\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<T}\![(\widehat{D}(t))^{2}]\!]=\mathbf{E}(\lbrace\lbrace D_{\epsilon}^{2})+ k\mathbf{E} k{\displaystyle \int_{t_{\epsilon}}^{t}} (D(v))^{4}(D(v)-1)dv 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<T} [(\widehat{D}(t))^{2}\rbrace\rbrace <= \mathbf{E}[|D_{\epsilon}|^{2}]+\mathbf{E} [{\displaystyle \int_{t_{\epsilon}}^{T}} \bar{C}[1+(1+ | D(v)|)^{2}dv

 <= \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<T} [(\widehat{D}(t))^{2}\rbrace\rbrace<= (\mathbf{E}[|D_{\epsilon}|^{2}+Q(T-t_{\epsilon}) exp(Qt)  Given Q>0,\exists \bar{Q}>0

such that

 \mathbf{E}\lbrace\lbrace\sup_{t<T}[(\widehat{D}(t))^{2}\rbrace\rbrace <= \bar{Q}(1+\mathbf{E}[|D_{\epsilon}|^{2}) <= < = (\mathbf{E}[|D_{\epsilon}|^{2}+Q(T-t_{\epsilon}) exp(Qt) 

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.

Advertisements