lim_{T\rightarrow\infty}(q_{p}|D_{\epsilon}|^{p}+C(T-t)_{\epsilon}^{p/2}) exp(CT)=\infty$

and the results are estabished.