We can extend the work for the heat equation to generic parabolic partial differential equations. We do this by first introducing viscosity solutions to Kolmogorov PDEs as given in Crandall \& Lions \cite{crandall_lions} and further extended, esp. in \cite{Beck_2021}.
%\subsection{The case without $f$}
%\subsection{Linear Algebra Preliminaries}
%\begin{lemma}
% For a matrix $A \in \R^{m\times n}$, $A[A]^*$ is symmetric.
% \textit{(i)} Assume $\lambda \in \mathbb{C}$ is an eigenvalue for the symmetric matrix A. This indicates that $Av = \lambda v$, where $v\neq 0$ for some $v$ in our vector space and further that:
% This indicates that $\lambda = \overline{\lambda}$ whence $\lambda$ is real.
% \medskip
%
% \textit{(ii)} Assume $A$ is not diagonalizable. Then there must exist some eigenvalue $\lambda_i$ of order 2 or more. This would indicate that there is some repeated eigenvalue $\lambda$, and $v\neq 0$ such that:
% Leading to a contradiction. Thus there are no generalized eigenvectors of order 2 or higher, and so $A$ must be diagonalizable.
%\end{proof}
\section{Some Preliminaries}
We take work previously pioneered by \cite{Ito1942a} and \cite{Ito1946}, and then seek to re-apply concepts first applied in \cite{Beck_2021} and \cite{BHJ21}.
\begin{lemma}\label{lemma:2.7}
Let $d,m \in\N$, $T \in(0,\infty)$. Let $\mu\in C^{1,2}([0,T]\times\R^d, \R^d)$ and $\sigma\in C^{1,2}([0,T]\times\R^d, \R^{d\times m})$ satisfying that they have non-empty compact supports and let $\mathfrak{S}=\supp(\mu)\cup\supp(\sigma)\subseteq[0,T]\times\R^d$. Let $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t )_{t \in[0,T]})$ be a filtered probability space satisfying usual conditions. Let $W:[0,T ]\times\Omega\rightarrow\R^m$ be a standard $(\mathbb{F}_t)_{t\in[0,T]}$ -Brownian motion, and let $\mathcal{X}:[0,T]\times\Omega\rightarrow\R^d$ be an $(\mathbb{F}_t)_{t\in[0,T]}$-adapted stochastic process with continuous sample paths satisfying for all $t \in[0,T]$ with $\mathbb{P}$-a.s. that:
\item$\lb\lp\mathbb{P}\lp\mathcal{X}_0\not\in\mathfrak{S}\rp=1\rp\implies\lp\mathbb{P}\lp\forall t \in[0,T]: \mathcal{X}_t =\mathcal{X}_0\rp=1\rp\rb$
\item$\lb\lp\mathbb{P}\lp\mathcal{X}_0\in\mathfrak{S}\rp=1\rp\implies\lp\mathbb{P}\lp\forall t \in[0,T]: \mathcal{X}_t \in\mathfrak{S}\rp=1\rp\rb$
\end{enumerate}
\end{lemma}
\begin{proof}
Assume that $\mathbb{P}(\mathcal{X}_0\not\in\mathfrak{S})=1$, meaning that the particle almost surely starts outside $\mathfrak{S}$. It is then the case that $\mathbb{P}(\forall t \in[0, T]: \|\mu(t,\mathcal{X}_0)\|_E +\|\sigma(t,\mathcal{X}_0)\|_F =0)=1$ as the $\mu$ and $\sigma$ are outside their supports, and we integrate over zero over time.
is an $(\mathbb{F}_t)_{t \in[0,T]}$ adapted stochastic process with continuous sample paths satisfying that for all $t \in[0,T]$ with $\mathbb{P}$-almost surety that:
Note that since $\mu\in C^{1,2}([0, T]\times\R^d, \R^d)$ and $\sigma\in C^{1,2}([0, T]\times\R^d, \R^{d \times m})$, and since continuous functions are locally Lipschitz, and since this is especially true in the space variable for $\mu$ and $\sigma$, the fact that $\mathfrak{S}$ is compact and continuous functions over compact sets are Lipschitz and bounded, and \cite[Theorem~5.2.5]{karatzas1991brownian} allows us to conclude that strong uniqueness holds, that is to say:
\begin{align}
\mathbb{P}\lp\forall t \in [0,T]: \mathcal{X}_t = \mathcal{X}_0 \rp = \mathbb{P}\lp\forall t \in [0,T]: \mathcal{X}_t = \mathcal{Y}_t \rp=1
\end{align}
establishing the case (i).
Assume now that $\mathbb{P}(\mathcal{X}_0\in\mathfrak{S})=1$ that is to say that the particle almost surely starts inside $\mathfrak{S}$. We define $\tau: \Omega\rightarrow[0,T]$ as $\tau=\inf\{t \in[0,T]: \mathcal{X}_t \not\in\overline{\mathfrak{S}}\}$. $\tau$ is an $(\mathbb{F}_t)_{t\in[0,T]}$-adapted stopping time. On top of $\tau$ we can define $\mathcal{Y}:[0,T]\times\Omega\rightarrow\R^d$, for all $t\in[0,T]$, $\omega\in\Omega$ as $\mathcal{Y}_t(\omega)=\mathcal{X}_{\min\{t,\tau\}}(\omega)$. $\mathcal{Y}$ is thus an $(\mathbb{F}_t)_{t \in[0,T]}$-adapted stochastic process with continuous sample paths. Note however that for $t > \tau$ it is the case $\|\mu(t, \mathcal{Y}_t)+\sigma(t, \mathcal{Y}_t)\|_E=0$ as we are outside their supports. For $t < \tau$ it is also the case that $\mathcal{Y}_t =\mathcal{X}_t$. This yields with $\mathbb{P}$-a.s. that:
Thus another application of \cite[Theorem~5.2.5]{karatzas1991brownian} and the fact that within our compact support $\mathfrak{S}$, the continuous functions $\mu$ and $\sigma$ are Lipschitz and hence locally Lipschitz, and also bounded gives us:
\begin{align}
\mathbb{P}(\forall t \in [0,T]: \mathcal{X}_t = \mathcal{Y}_t) =1
\end{align}
Proving case (ii).
\end{proof}
\begin{lemma}\label{lem:3.4}
Let $d,m \in\N$, $T\in(0,\infty)$. Let $g \in C^2(\R^d, \R)$. Let $\mu\in C^{1,3}([0,T]\times\R^d,\R^d)$ and $\sigma\in C^{1,3}([0,T]\times\R^d,\R^{d \times m})$ have non-empty compact supports and let $\mathfrak{S}=\supp(\mu)\cup\supp(\sigma)$. Let $(\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t \in[0,T]})$ be a stochaastic basis and let $W: [0,T]\times\Omega\rightarrow\R^m$ be a standard $(\mathbb{F}_t)_{t\in[0,T]}$-Brownian motion. For every $t\in[0,T]$ , $x\in\R^d$, let $\mathcal{X}^{t,x}=(\mathcal{X}^{t,x}_s)_{s\in[t,T]}: [t,T]\times\Omega\rightarrow\R^d$ be an $(\mathbb{F}_s)_{s\in[t,T]}$-adapted stochastic process with continuous sample paths satisfying for all $s\in[t,T]$ with $\mathbb{P}$-almost surety that:
\begin{align}\label{2.13}
\mathcal{X}^{t,x}_s = x + \int^s_t \mu(r,\mathcal{X}^{t,x}_r) dr + \int^s_t \sigma(r,\mathcal{X}^{t,x}_s)dW_r
\end{align}
also let $u:\R^d \rightarrow\R$ satisfy for all $t \in[0,T]$, $x \in\R^d$ that:
\begin{align}\label{2.14}
u(t,x) = \E\lb g(\mathcal{X}^{t,x}_T) \rb
\end{align}
then it is the case that we have:
\begin{enumerate}[label = (\roman*)]
\item$u \in C^{1,2}([0,T]\times\R^d, \R)$ and
\item for all $t \in[0,T]$, $x \in\R^d$ that $u(T,x)= g(x)$ and:
\begin{align}
\lp\frac{\partial}{\partial t} u \rp\lp t,x \rp + \frac{1}{2}\Trace\lp\sigma\lp t,x \rp\lb\sigma\lp t,x \rp\rb^* \lp\Hess_x u \rp\lp t,x \rp\rp + \la\mu\lp t,x \rp, \lp\nabla_x u \rp\lp t,x \rp\ra = 0
\end{align}
\end{enumerate}
\end{lemma}
\begin{proof}
We break the proof down into two cases, inside the support $\mathfrak{S}=\supp(\mu)\cup\supp(\sigma)$ and outside the support: $[0, T]\times(\mathbb{R}^d \setminus\mathfrak{S})$.
For the case inside $\mathfrak{S}$. Note that we may deduce from Item $(i)$ of Lemma \ref{lemma:2.7} that for all $t \in[0,T]$, $x \in\R^d \setminus\mathfrak{S}$ it is the case that $\mathbb{P}(\forall s \in[t,T]: \mathcal{X}^{t,x}_s =x)=1$. Thus for all $t \in[0,T]$, $x \in\R^d \setminus\mathfrak{S}$ we have, deriving from (\ref{2.14}):
\begin{align}\label{(2.19)}
u(t,x) = \E\lb g \lp\mathcal{X}^{t,x}_T \rp\rb = g(x)
\end{align}
Note that $g(x)$ only has a space parameter and so derivatives w.r.t. $t$ is $0$. Inhereting from the regularity properties of $g$ and (\ref{(2.19)}), we may assume for all $t \in[0,T]$, $x \in\R^d \setminus\mathfrak{S}$, that $u |_{[0,T]\times(\R^d \setminus\mathfrak{S})}\in C^{1,2}([0,T]\times(\R^d \setminus\mathfrak{S}))$. Note that the hypotheses that $\mu\in C^{1,3}([0,T]\times\R^d, \R^d)$ and $\sigma\in C^{1,3}([0,T]\times\R^d, \R^{d \times m})$ allow us to apply Theorem~7.4.3, Theorem~7.4.5 and Theorem~7.5.1 from \cite{da_prato_zabczyk_2002} for $t \in[0,T]$, $x \in\R^d \setminus\mathfrak{S}$, to give us:
\begin{enumerate}[label = (\roman*)]
\item$u \in C^{1,2}([0,T]\times\R^d,\R)$.
\item\begin{align}
0 &= \lp\frac{\partial}{\partial t} u \rp\lp t,x \rp\nonumber\\
Now consider the case within support $\mathfrak{S}$. Note that by hypothesis $\mu$ and $\sigma$ must at least be locally Lipschitz. Thus \cite[Theorem~5.2.5]{karatzas1991brownian} allows us to conclude that within $\mathfrak{S}$ the pair $(\mu,\sigma)$ for our our stochastic process $\mathcal{X}^{t,x}_s$ defined in (\ref{2.13}) must exhibit a strong uniqueness property.
\medskip
Further note that Item $(ii)$ from Lemma \ref{lemma:2.7} tells us that:
\begin{align}
\mathbb{P}(\forall t \in [0,T]: \mathcal{X}^{t,x}_s \in\mathfrak{S}) = 1.
\end{align}
Note that again the hypotheses that $\mu\in C^{1,3}([0,T]\times\R^d, \R^d)$ and $\sigma\in C^{1,3}([0,T]\times\R^d, \R^{d \times m})$, and $g \in C^2(\R^d)$ allow us to apply Theorem~7.4.3, Theorem~7.4.5 and Theorem~7.5.1 from \cite{da_prato_zabczyk_2002} for $t \in[0,T]$, $x \in\mathfrak{S}$, to give us:
Note that (\ref{2.13}) and (\ref{2.14}) together prove that $u(T,x)= g(x)$. This completes the proof.
\end{proof}
\section{Viscosity Solutions}
\begin{definition}[Symmetric Matrices] Let $d \in\N$. The set of symmetric matrices is denoted $\mathbb{S}_d$ given by $\mathbb{S}_d =\{ A \in\mathbb{S}_d : A^*= A \}$.
\end{definition}
\begin{definition}[Upper semi-continuity]
A function $f: U \rightarrow\R$ is upper semi-continuous at $x_0$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that:
\begin{align}
f(x) < f(x_0) + \ve\text{ for all } x \in B\lp x_0, \delta\rp\cap U
\end{align}
\end{definition}
\begin{definition}[Lower semi-continuity]
A function $f: U \rightarrow\R$ is lower semi-continuous at $x_0$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that:
\begin{align}
f(x) > f(x_0) - \varepsilon\text{ for all } x \in B\lp x_0, \delta\rp\cap U
\end{align}
\end{definition}
\begin{corollary}\label{sumofusc}
Given two upper semi-continuous functions $f,g: \R^d \rightarrow\R$, their sum $(f+g): \R^d \rightarrow\R$ is also upper semi-continuous.
\end{corollary}
\begin{proof}
From definitions, at any given $x_0\in\R^d$, for any $\ve\in(0, \infty)$ there exist neighborhoods $U$ and $V$ around $x_0$ such that:
\begin{align}
\lp\forall x \in U \rp\lp f(x) \leqslant f(x_0) + \varepsilon\rp\\
\lp\forall x \in V \rp\lp g(x) \leqslant g(x_0) + \varepsilon\rp
\end{align}
and hence:
\begin{align}
\lp\forall x \in U \cap V \rp\lp f(x) + g(x) \leqslant f(x_0)+g(x_0) + 2\varepsilon\rp
\end{align}
\end{proof}
\begin{corollary}\label{neglsc}
Given an upper semi-continuous function $f:\R^d \rightarrow\R$, it is the case that $(-f):\R^d \rightarrow\R$ is lower semi-continuous.
\end{corollary}
\begin{proof}
Let $f:\R^d \rightarrow\R$ be upper semi-continuous. At any given $x_0\in\R^d$, for any $\ve\in(0, \infty)$ there exists a neighborhood $U$ around $x_0$ such that:
\begin{align}
\lp\forall x \in U \rp\lp f(x) \leqslant f(x_0) + \ve\rp
\end{align}
This also means that:
\begin{align}
&\lp\forall x \in U \rp\lp -f(x) \geqslant -f(x_0) - \ve\rp\nonumber\\
\end{align}
This completes the proof.
\end{proof}
\begin{definition}[Degenerate Elliptic Functions] Let $d \in\N$, $T \in\lp0, \infty\rp$, let $\mathcal{O}\subseteq\R^d$ be a non-empty open set, and let $\la\cdot, \cdot\ra: \R^d \times\R^d \rightarrow\R$ be the standard Euclidean inner product on $\R^d$. $G$ is degenerate elliptic on $\lp0,T \rp\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ if and only if:
\begin{enumerate}[label = (\roman*)]
\item$G: \lp0,T \rp\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ is a function, and
\item for all $t \in\lp0 ,T \rp$, $x \in\mathcal{O}$, $r \in\R$, $p \in\R^d$, $A,B \in\mathbb{S}_d$, with $\forall y \in\R^d$: $\la Ay,y \ra\leqslant\la By, y \ra$ that $G(t,x,r,p,A)\leqslant G(t,x,r,p,B)$.
\end{enumerate}
\end{definition}
\begin{remark}
Let $t \in(0,T)$, $x \in\R^d$, $r\in\R$, $p\in\R^d$, $A \in\mathbb{S}_d$. Let $u \in C^{1,2}([0,T]\times\R^d, \R)$, and let $\sigma: \R^d \rightarrow\R^{d\times d}$ and $\mu: \R^d \rightarrow\R^d$ be infinitely often differentiable. The function $G:(0,T)\times\R^d \times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ given by:
where $\lp t,x,u(t,x),\mu(x), \sigma(x)\lb\sigma(x)\rb^*\rp\in\lp0,T\rp\times\R^d \times\R\times\R^d \times\mathbb{S}_d$, is degenerate elliptic.
\end{remark}
\begin{lemma}\label{negdegel}
Given a function $G: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ that is degerate elliptic on $(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ it is also the case that $H: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ given by $H(t,x,r,p,A)=-G(t,x,-r, -p,-A)$ is degenerate elliptic on $(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$.
\end{lemma}
\begin{proof}
Note that $H$ is a function. Assume for $y\in\R^d$ it is the case that $\la Ay, y \ra\leqslant\la By,y \ra$ then it is also the case by (\ref{bigsum}) that $\la-Ay,y\ra\geqslant\la-By, y \ra$ for $y\in\R^d$. However since $G$ is monotoically increasing over the subset of $(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ where $\la Ay,y \ra\leqslant\la By,y \ra$ then it is also the case that $H(t,x,r,p,A)=-G(t,x,-r,-p, -A)\geqslant-G(t,x,-r,-p,-B)=H(t,x,r,p,B)$.
Let $d \in\N$, $T \in\lp0, \infty\rp$, let $\mathcal{O}\subseteq\R^d$ be a non-empty open set, and let $G: \lp0, T \rp\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ be degenrate elliptic. Then we say that $u$ is a viscosity solution of $\lp\frac{\partial}{\partial t} u \rp\lp t,x \rp+ G \lp t,x,u(t,x), \lp\nabla_x u \rp\lp t,x \rp, \lp\Hess_x u \rp\lp t,x \rp\rp\geqslant0$ for $\lp t,x, \rp\in\lp0,T \rp\times\mathcal{O}$ if and only if there exists a set $A$ such that:
\begin{enumerate}[label = (\roman*)]
\item we have that $\lp0,T \rp\times\mathcal{O}\subseteq A$.
\item we have that $u: A \rightarrow\R$ is an upper semi-continuous function from $A$ to $\R$, and
\item we have that for all $t\in\lp0, T \rp$, $x \in\mathcal{O}$, $\phi\in C^{1,2}\lp\lp0,T \rp\times\mathcal{O}, \R\rp$ with $\phi(t,x)= u (t,x)$ and $\phi\geqslant u$ that:
Let $d \in\N$, $T \in\lp0, \infty\rp$, let $\mathcal{O}\subseteq\R^d$ be a non-empty open set, and let $G: \lp0, T \rp\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ be degenrate elliptic. Then we say that $u$ is a viscosity solution of $\lp\frac{\partial}{\partial t} u \rp\lp t,x \rp+ G \lp t,x,u(t,x), \lp\nabla_x u \rp\lp t,x \rp, \lp\Hess_x u \rp\lp t,x \rp\rp\leqslant0$ for $\lp t,x, \rp\in\lp0,T \rp\times\mathcal{O}$ if and only if there exists a set $A$ such that:
\begin{enumerate}[label = (\roman*)]
\item we have that $\lp0,T \rp\times\mathcal{O}\subseteq A$.
\item we have that $u: A \rightarrow\R$ is an upper semi-continuous function from $A$ to $\R$, and
\item we have that for all $t\in\lp0, T \rp$, $x \in\mathcal{O}$, $\phi\in C^{1,2}\lp\lp0,T \rp\times\mathcal{O}, \R\rp$ with $\phi(t,x)= u (t,x)$ and $\phi\leqslant u$ that:
Let $d \in\N$, $T \in\lp0, \infty\rp$, $\mathcal{O}\subseteq\R^d$ be a non-empty open set and let $G: \lp0, T \rp\times\mathcal{O}\times R \times\R^d \times\mathbb{S}_d \rightarrow\R$ be degenerate elliptic. Then we say that $u_d$ is a viscosity solution to $\lp\frac{\partial}{\partial t} u_d \rp(t,x)+ G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))$ if and only if:
\begin{enumerate}[label = (\roman*)]
\item$u$ is a viscosity subsolution of $\lp\frac{\partial}{\partial t} u_d \rp(t,x)+ G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))=0$ for $(t,x)\in(0,T)\times\mathcal{O}$
\item$u$ is a viscosity supersolution of $\lp\frac{\partial}{\partial t} u_d \rp(t,x)+ G(t,x,u(t,x), \nabla_x(x,t), (\Hess_x u_d)(t,x))=0$ for $(t,x)\in(0,T)\times\mathcal{O}$
\end{enumerate}
\end{definition}
%
%\begin{lemma}
% Let $T\in (0, \infty)$, let $\left( \Omega, \mathcal{F}, \mathbb{P} \right)$ be a probability space, let $\sigma_d: \R^d \rightarrow \R^{d\times d}$, $d\in \N$, be infinitely often differentiable functions, let $u_d \in C^{1,2} \left( \left[ 0,T \right] \times \R^d, \R \right)$, $d\in \N$, satisfy for all $d\in \N$, $t \in \left[ 0,T \right]$, $x \in \R^d$ that:
%Let $W^d: [0,T] \times \Omega \rightarrow \R^d$, $d\in \N$, be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: [t,T] \times \Omega \rightarrow \R^d$, $d \in \N$, $t \in [0,T]$, $x \in \R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in \N$, $t\in [0,T]$, $s \in [t,T]$, $x \in \R^d$, we have $\mathbb{P}$-a.s. that:
%Let $\mathfrak{W}^{i,j,d}_{t-s} = \sum^d_{i=1} \left( c_{j, i} W_{i,t-s}\right)$, given that the product of a Brownian motion with a constant is a Brownian motion and the sum of Brownian motion is also a Brownian motion, we have that: $\mathfrak{W}^{i,j,d}_{t-s} = \sum^d_{i=1} \left( c_{j, i} W_{i,t-s}\right)$ is also a Brownian motion.
%\medskip
%
%For each row $j$ we therefore have $x_k + \sqrt{2} \mathfrak{W}^{i,j,d}_{t-s}$
Let $d\in\N$, $T \in\lp0,\infty\rp$, $\mathfrak{t}\in\lp0,T \rp$, let $\mathcal{O}\subseteq\R^d$ be an open set, let $\mathfrak{r}\in\mathcal{O}$, $\phi\in C^{1,2}\lp\lp0,T \rp\times\mathcal{O},\R\rp$, let $G: \lp0,T \rp\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ be degenerate elliptic and let $u_d: (0,T)\times\mathcal{O}\rightarrow\R$ be a viscosity solution of \\
$\lp\frac{\partial}{\partial t} u_d \rp\lp t,x \rp+ G \lp t,x,u(t,x), \lp\nabla_x u_D \rp\lp t,x \rp, \lp\Hess_x u_d \rp\lp t,x \rp\rp\geqslant0$ for $(t,x)\in(0,T)\times\mathcal{O}$, and assume that $u-\phi$ has a local maximum at $(\mathfrak{t}, \mathfrak{r})\in(0,T)\times\mathcal{O}$, then:
That $u$ is upper semi-continuous ensures that there exists as a neighborhood $U$ around $(\mathfrak{t}, \mathfrak{r})$ and $\psi\in C^{1,2}((0,T)\times\mathcal{O},\R)$ where:
\begin{enumerate}[label = (\roman*)]
\item for all $(t,x)\in(0,T)\times\mathcal{O}$ that $u(\mathfrak{t}, \mathfrak{r})-\psi(\mathfrak{t}, \mathfrak{r})\geqslant u(t,x)-\psi(t,x)$
\item for all $(t,x)\in U$ that $\phi(t,x)=\phi(t,x)$.
Let $d \in\N$, $T \in(0, \infty)$, let $\mathcal{O}\subseteq\R^d$ be a non-empty open set, let $u_n:(0,T)\times\mathcal{O}\rightarrow\R$, $n \in\N_0$ be functions, let $G_n: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$, $n \in\N$ be degenerate elliptic, assume that $G_\infty$ is upper semi-continuous for all non-empty compact $\mathcal{K}\subseteq(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ that:
Let $(t_0, x_0)\in(0,T)\times\mathcal{O}$. Let $\phi_\epsilon\in C^{1,2}((0,T)\times\mathcal{O}, \R)$ satisfy for all $\epsilon\in(0, \infty)$, $s \in(0,T)$, $y \in\mathcal{O}$ that $\phi_0(t_0,x_0)= u_0(t_0,x_0)$, $\phi_0(t_0,x_0)\geqslant u_0(t_0,x_0)$, and:
\phi_\varepsilon(s,y) = \phi_o(s,y) + \varepsilon\lp\lv s - t_0 \rv + \| y - x_0 \|_E \rp
\end{align}
Let $\delta\in(0,\infty)$ be such that $\{(s,y)\in\R^d \times\R: \max\lp|s-t_0|^2, \|y-x_0\|_E^2\rp\leqslant\delta\}$. Note that this and (\ref{limitofun})
then imply for all $\varepsilon\in(0,\infty)$ there exists an $\nu_\varepsilon\in\N$ such that for all $n \geqslant\nu_\varepsilon$, and $\max\lp|s-t_0|, \|y-x_0\|_E \rp\leqslant\delta$, it is the case that:
Note that this combined with (\ref{phieps}) tells us that for all $\varepsilon\in(0,\infty)$, $n \in\N\cap[\nu_\epsilon, \infty)$, $s\in(0,T)$, $y\in\mathcal{O}$, with $|s-t_0| < \delta$, $\|y-x_0\|_E \leqslant\delta$, $|s-t_0| +\|y-x_0\|_E > \delta$ that:
Note that Corollary \ref{sumofusc} implies that for all $\epsilon\in(0,\infty)$ and $n\in\N$ that $u_n -\phi_\varepsilon$ is upper semi-continuous. There therefore exists for all $\epsilon\in(0,\infty)$ and $n \in\N$, a $\tau^\varepsilon_n \in(t_0-\delta, t_0+\delta)$ and a $\rho^\varepsilon_n$, where $\|\rho_n^\varepsilon- x_0\|\leqslant\delta$ such that:
However note also that since $G_0$ is upper semi-continuous, further the fact that, $\phi_0\in\lp\lp0,T\rp\times\mathcal{O}, \R\rp$, and then $(\ref{limitofun})$, and $(\ref{phieps})$, imply for all $\ve\in(0,\infty)$ we have that:
This establishes $(\ref{hessungeq0})$ which establishes the lemma.
\end{proof}
\begin{corollary}\label{unleq0}
Let $d \in\N$, $T \in(0, \infty)$, let $\mathcal{O}\subseteq\R^d$ be a non-empty open set, let $u_n:(0,T)\times\mathcal{O}\rightarrow\R$, $n \in\N_0$ be functions, let $G_n: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$, $n \in\N_0$ be degenerate elliptic, assume that $G_0$ is lower semi-continuous for all non-empty compact $\mathcal{K}\subseteq(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ that:
Let $v_n:(0,T)\times\mathcal{O}\rightarrow\R$, $n \in\N_0$ and $H_n:(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ satisfy for all $n\in\N_0$, $t\in(0,T)$, $x \in\mathcal{O}$, $r\in\R$, $p \in\R^d$, $A \in\mathbb{S}_d$ that $v_n(t,x)=-u_n(t,x)$ and that $H_n(t,x)=-G_n(t,x,-r,-p,-A)$.
\medskip
Note that Corollary \ref{neglsc} gives us that $H_0$ is upper semi-continuous. Note also that since it is the case that for all $n\in\N_0$, $G_n$ is degenerate elliptic then it is also the case by Lemma \ref{negdegel} that $H_n$ is degenerate elliptic for all $n\in\N_0$. These together with (\ref{viscsolutionvn}) ensure that for all $n\in\N$, $v_n$ is a viscosity solution of:
Let $d \in\N$, $T \in(0,\infty)$, let $\mathcal{O}\subseteq\R^d$ be a non-empty set, let $u_n: (0,T)\times\mathcal{O}\rightarrow\R$, $n \in\N_0$, be functions, let $G_n: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$, $n \in\N_0$ be degenerate elliptic, assume also that $G_0: (0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ be consinuous and assume for all non-empty compact $\mathcal{K}\subseteq(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ it is the case that:
Let $d,m \in\N$, $T \in(0,\infty)$. Let $\mathcal{O}\subseteq\R^d$ be a non-empty compact set, and for all $n\in\N_0$, $\mu_n \in C([0,T]\times\mathcal{O}, \R)$, $\sigma_n \in C([0,T]\times\mathcal{O}, \R^{d \times m})$
Let $\lp\Omega, \mathcal{F}, \mathbb{R}\rp$ be a stochastic basis and let $W: [0,T]\times\Omega\rightarrow\R^m$ be a standard $(\mathbb{F}_t)_{t\in[0,T]}$-Brownian motion for every $t\in[0,T]$, $x \in\mathcal{O}$, let $\mathcal{X}^{t,x}=(\mathcal{X}^{t,x}_s)_{s\in[t,T]}: [t,T]\times\Omega\rightarrow\R^d$ be an $(\mathbb{F}_s)_{s\in[t,T]}$ adapted stochastic process with continuous sample paths, satisfying for all $s \in[t, T]$ we have $\mathbb{P}$-a.s.
\begin{align}\label{xnasintuvxn}
\mathcal{X}^{n,t,x}_s = x + \int^s_t \mu_n(r,\mathcal{X}^{n,t,x}_s) dr + \int^s_t \sigma_n(r,\mathcal{X}^{n,t,x}_r) dW_r
&\lp\int^s_t \E\lb\left\|\sigma_n(r,\mathcal{X}^{n,t,x}_r) - \sigma_0(r,\mathcal{X}^{0,t,x})\right\|_F^2 \rb dr \rp^\frac{1}{2}
\end{align}
Applying Lemma \ref{absq} followed by the Cauchy-Schwarz Inequality then gives us for all $n \in\N$, $t\in[0,T]$, $s \in[t,T]$, and $x\in\mathcal{O}$ that:
Applying $\limsup_{n\rightarrow\infty}$ to both sides and applying (\ref{limsupis0}) gives us for all $n \in\N$, $t \in[0,T]$, $s\in[t,T]$, $x \in\mathcal{O}$ that:
Let $d,m \in\N$, $T \in(0,\infty)$, let $\mathcal{O}\subseteq[0,T]\times\R^d$, let $\mu\in C([0,T]\times\mathcal{O},\R^d)$ and $\sigma\in C([0,T]\times\mathcal{O}, \R^{d\times m})$ have compact supports such that $\supp(\mu)\cup\supp(\sigma)\subseteq[0,T]\times\mathcal{O}$
let $g\in C(\R^d,\R)$. Let $\lp\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t\in[0,T]}\rp$ be a stochastic basis, let $W:[0,T]\times\Omega\rightarrow\R^m$ be a standard $(\mathbb{F}_t)_{t\in[0,T]}$ Brownian motion, for every $t\in[0,T]$, $x\in\R^d$, let $\mathcal{X}^{t,x}=(\mathcal{X}^{t,x}_s)_{s\in[t,T]}: [t,T]\times\Omega\rightarrow\R^d$ be an $(\mathbb{F}_s)_{s\in[t,T]}$ adapted stochastic process with continuous sample paths satisfying for all $s\in[t,T]$ with $\mathbb{F}$-a.s. that:
\begin{align}\label{2.59}
\mathcal{X}^{t,x}_s = x + \int^s_t \mu\lp r, \mathcal{X}^{t,x}_r \rp dr + \int^s_t \sigma\lp r, \mathcal{X}^{t,x}_r \rp dW_r
\end{align}
and further let $u:[0,T]\times\R^d \rightarrow\R$ satisfy for all $t\in[0,T]$, $x\in\R^d$ that:
and where $u(T,x)= g(x)$ for $(t,x)\in(0,T)\times\mathcal{O}$.
\end{lemma}
\begin{proof}
Let $\mathcal{S}=\supp(\mu)\cup\supp(\sigma)\subseteq[0,T]\times\mathcal{O}$ be bounded in space by $\rho\in(0,\infty)$, as $\mathcal{S}\subseteq[0,T]\times(-\rho, \rho)^d$. This exists as the supports are compact and thus by Hiene-B\"orel is closed and bounded. Let $\mathfrak{s}_n, \mathfrak{m}_n \in C^\infty([0,T]\times\R^d, \R^{d\times n})$ where $\bigcup_{n\in\N}\lb\supp(\mathfrak{s}_n)\cup\supp(\mathfrak{m}_n)\rb\subseteq[0,T]\times(-\rho, \rho)^d$ satisfy for $n\in\N$ that:
We construct a set of degenerate elliptic functions, $G^{n}:(0,T)\times\R^d \times\R\times\R^d \times\mathbb{S}_d \rightarrow\R$ with $n \in\N_0$ such that:
Further let $\mathfrak{X}^{n,t,x}=(\mathfrak{X}^{n,t,x}_s)_{s\in[t,T]}:[t,T]\times\Omega\rightarrow\R^d$ be an $(\mathbb{F}_s)_{s\in[t,T]}$-adapted stochastic process with continuous sample paths that satisfy:
Note that \cite[Lemma~2.2]{BHJ21} with $g \curvearrowleft\mathfrak{g}_k$, $\mu\curvearrowleft\mathfrak{m}_n$, $\sigma\curvearrowleft\mathfrak{s}_n$, $\mathcal{X}^{t,x}\curvearrowleft\mathcal{X}^{n,t,x}$ gives us $\mathfrak{u}^n \in C^{1,2}([0,T]\times\R^d, \R)$, and $\mathfrak{u}^n(t,x)=\mathfrak{g}_k(x)$ where:
And by Definitions \ref{def:viscsubsolution}, \ref{def:viscsupsolution}, and \ref{def:viscsolution} we have that $\mathfrak{u}^n$ is a viscosity solution of
Since for all $n\in\N$, it is the case that $\mathcal{S}=\lp\supp(\mathfrak{m}_n)\cup\supp(\mathfrak{s}_n)\cup\supp(\mu)\cup\supp(\sigma)\rp\subseteq[0,T]\times(-\rho, \rho)^d$ and because of (\ref{2.59}) of (\ref{2.66}) we have that \cite[Lemma~3.2, Item~(ii)]{Beck_2021} which yields that for all $n\in\N$, $t\in[0,T]$, $x \in\R^d\setminus(-\rho, \rho)^d$ that $\mathbb{P}(\forall s \in[t,T]: \mathfrak{X}^{n,t,x}_s =x=\mathcal{X}^{t,x}_s)=1$. This in turn shows that for all $n\in\N$, $x \in\R^d \setminus(-\rho,\rho)^d$ that $\mathfrak{u}^n(t,x)=\mathfrak{u}^0(t,x)$ which along with (\ref{ungn}) and (\ref{u0gn}) yields that:
But since we've established (\ref{2.62}) we have that for a non-empty compact set $\mathcal{C}\subseteq(0,T)\times\mathcal{O}\times\R\times\R^d \times\mathbb{S}_d$ that:
for $(t,x)\in[0,T]\times\R^d$. Finally (\ref{2.59}) and (\ref{2.60}) allows us to conclude that for all $x \in\R^d$ it is the case that $u(T,x)= g(x)$. This concludes the proof.
\end{proof}
\begin{lemma}
Let $d,m \in\N$, $T \in(0,\infty)$, further let $\mathcal{O}\subseteq\R^d$ be a non, empty compact set. Let every $r \in(0,\infty)$ satisfy the condition that $O_r \subseteq\mathcal{O}$, where $O_r =\{x \in\mathcal{O}: \lp\|x\|_E \leqslant r \text{ and }\{y \in\R^d: \| y-x\|_E < \frac{1}{r}\}\subseteq\mathcal{O}\rp\}$ let $g \in C(\mathcal{O},\R)$, $\mu\in C([0,T]\times\mathcal{O},\R)$, $V \in C^{1,2}([0,T]\times\mathcal{O},(0,\infty))$, assume that for all $t \in[0,T]$, $x \in\mathcal{O}$ that:
\begin{align}
\sup\lp\left\{\frac{\|\mu(t,x)-\mu(t,y)\|_E+\|\sigma(t,x)-\sigma(t,y)\|_F}{\|x-y\|_E}:t\in [0,T], x,y\in O_r, x \neq y \right\}\cup\{0\}\rp < \infty
assume that $\sup_{r \in(0,\infty)}\lb\inf_{x \in\mathcal{O}\setminus O_r} V(t,x)\rb=\infty$ and $\inf_{r \in(0, \infty)}\lb\sup_{t \in[0,T]}\sup_{x \in\mathcal{O}\setminus O_r}\lp\frac{g(x)}{V(T,x)}\rp\rb=0$.
Let $\lp\Omega, \mathcal{F}, \mathbb{P}, (\mathbb{F}_t)_{t\in[0,T]}\rp$ be a stochastic basis and let $W: [0,T]\times\Omega\rightarrow\R^m$ be a standard $(\mathbb{F}_t)_{t \in[0,T]}$-Brownian motion, for every $t \in[0,T]$, $x \in\mathcal{O}$ let $\mathcal{X}^{t,x}=(\mathcal{X}^{t,x}_s)_{s\in[t,T]}: [t,T]\times\Omega\rightarrow\mathcal{O}$ be an $(\mathbb{F}_s)_{s\in[t,T]}$-adapted stochastic process with continuous sample paths satisfying that for all $s\in[t, T]$, we have $\mathbb{P}$-a.s. that:
\begin{align}\label{2.79}
\mathcal{X}^{t,x}_s = x+\int^s_t \mu(r,\mathcal{X}^{t,x}_r) dr + \int^s_t \sigma(r, \mathcal{X}^{t,x}_n) dW_r
\end{align}
also let $u:[0,T]\times\R^d \rightarrow\R$ satisfy for all $t \in[0,T]$, $x \in\R^d$ that:
\begin{align}
u(t,x) = \E\lb u(T,\mathcal{X}^{t,x}_T) \rb
\end{align}
It is then the case that $u$ is a viscosity solution to:
for $(t,x)\in(0,T)\times\mathcal{O}$ with $u(T,x)= g(x)$.
\end{lemma}
\begin{proof}
Let it be the case, that throughout the proof, for $n\in\N$, we have that $\mathfrak{g}_n \in C(\R^d, \R)$, compactly supported and that $\lb\bigcup_{n\in\N}\supp(\mathfrak{g}_m)\rb\subseteq[0, T]\times\mathcal{O}$ and further that:
Let is also be the case that for $n\in\N$, $\mathfrak{m}_n \in C([0,T]\times\R^d, \R^d)$ and $\mathfrak{s}_n \in C([0,T]\times\R^d, \R^{d \times m})$ satisfy:
Next for every $n\in\N$, $t\in[0,T]$ and $x\in\R^d$ let it be the case that $\mathfrak{X}^{n,t,x}_s =(\mathfrak{X}^{n,t,x}_s)_{s\in[t,T]}: [t,t]\times\Omega. \rightarrow\R^d$ be a stochastic process with continuous sample paths satisfying:
\begin{align}\label{2.86}
\mathfrak{X}^{n,t,x}_s = x + \int^s_t \mathfrak{m}_n(r, \mathfrak{X}^{n,t,x}_s) dr + \int^s_t \mathfrak{s}_n(r, \mathfrak{X}^{n,t,x}_s) dW_r
\end{align}
Let $\mathfrak{u}^n: [0,T]\times\R^d \rightarrow\R$, $k \in\N$, $n \in\N_0$, satisfy for all $n\in\N$, $t \in[0,T]$, $x \in\R^d$ that:
and finally let, for every $n\in\N$, $t \in[0,T]$, $x \in\mathcal{O}$, there be $\mathfrak{t}^{t,x}_n: \Omega\rightarrow[t,T]$ which satisfy $\mathfrak{t}^{t,x}_n =\inf\lp\{ s \in[t,T], \max\{V(s,\mathfrak{X}^{t,x}_s),V(s,\mathcal{X}^{t,x}_s)\}\geqslant n \}\cup\{T\}\rp$. We may apply Lemma \ref{2.19} with $\mu\curvearrowleft\mathfrak{m}_n$, $\sigma\curvearrowleft\mathfrak{s}_n$, $g \curvearrowleft\mathfrak{g}_k$ to show that for all $n,k \in\N$ we have that $\mathfrak{u}^{n,k}$ is a viscosity solution to:
&\leqslant 2 \lb\sup_{y \in\mathcal{O}}\lv\mathfrak{g}_k(y) \rv\rb\mathbb{P}\lp\mathfrak{t}^{t,x}_n < T \rp\nonumber
\end{align}
Note that this combined with \cite[Lemma~3.1]{Beck_2021} implies for all $t \in[0,T]$, $x \in\mathcal{O}$, $n \in\N$ we have that $\E\lb V \lp\mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n}\rp\rb\leqslant V(t,x)$, which then further proves that:
\begin{align}
\lv\mathfrak{u}^{n,k}(t,x) - \mathfrak{u}^{0,k}(t,x) \rv&\leqslant 2 \lb\sup_{y\in\mathcal{O}}\lv\mathfrak{g}_k(y) \rv\rb\mathbb{P}\lp\mathfrak{t}^{t,x}_n < T \rp\nonumber\\
&\leqslant 2 \lb\sup_{y \in\mathcal{O}}\lv\mathfrak{g}_k(y) \rv\rb\mathbb{P}\lp V \lp\mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n}\rp\geqslant n\rp\nonumber\\
%TODO: Ask Dr. Padgett how this came about
&\leqslant\frac{2}{n}\lb\sup_{y\in\mathcal{O}}\lv\mathfrak{g}_k(y) \rv\rb\E\lb V \lp\mathfrak{t}^{t,x}_n, \mathcal{X}^{t,x}_{\mathfrak{t}^{t,x}_n}\rp\rb\nonumber\\
&\leqslant\frac{2}{n}\lb\sup_{y \in\mathcal{O}}\lv\mathfrak{g}_k(y) \rv\rb V \lp t,x, \rp\nonumber
\end{align}
Together these imply that for all $k\in\N$ and compact $\mathcal{K}\subseteq[0,T]\times\mathcal{O}$:
But again note that since have that $\sup_{r\in(0,\infty)}\lb\inf_{t \in[0,T], x \in\R^d \setminus O_r} V(t,x)\rb=\infty$ and (\ref{2.84}) tell us that for all compact $\mathcal{K}\subseteq[0,T]\times\mathcal{O}$ we have that:
Note that (\ref{2.89}), (\ref{2.90}) and Corollary \ref{unneq0} tell us that for all $k \in\N$ we have that $\mathfrak{u}^{0,k}$ is a viscosity solution to:
for $(t,x)\in(0,T)\times\mathcal{O}$. However note that (\ref{2.79}),(\ref{2.82}), (\ref{2.88}) prove that for all compact $\mathcal{K}\subseteq[0,T]\times\mathcal{O}$ we have:
This together with (\ref{2.93}), (\ref{2.82}), Corollary \ref{unneq0} shows that $u_0$ is a viscosity solution to:
\begin{align}\label{2.95}
\lp\frac{\partial}{\partial t} u \rp\lp t,x \rp + \frac{1}{2}\Trace\lp\sigma(t,x) \lb\sigma(t,x) \rb^*\lp\Hess_x u \rp (t,x) \rp + \la\mu(t,x), \lp\nabla_x u \rp\ra = 0
\end{align}
for $(t,x)\in(0,T)\times\mathcal{O}$. By (\ref{2.81}) we are ensured that for all $x\in\R^d$ we have that $u(T,x)= g(x)$ which together with proves the proposition.
% From Feynman-Kac, especially from \cite[(1.5)]{hutzenthaler_strong_2021} and setting $f=0$ in the notation of \cite[(1.5)]{hutzenthaler_strong_2021} we have that:
% Looking closely at (2.3), where according to the notation of (2.2), the time reversal property of Brownian motions, (2.5), and the shift-invariance of Brownian motions we have that:
% The independence of the Brownian motions then indicates that (2.5) and (2.6) are equal.
% This completes the proof of Lemma 2.1.
% \end{proof}
\begin{theorem}[Existence and characterization of $u_d$]\label{thm:3.21}
Let $T \in(0,\infty)$. Let $\lp\Omega, \mathcal{F}, \mathbb{P}\rp$ be a probability space. Let $\sigma_d \in C \lp\R^d,\R^{d\times d}\rp$ and $\mu_d\in C \lp\R^d, \R^d \rp$ for $d \in\N$, let $u_d \in C^{1,2}\lp\lb0,T \rb\times\R^d, \R\rp$ satisfy for all $d \in\N$, $t \in\lb0,T \rb$ , $x \in\R^d$ that:
let $\mathcal{W}^d:[0,T]\times\Omega\rightarrow\R^d$, $d \in\N$ be a standard Brownian motions and let $\mathcal{X}^{d,t,x}: \lb t, T\rb\times\Omega\rightarrow\R^d$, $d \in\N$, $ t\in\lb0,T \rb$, be a stochastic process with continuous sample paths satisfying for all $d \in\N$, $t \in\lb0,T \rb$, $s \in\lb t,T \rb$, $x \in\R^d$, we have $\mathbb{P}$-a.s. that:
\begin{align}\label{3.102}
\mathcal{X}^{d,t,x} = x + \int^t_s \mu_d \lp\mathcal{X}^{d,t,x}_r \rp dr + \int^t_s \sigma\lp\mathcal{X}^{d,t,x}_r \rp d\mathcal{W}^d_r
\end{align}
Then for all $d \in N$, $t \in\lb0,T \rb$, $x \in\R$, it holds that:
\begin{corollary}\label{lem:3.19} Let $T \in(0,\infty)$,\\ let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space, let $u_d \in C^{1,2}\left(\left[0,T \right]\times\R^d, \R\right)$, $d \in\N$ satisfy for all $d \in\N$, $t \in[0,T]$, $x \in\R^d$ that:
Let $\mathcal{W}^d: [0,T]\times\Omega\rightarrow\R^d$, $d \in\N$ be standard Brownian motions, and let $\mathcal{X}^{d,t,x}: [t,T]\times\Omega\rightarrow\R^d$, $d\in\N$, $t \in[0,T]$, $x \in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d \in N$, $t \in[0,T]$, $s \in[t,T]$, $x \in\R^d$ we have $\mathbb{P}$-a.s. that:
\begin{align}
\mathcal{X}^{d,t,x}_s = x + \int^s_t d\mathcal{W}^d_r = x + \mathcal{W}^d_{t-s}
\end{align}
Then for all $d\in\N$, $t \in[0,T]$, $x \in\R^d$ it holds that:
This is a special case of Theorem \ref{thm:3.21}. It is the case where $\sigma_d(x)=\mathbb{I}_d$, the uniform identity function where $\mathbb{I}_d$ is the identity matrix in dimension $d$ for $d \in\N$, and $\mu_d(x)=\mymathbb{0}_{d}$ where $\mymathbb{0}_d$ is the zero vector in dimension $d$ for $d \in\N$.
Let $T \in\lp0,\infty\rp$, let $\lp\Omega, \mathcal{F}, \mathbb{P}\rp$, be a probability space, let $\alpha_d\in C^2_b \lp\R^d,\R\rp$, and $\alpha\in\mathcal{O}\lp x^2\rp$ for $d \in N$, be infinitey often differentiable function, let $u_d \in C^{1,2}\lp\lb0,T \rb\times\R^d,\R\rp$, $d \in\N$, satisfy for all $d \in\N$, $t \in\lb0,T \rb$, $x \in\R^d$, that:
Let $\mathcal{W}^d: \lb0,T \rb\times\Omega\rightarrow\R^d$ be standard Brownian motions and let $\mathcal{X}^{d,t,x}: \lb t,T \rb\times\Omega\rightarrow\R^d$, $d \in\N$, $t \in\lb0,T \rb$, $x \in\R^d$ be a stochastic process with continuous sample paths satisfying that for all $d \in\N$, $ t\in\lb0,T \rb$, $s \in\lp t,T \rb$, $x \in\R^d$, we have $\mathbb{P}$-a.s. that:
\begin{align}
\mathcal{X}^{d,t,x}_s = x + \int^t_s \frac{1}{2} d \mathcal{W}^d_r =\frac{1}{2}\mathcal{W}^d_{t-r}
\end{align}
Then for all $d \in\N$, $t \in\lb0,T \rb$, $x \in\R^d$ it holds that:
\begin{proof} Let $v_d: \R^d \rightarrow\R$ be continuous. Throughout the proof let $u_d \lp t,x\rp= e^{-t\alpha_d(x)}v_d(t,x)$ for all $d \in\N$, $t \in[0,T]$, $x \in\R^d$. For notational simplicity, we will drop the $d,t,x$ wherever it is obvious. Therefore the derivatives become:
% v_t = 2t\alpha_xv_x - tv \lb t(\alpha_x)^2-\alpha_{xx} \rb &-v_{xx}
%\end{align}
Let $\sigma(t,x)=\mathbb{I}_d$, i.e. the uniform identity function. Let $\mu(t,x)=-t\nabla_x \alpha$ for $t \in[0,T], x\in\R^d$, and for fixed $\alpha$. Let $f(t,x,v)=-\frac{1}{2}tv\nabla_x^2\alpha$ for $t \in[0,T], x \in\R^d$.
\begin{claim}
It is the case that for for all $x \in\R^d$ and $t \in[0,T]$ that $\la x, \mu(t,x)\ra\leqslant L \lp1+\|x\|_E \rp$ for some constant $L \in(0,\infty)$.
\end{claim}
\begin{proof}
Since $\alpha$ has bounded first and second derivatives let:
&\leqslant T \lp\| x \|_E \mathfrak{B}\rp\nonumber\\
&\leqslant T \lp\mathfrak{B} +d \rp\|x\|_E \nonumber\\
&= L\| x\|_E \leqslant L\lp 1 + \|x \|_E^2 \rp
\end{align}
It also follows that $\|\sigma(t,x)\|_F =\sqrt{d}\leqslant L \leqslant L(1+\|x\|_E)$.
\end{proof}
\begin{claim}
It is the case that for all $x,y \in\R^d$, and $t \in[0,T]$ that: $\|\mu(t,x)-\mu(t,y)\|_E +\|\sigma(t,x)-\sigma(t,y)\|_E \leqslant\mathfrak{C}\lp\|x\|_E+\|y\|_E \rp\lp\|x-y\|_E\rp$ for some constant $\mathfrak{C}\in(0,\infty)$.
\end{claim}
\begin{proof}
The fact that for all $x,y \in\R^d$ and $t \in[0,T]$ it is the case that $\|\sigma(t,x)-\sigma(t,y)\|_F=0$, the fact that for all $x,y \in\R^d$ it is the case that $(\|x\|_E+\|y\|_E)(\|x-y\|_E)\geqslant0$ and (\ref{3.3.13}) tells us that:
Now consider a function $\mathfrak{f}\in C \lp[0,T]\times\R^d,\R^d \rp$, where for all $x,y \in\R^d$ it is the case that $\mathfrak{f}(x)-\mathfrak{f}(y)\leqslant\mathscr{C}\lp\|x\|_E+\|y\|_E \rp\lp\|x+y\|_e \rp$. Note then that setting $y=x+h$ gives us:
\left| \nabla_x\mathfrak{f}\lp x \rp\right| &\leqslant 2\mathscr{C}\|x\|_E = \mathscr{K}\|x\|_E
\end{align}
This suggests that $\nabla_x\mathfrak{f}\in O \lp x \rp$ and in particular that $\mathfrak{f}\in O \lp x^2\rp$. However with $\mathfrak{f}\curvearrowleft\mu$ we first notice that because $\mu\leqslant2T\mathfrak{B}$ in (\ref{3.3.15}) it must also be that case that $\mu\in O(1)$ by Corollary \ref{1.1.20.1}. However since $O(c)\subseteq O(x)\subseteq O \lp x^2\rp$ by Corollary \ref{1.1.20.2} it is also the case that $\mu\in O \lp x^2\rp$, and hence there exists a $\mathfrak{C}$ satisfying the claim. This proves the claim.
\end{proof}
\begin{claim}\label{3.3.5}
It is the case that $\lv f(t,x,v)- f(t,x,w)\rv\leqslant L \lv v-w \rv$
\end{claim}
\begin{proof}
Note that by the absolute homogeneity property of norms, we have:
We realize that (\ref{3.3.12}) is a case of \cite[Corollary~3.9]{bhj20} where it is the case that: $u(t,x)\curvearrowleft v(t,x)$, where $\sigma_d(x)=\mathbb{I}_d$ for all $x \in\R^d$, $d \in\N$, where $\mu(t,x)=-t\nabla_x\alpha$ for fixed $\alpha$ and for all $t \in[0,T]$, $x \in\R^d$, and where $f \lp t,x,u \lp t,x \rp\rp=-\frac{1}{2}tu\nabla_x^2\alpha$ for fixed $\alpha$ and for all $t\in[0,T]$, $x \in\R^d$.
We thus have that there exists a unique, at most polynomially growing viscosity solution $v \in C\lp\lb0,T \rb\times\R^d, \R\rp$ given as:
\begin{align}
v(t,x) &= \E\lb v \lp T, \mathcal{Y}^{t,x}_T \rp+ \int^T_t f \lp s,\mathcal{Y}^{t,x}_s, v \lp s,\mathcal{Y}^{t,x}_s \rp\rp ds \rb\label{3.2.21}
\end{align}
Let $\mathcal{V}: \lb0,T \rb\times\Omega\rightarrow\R^m$ be a standard $\lp\mathbb{F}_t\rp_{t \in\lb0,T \rb}$-Brownian motion. Note that this also implies that the $\mathcal{Y}$ in (\ref{3.2.21}) is characterized as:
\begin{align}
\mathcal{Y}^{t,x}_s = x + \int^s_t \mu\lp r,\mathcal{Y}^{t,x}_r \rp dr + \int^s_t \sigma\lp s, \mathcal{X}^{t,x}_r \rp d\mathcal{V}_r
\end{align}
With substitution, this is then:
\begin{align}
\mathcal{Y}^{t,x}_s &= x + \int^s_t -r\nabla_x\alpha\lp\mathcal{Y}^{t,x}_r \rp dr + \int^s_t \mathbb{I} d\mathcal{V}_r \nonumber\\
\mathcal{Y}^{t,x}_s &=x - \int^s_t r\nabla_x\alpha\lp\mathcal{Y}^{t,x}_s \rp dr + \mathcal{V}_{s-t}\nonumber
\end{align}
Note that our initial substitution tells us: $v(t,x)= e^{t\alpha(x)}u(t,x)$. And so we have that:
\begin{align}
v(t,x) &= \E\lb v\lp T, \mathcal{X}_T^{t,x}\rp + \int^T_t f \lp s, \mathcal{X}^{t,x}_s, v \lp s, \mathcal{X}^{t,x}_s \rp\rp ds\rb\label{3.3.20}\\