---
abstract: 'We show that the value function in a stochastic differential game does not change if we keep the same space $(\Omega,{\mathcal{F}})$ but introduce probability measures by means of Girsanov’s transformation [*depending*]{} on the policies of the players.  We also show that the value function does not change if we allow the driving Wiener processes to depend on the policies of the players.  Finally, we show that the value function does not change if we perform a random time change with the rate depending on the policies of the players.' 
address: '127 Vincent Hall, University of Minnesota, Minneapolis, MN, 55455'
author:
- 'N.V. Krylov'
title: On the independence of the value function for stochastic differential games of the probability space
---

[^1]

Introduction
============

Let ${\mathbb{R}}^{d}=\{x=(x^{1},...,x^{d})\}$ be a $d$-dimensional Euclidean space and let $d_{1}\geq d$ be an integer.  Assume that we are given separable metric spaces $A$ and $B$, and let, for each $\alpha\in A$, $\beta\in B$, the following functions on ${\mathbb{R}}^{d}$ be given:

\(i) $d\times d_{1}$ matrix-valued $\sigma^{\alpha\beta}( x)
=\sigma(\alpha,\beta, x)=
(\sigma^{\alpha\beta}_{ij}( x))$,

\(ii) ${\mathbb{R}}^{d}$-valued $b^{\alpha\beta}( x)=
b(\alpha,\beta, x)=
(b^{\alpha\beta}_{i }( x))$, and

\(iii) real-valued functions $c^{\alpha\beta}( x)=c(\alpha,\beta, x)\geq0$, $f^{\alpha\beta}( x)=f(\alpha,\beta, x)$, and $g(x)$.

Under natural assumptions which will be specified later, on a probability space $(\Omega,{\mathcal{F}},P)$ carrying a $d_{1}$-dimensional Wiener process $w_{t}$ one associates with these objects and a bounded domain $G\subset{\mathbb{R}}^{d}$ a stochastic differential game with the diffusion term $\sigma^{\alpha\beta}(x)$, drift term $b^{\alpha\beta}(x)$, discount rate $c^{\alpha\beta}(x)$, running cost $f^{\alpha\beta}(x)$, and the final cost $g(x)$ payed when the underlying process first exits from $G$.

After the order of players is specified in a certain way it turns out (see our Remark \[remark 3.14.1\]) that the value function $v(x)$ of this differential game is a unique continuous in $\bar{G}$ viscosity solution of the Isaacs equation $$\label{3.14.3}
H[v]=0$$ in $G$ with boundary condition $v=g$ on $\partial G$, where for a sufficiently smooth function $u=u(x)$ $$\label{1.16.1}
H[u](x)={\operatornamewithlimits{sup\,\,\,inf}}_{\alpha\in A\,\,\beta\in B}
[L^{\alpha\beta} u(x)+f^{\alpha\beta} (x)],$$ $$L^{\alpha\beta} u(x):=a^{\alpha\beta}_{ij}( x)D_{ij}u(x)+
b ^{\alpha\beta}_{i }( x)D_{i}u(x)-c^{\alpha\beta} ( x)u(x),$$ $$a^{\alpha\beta}( x):=(1/2)\sigma^{\alpha\beta}( x)
(\sigma^{\alpha\beta}( x))^{*},\quad
D_{i}=\partial/\partial x^{i},\quad D_{ij}=D_{i}D_{j}.$$

We will assume that $\sigma$ and $b$ are uniformly Lipschitz with respect to $x$, $\sigma\sigma^{*}$ is uniformly nondegenerate, and $c$ and $f$ are uniformly bounded.  In such a situation uniqueness of continuous viscosity solutions or even continuous $L_{p}$ viscosity solutions of is shown in [@JS05] and therefore the fact of the independence of $v$ of the probability space seems to be obvious. 

Roughly speaking, the goal of this paper is to show that the value function does not change even if we keep the same space $(\Omega,{\mathcal{F}})$ but introduce probability measures by means of Girsanov’s transformation [*depending*]{} on the policies of the players.  We also show that the value function does not change if we allow the driving Wiener processes to depend on the policies of the players.  Finally, we show that the value function does not change if we perform a random time change with the rate depending on the policies of the players. 

These facts are well known for controlled diffusion processes and play there a very important role, in particular, while estimating the derivatives of the value function.  A rather awkward substitute of them for stochastic differential games was used for the same purposes in [@Kr_14_1].  Applying the results presented here one can make many constructions in [@Kr_14_1] more natural and avoid introducing auxiliary “shadow” processes. 

However, not all proofs in [@Kr_14_1] can be simplified using our present methods.  We deliberately avoided discussing the way to use the external parameters in contrast with [@Kr_14_1] just to make the presentation more transparent. 

Our proofs do not use anything from the theory of viscosity solutions and are based on a version of Świ[ȩ]{}ch’s ([@Sw96]) idea as presented in [@Kr_13_1] and a general solvability theorem in class $C^{1,1}$ of Isaacs equations from [@Kr_12]. 

The article is organized as follows.  In Section \[section 2.26.3\] we present our main result, Theorem \[theorem 1.14.1\].  We prove it in Section \[section 2.11.1\] under the additional assumption that the corresponding Isaacs equation has a smooth solution.  Then in Section \[section 4.6.1\] we allow the solutions to belong to the Sobolev class $W^{2}_{d}$. Section \[section 2.25.1\] contains a general approximation result, which allows us in Section \[section 4.6.2\] to use a result from [@Kr_12] (see Theorem \[theorem 9.23.1\]) and conclude the proof of Theorem \[theorem 1.14.1\] in the general case. 

Main result 
============

\[section 2.26.3\]

We start with our assumptions. 

\[assumption 1.9.1\]

\(i) The functions $\sigma^{\alpha\beta}(x)$, $b^{\alpha\beta}(x)$, $c^{\alpha\beta}(x)$, and $f^{\alpha\beta}(x)$ are continuous with respect to $\beta\in B$ for each $(\alpha, x)$ and continuous with respect to $\alpha\in A$ uniformly with respect to $\beta\in B$ for each $x$. The function $g(x)$ is bounded and continuous. 

\(ii) The functions $c^{\alpha\beta}( x )$ and $f^{\alpha\beta}( x )$ are uniformly continuous with respect to $x$ uniformly with respect to $(\alpha,\beta )\in A\times B $ and for any $x \in{\mathbb{R}}^{d}$ and $(\alpha,\beta )\in A\times B $ $$\|\sigma^{\alpha\beta}( x )\|,|b^{\alpha\beta}( x )|,
|c^{\alpha\beta}( x )|,|f^{\alpha\beta}( x )|
\leq K_{0},$$ where $K_{0}$ is a fixed constants and for a matrix $\sigma$ we denote $\|\sigma\|^{2}
={\text{\rm tr}\,}\sigma\sigma^{*}$,

\(iii) For any $(\alpha,\beta )\in A\times B $ and $x,y\in{\mathbb{R}}^{d}$ we have $$\|\sigma^{\alpha\beta}( x )-\sigma^{\alpha\beta}( y )\|+
|b^{\alpha\beta}( x )-b^{\alpha\beta}( y )|\leq K_{0}|x-y|.$$

Let $(\Omega,{\mathcal{F}},P)$ be a complete probability space, let $\{{\mathcal{F}}_{t},t\geq0\}$ be an increasing filtration of $\sigma$-fields ${\mathcal{F}}_{t}\subset {\mathcal{F}}$ such that each ${\mathcal{F}}_{t}$ is complete with respect to ${\mathcal{F}},P$.

The set of progressively measurable $A$-valued processes $\alpha_{t}=\alpha_{t}(\omega)$ is denoted by ${\mathfrak{A}}$. Similarly we define ${\mathfrak{B}}$ as the set of $B$-valued progressively measurable functions.  By $ {\mathbb{B}}$ we denote the set of ${\mathfrak{B}}$-valued functions $ {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})$ on ${\mathfrak{A}}$ such that, for any $T\in(0,\infty)$ and any $\alpha^{1}_{\cdot},
\alpha^{2}_{\cdot}\in{\mathfrak{A}}$ satisfying $$\label{4.5.4}
P(  \alpha^{1}_{t}=\alpha^{2}_{t} 
 \quad\text{for almost all}\quad t\leq T)=1,$$ we have $$P(  {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(\alpha^{1}_{\cdot})={\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(\alpha^{2}_{\cdot}) 
\quad\text{for almost all}\quad t\leq T)=1.$$

A function $p^{\alpha_{\cdot}\beta_{\cdot}}_{t}=
p^{\alpha_{\cdot}\beta_{\cdot}}_{t}(\omega)$ given on ${\mathfrak{A}}\times{\mathfrak{B}}\times\Omega\times[0,\infty)$ with values in some measurable space is called [*a control adapted process*]{} if, for any $(\alpha_{\cdot},\beta_{\cdot})\in{\mathfrak{A}}\times{\mathfrak{B}}$, it is progressively measurable in $(\omega,t)$ and, for any $T\in(0,\infty)$, we have $$P( p^{\alpha^{1}_{\cdot}\beta^{1}_{\cdot}}_{t}
=p^{\alpha^{2}_{\cdot}\beta^{2}_{\cdot}}_{t}
\quad\text{for almost all}\quad t\leq T)=1$$ as long as $$P(  \alpha^{1}_{t}=\alpha^{2}_{t} ,
\beta^{1}_{t}=\beta^{2}_{t}
\quad\text{for almost all}\quad t\leq T)=1.$$

\[assumption 2.9.1\] For each $\alpha_{\cdot}\in{\mathfrak{A}}$ and $\beta_{\cdot}\in{\mathfrak{B}}$ we are given control adapted processes

\(i) $w^{\alpha_{\cdot}\beta_{\cdot}}_{t},t\geq0$, which are standard $d_{1}$-dimensional Wiener process relative to to the filtration $\{{\mathcal{F}}_{t},t\geq0\}$,

\(ii) $r^{\alpha_{\cdot}\beta_{\cdot}}_{t},t\geq0$, and $\pi^{\alpha_{\cdot}\beta_{\cdot}}_{t},t\geq0$, which are real-valued and ${\mathbb{R}}^{d_{1}}$-valued, respectively,

\(iii) for all values of the arguments $$\delta_{1}^{-1}\geq r^{\alpha_{\cdot}\beta_{\cdot}}_{t}\geq\delta_{1},\quad
|\pi^{\alpha_{\cdot}\beta_{\cdot}}_{t}|\leq K_{1},$$ where $\delta_{1}>0$ and $K_{1}\in(0,\infty)$ are fixed constants. 

Finally we introduce $$a^{\alpha\beta}( x):=(1/2)\sigma^{\alpha\beta}( x)
(\sigma^{\alpha\beta}( x))^{*},$$ fix a domain $G\subset{\mathbb{R}}^{d}$, and impose the following. 

\[assumption 3.19.1\] $G$ is a bounded domain of class $C^{2}$ and there exists a constant $\delta\in(0,1)$ such that for any $\alpha\in A$, $\beta\in B$, and $x,\lambda\in {\mathbb{R}}^{d}$ $$\delta|\lambda|^{2}\leq a^{\alpha\beta}_{ij}( x)
\lambda^{i}\lambda^{j}\leq \delta^{-1}|\lambda|^{2}.$$

\[remark 2.22.2\]

As is well known, if Assumption \[assumption 3.19.1\] is satisfied, then there exists a bounded from above $\Psi\in   C^{2}_{loc}({\mathbb{R}}^{d})$ such that $\Psi>0$ in $G$, $\Psi=  0$ on $\partial G$, and for all $\alpha\in A$, $\beta\in B$, and $x\in G$ $$\label{3.20.1}
  L^{\alpha\beta}\Psi( x)
+c^{\alpha\beta}\Psi( x)\leq-1.$$

For $\alpha_{\cdot}\in{\mathfrak{A}}$, $ \beta_{\cdot} 
\in{\mathfrak{B}}$, and $x\in{\mathbb{R}}^{d}$ consider the following Itô equation $$x_{t}=x+\int_{0}^{t}
r^{\alpha_{\cdot}\beta_{\cdot}}_{s}
\sigma^{\alpha_{s}\beta_{s} }(  x_{s})\,dw^{\alpha_{\cdot}\beta_{\cdot}}_{s}$$ $$\label{5.11.1}
+\int_{0}^{t}[r^{\alpha_{\cdot}\beta_{\cdot}}_{s}]^{2}\big[b^{\alpha_{s}
\beta_{s} }(  x_{s})+\sigma^{\alpha_{s}\beta_{s} }(  x_{s})
\pi^{\alpha_{\cdot}\beta_{\cdot}}_{s}\big]\,ds.$$

Observe that equation satisfies the usual hypothesis, that is for any $\alpha_{\cdot}\in{\mathfrak{A}}$, $ \beta_{\cdot} 
\in{\mathfrak{B}}$, $x\in{\mathbb{R}}^{d}$, and $T\in(0,\infty)$ it has a unique solution on $[0,T]$ denoted by $x^{\alpha_{\cdot} 
\beta_{\cdot} x}_{t} $ and $x^{\alpha_{\cdot} 
\beta_{\cdot} x}_{t} $ is a control adapted process for each $x$.

Set $$\phi^{\alpha_{\cdot}\beta_{\cdot} x}_{t}
=\int_{0}^{t}[r^{\alpha_{\cdot}\beta_{\cdot}}_{s}]^{2}
c^{\alpha_{s}
\beta_{s} }( x^{\alpha_{\cdot}\beta_{\cdot}  x}_{s}) 
\,ds,$$ $$\psi^{\alpha_{\cdot}\beta_{\cdot} x}_{t}
=-(1/2)
\int_{0}^{t}[r^{\alpha_{\cdot}\beta_{\cdot}}_{s}]^{2}
|\pi^{\alpha_{\cdot}\beta_{\cdot}}_{s}|^{2}
\,ds-\int_{0}^{t}
r^{\alpha_{\cdot}\beta_{\cdot}}_{s}
\pi^{\alpha_{\cdot}\beta_{\cdot}}_{s}\,dw^{\alpha_{\cdot}\beta_{\cdot}}_{s},$$ define $\tau^{\alpha_{\cdot}\beta_{\cdot} x}$ as the first exit time of $x^{\alpha_{\cdot} 
\beta_{\cdot} x}_{t}$ from $G$, and introduce $$\label{2.12.2}
v(x)={\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}\big[\int_{0}^{\tau}
r_{t}^{2}f( x_{t})e^{-\phi_{t}
-\psi_{t}}\,dt+g(x_{\tau})e^{-\phi_{\tau}
-\psi_{\tau}}\big],$$ where the indices $\alpha_{\cdot}$, ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}$, and $x$ at the expectation sign are written to mean that they should be placed inside the expectation sign wherever and as appropriate, that is $$E_{x}^{\alpha_{\cdot}\beta_{\cdot}}\big[\int_{0}^{\tau}r^{2}_{t}
f( x_{t})e^{-\phi_{t}-\psi_{t}}\,dt+g(x_{\tau})e^{-\phi_{\tau}
-\psi_{\tau}}\big]$$ $$:=
E \big[
 g(x^{\alpha_{\cdot}\beta_{\cdot}  x}
_{\tau^{\alpha_{\cdot}\beta_{\cdot}  x}}
)
e^{-\phi^{\alpha_{\cdot}\beta_{\cdot}  x}
_{\tau^{\alpha_{\cdot}\beta_{\cdot}  x}}
-\psi^{\alpha_{\cdot}\beta_{\cdot}  x}
_{\tau^{\alpha_{\cdot}\beta_{\cdot}  x}}}$$ $$+\int_{0}^{\tau^{\alpha_{\cdot}\beta_{\cdot}  x}}
[r^{\alpha_{\cdot}\beta_{\cdot}  x}_{t}]^{2}
f^{\alpha_{t}\beta_{t}   }
( 
x^{\alpha_{\cdot}\beta_{\cdot}  x}_{t})
e^{-\phi^{\alpha_{\cdot}\beta_{\cdot}  x}_{t}
-\psi^{\alpha_{\cdot}\beta_{\cdot}  x}_{t}}\,dt\big].$$ Observe that, formally, the value $x_{\tau}$ may not be defined if $\tau=\infty$. In that case we set the corresponding terms to equal zero.  The above definitions make perfect sense due to our Remark \[remark 2.20.1\]. 

Here is our main result. 

\[theorem 1.14.1\]

Under the above assumptions the function $v(x)$ is independent of the choice of the probability space, filtration and control adapted process $(r,\pi,w)^{\alpha_{\cdot}\beta_{\cdot}}_{t}$, it is bounded and continuous in $\bar{G}$.

\[remark 3.14.1\] Once we know that $v(x)$ is independent of the choice of the probability space, filtration and control adapted process $(r,\pi,w)^{\alpha_{\cdot}\beta_{\cdot}}_{t}$, we can take any probability space carrying a $d_{1}$-dimensional Wiener process $w_{t}$ and construct $v(x)$ by setting $w^{\alpha_{\cdot}\beta_{\cdot}}_{t}
=w_{t}$, $r\equiv1$, $\pi\equiv0$. In that case we are in the position to apply the results of [@FS89], [@Ko09], and [@Kr_13] according to which $v$ is continuous in $\bar{G}$ and satisfies the dynamic programming principle.  Then it is a standard fact that $v$ is a viscosity solution of (see, for instance, [@FS89], [@Ko09], [@Sw96]).  Indeed, if a smooth function $\psi(x)$ is such that $\psi(x)\geq v(x)$ in a neighborhood of $x_{0}\in G$ and $\psi(x_{0})= v(x_{0})$, then by defining $\gamma^{\alpha_{\cdot}\beta_{\cdot}}_{\varepsilon}$, $\varepsilon>0$ as the first exit time of $x^{\alpha_{\cdot}\beta_{\cdot}x_{0}}_{t}$ from an $\varepsilon$-neighborhood of $x_{0}$ for all small $\varepsilon$ we have $$\psi(x_{0})=v(x_{0})
={\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x_{0}}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}
\big[\int_{0}^{\gamma_{\varepsilon}}
 f( x_{t})e^{-\phi_{t}}\,dt+v(x_{\gamma_{\varepsilon}})
e^{-\phi_{\gamma_{\varepsilon}}}\big]$$ $$\leq {\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x_{0}}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}
\big[\int_{0}^{\gamma_{\varepsilon}}
 f( x_{t})e^{-\phi_{t}}\,dt+\psi(x_{\gamma_{\varepsilon}})
e^{-\phi_{\gamma_{\varepsilon}}}\big].$$ On the other hand set $H[\psi]=-h$ and observe that by Theorem 4.1 of [@Kr_13_1] $$\psi(x_{0})=
{\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x_{0}}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}
\big[\int_{0}^{\gamma_{\varepsilon}}
 (f+h)( x_{t})e^{-\phi_{t}}\,dt+\psi(x_{\gamma_{\varepsilon}})
e^{-\phi_{\gamma_{\varepsilon}}}\big].$$ It follows that $$\label{3.14.4}
{\operatornamewithlimits{inf\,\,\,inf}}_{\alpha_{\cdot}\in{\mathfrak{A}}\,\,\,\beta_{\cdot}\in{\mathfrak{B}}}
E_{x_{0}}^{\alpha_{\cdot}\beta _{\cdot}}
 \int_{0}^{\gamma_{\varepsilon}}
 h( x_{t})e^{-\phi_{t}}\,dt\leq0,$$ and if we assume that $H[\psi](x_{0})<0$, then $h>0$ in an $\varepsilon$-neighborhood of $x_{0}$ and is impossible, since $c$ is bounded and $\sigma$ and $b$ are bounded so that $E_{x_{0}}^{\alpha_{\cdot}\beta _{\cdot}}
\gamma_{\varepsilon}$ is bounded away from zero.  Hence $H[\psi](x_{0})\geq0$ and $v$ is a viscosity subsolution by definition.  Similarly one shows that it is a viscosity supersolution. 

Provided that we know that continuous viscosity solutions are unique the above argument proves the fact that the value function is independent of the probability space (if we drop out $r$ and $\pi$ and take $w$ independent of the policies).  Jensen [@Je_88] proved uniqueness for [*Lipschitz*]{} continuous viscosity solutions to the fully nonlinear second order elliptic PDE not explicitly depending on $x$ on a bounded domain.  Related results in the same year with $H$ depending on $x$ were published in Jensen-Lions-Souganidis [@JLS_88]. 

In what concerns uniformly nondegenerate Isaacs equations, Trudinger in [@Tr_89] proves the existence and uniqueness of continuous viscosity solutions for Isaacs equations if the coefficients are continuous and $a$ is $1/2$ Hölder continuous uniformly with respect to $\alpha,\beta$ (see Corollary 3.4 there).  Uniqueness is also stated for Isaacs equations with Lipschitz continuous $a$ as Corollary 5.11 in [@Je_98].  Jensen and Świ[ȩ]{}ch in [@JS05] further relaxed the requirement on $a$ and proved uniqueness of continuous even $L_{p}$-viscosity solutions. 

We will use Theorem \[theorem 1.14.1\] to prove in a subsequent article a result to state which we need a few new objects.  In the end of Section 1 of [@Kr_12] a function $P(u_{ij},u_{i},u)$ is constructed defined for all symmetric $d\times d$ matrices $(u_{ij})$, ${\mathbb{R}}^{d}$-vectors $(u_{i})$, and $u\in{\mathbb{R}}$ such that it is positive-homogeneous of degree one, is Lipschitz continuous, and at all points of differentiability of $P$ for all values of arguments we have $P_{u}\leq0$ and $$\hat\delta|\lambda|^{2}\leq P_{u_{ij}}
\lambda^{i}\lambda^{j}\leq \hat{\delta}^{-1}|\lambda|^{2},$$ where $\hat{\delta}$ is a constant in $(0,1)$ depending only on $d,K_{0}$, and $\delta$. For smooth enough functions $u(x)$ introduce $$P[u](x)=P(D_{ij}u(x),D_{i}u(x),u(x))$$

We now state part of Theorem 1.1 of [@Kr_12] which we need even in the present article. 

\[theorem 9.23.1\] Let $g\in C^{1,1}({\mathbb{R}}^{d})$. Then for any $K\geq0$ the equation $$\label{9.23.2}
\max(H[u],P[u]-K)=0$$ in $G$ (a.e.)  with boundary condition $u=g$ on $\partial G$ has a unique solution $u\in C^{0,1}(\bar{G})\cap C^{1,1}_{loc}(G)$.

The result we are aiming at in a subsequent article consists of proving the conjecture stated in [@Kr_12]:

\[theorem 1.14.01\]

Denote by $u_{K}$ the function from Theorem \[theorem 9.23.1\] and assume that $G$ and $g$ are of class $C^{3}$. Then there exists a constant $N$ such that $|v-u_{K}|\leq N/K$ on $G$ for $K\geq1$.

A very week version of this theorem was already used in [@Kr_14_2] for establishing a rate of convergence of finite-difference approximations for solutions of Isaacs equations. 

We finish this section with a useful technical result. 

\[lemma 2.20.1\]

For any $\alpha_{\cdot}\in{\mathfrak{A}}$, $\beta_{\cdot}
\in{\mathfrak{B}}$, and $x\in{\mathbb{R}}^{d}$ the process $$\exp(-\psi_{t\wedge\tau^{\alpha_{\cdot}\beta_{\cdot}x}}
^{\alpha_{\cdot}\beta_{\cdot}x})$$ is a uniformly integrable martingale on $[0,\infty)$. Furthermore, there exists a constant $N$ independent of $\alpha_{\cdot}\in{\mathfrak{A}}$, $\beta_{\cdot}
\in{\mathfrak{B}}$, and $x\in{\mathbb{R}}^{d}$ such that $$\label{2.20.1}
E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
\int_{0}^{
\tau}
e^{-\psi_{s}}\,ds\leq N.$$ Finally, $$\label{2.20.2}
E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{\tau}}=1.$$

Proof.  Owing to by Itô’s formula $$E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
G(x_{t\wedge\tau})e^{-\psi_{t\wedge\tau}}
=G(x)+E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
\int_{0}^{t\wedge\tau}
r^{2}_{s}[LG+cG](x_{s})e^{-\psi_{s}}\,ds$$ $$\leq
 G(x)-\delta_{1}^{2}
E^{\alpha_{\cdot}\beta_{\cdot}}_{x}\int_{0}^{
t\wedge\tau}
e^{-\psi_{s}}\,ds,$$ and follows.  To prove use that $$1=E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{t\wedge\tau}}
=E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{\tau}}I_{\tau\leq t}
+E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{t}}I_{\tau> t},$$ where the last term decreases as $t$ increases, which is seen from the formula, and tends to zero as $t\to\infty$ since its integral with respect to $t$ over $[0,
\infty)$ is finite being equal to the left-hand side of . 

Finally, the first assertion of the lemma follows from due to the well-known properties of martingales.  The lemma is proved. 

\[remark 2.20.1\] In light of the proof of Lemma \[lemma 2.20.1\] $$0=\lim_{t\to\infty}E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{t}}I_{\tau> t}
=\lim_{t\to\infty}E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{\tau}}I_{\tau> t}=
E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
e^{-\psi_{\tau}}I_{\tau=\infty}.$$ Hence defining the terms containing $x_{\tau}$ as zero on the set where $\tau=\infty$ is indeed natural.  Lemma \[lemma 2.20.1\] shows that the function $v$ is well defined and one can rewrite its definition as $$v(x)={\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}\big[\int_{0}^{\tau}
r_{t}^{2}f( x_{t})e^{-\phi_{t}
}\,dt+g(x_{\tau})\big]e^{-\psi_{\tau}},$$ which calls for changes of probability measure by using Girsanov’s theorem. 

Proof of Theorem \[theorem 1.14.1\] in case that the Isaacs equation has a smooth solution
==========================================================================================

\[section 2.11.1\]

In this section we replace Assumption \[assumption 1.9.1\] (iii) with a weaker one. 

\[assumption 2.16.1\]

The functions $\sigma^{\alpha\beta}( x )$ and $b^{\alpha\beta}( x )$ are uniformly continuous with respect to $x$ uniformly with respect to $(\alpha,\beta )\in A\times B $. 

However, this time there is no guarantee that equation has a unique solution and we impose the following. 

\[assumption 5.23.2\] Equation satisfies the usual hypothesis, that is for any $\alpha_{\cdot}\in{\mathfrak{A}}$, $ \beta_{\cdot} 
\in{\mathfrak{B}}$, $x\in{\mathbb{R}}^{d}$, and $T\in(0,\infty)$ it has a unique solution on $[0,T]$ denoted by $x^{\alpha_{\cdot} 
\beta_{\cdot} x}_{t} $ and $x^{\alpha_{\cdot} 
\beta_{\cdot} x}_{t} $ is a control adapted process for each $x$.

We also assume that we are given two functions $\hat{u},\check{u} \in C^{2}(\bar{G})$.

\[theorem 2.11.1\]

\(i) If $H[\hat{u}]\leq0$ in $G$ and $\hat{u}\geq g$ on $\partial G$, then $v\leq\hat{u}$ in $\bar{G}$.

\(ii) If $H[\check{u}]\geq0$ in $G$ and $\check{u}\leq g$ on $\partial G$, then $v\geq\check{u}$ in $\bar{G}$.

\(iii) If $\hat{u}$ and $\check{u}$ are as in (i) and (ii) and $\hat{u}=\check{u}$, then $v$ is independent of the choice of the probability space, filtration, $r,\pi$, and $w$.

We need three lemmas. 

\[lemma 2.10.1\] Set $\kappa_{n}(t)=[nt]/n$. Then there exists a constant $N$ such that for all $n\geq1,x\in{\mathbb{R}}^{d},
\alpha_{\cdot}\in{\mathfrak{A}},\beta_{\cdot}\in{\mathfrak{B}}$ we have $$\label{2.10.3}
E^{\alpha_{\cdot}\beta_{\cdot}}_{x}\int_{0}^{\tau}
e^{-\phi_{t}-\psi_{t}}|x_{t}-x_{\kappa_{n}(t)}|^{2}\,dt\leq N/n.$$

Proof.  For each fixed $t$ while estimating $$E^{\alpha_{\cdot}\beta_{\cdot}}_{x} 
e^{-\phi_{t}-\psi_{t}}
|x_{t}-x_{\kappa_{n}(t)}|^{2}I_{\tau>t}$$ Girsanov’s theorem allows us to assume that $\pi\equiv 0$. In that case for simplicity of notation we will drop the indices $x,\alpha_{\cdot},\beta_{\cdot}$ and observe that $$|x_{t}-x_{\kappa_{n}(t)}|^{2}\leq 2\big|\int_{k/n}^{t}
\sigma(x_{s})\,dw_{s}\big|^{2}+
\big|\int_{k/n}^{t}
b(x_{s})\,ds\big|^{2}$$ so that for $t\in[k/n,(k+1)/n]$ $$E^{\alpha_{\cdot}\beta_{\cdot}}_{x}\{
e^{-\phi_{t} }|x_{t}-x_{k/n}|^{2}\mid{\mathcal{F}}_{k/n}\}
\leq  E^{\alpha_{\cdot}\beta_{\cdot}}_{x}\{
 |x_{t}-x_{k/n}|^{2}\mid{\mathcal{F}}_{k/n}\}\leq N/n,$$ where $N$ depends only on $d$ and $K_{0}$. Hence, owing also to the left-hand side of is dominated by $$\int_{0}^{\infty}E^{\alpha_{\cdot}\beta_{\cdot}}_{x} 
e^{-\phi_{t}}|x_{t}-x_{\kappa_{n}(t)}|^{2}I_{\tau>t}\,dt
\leq \int_{0}^{\infty}E^{\alpha_{\cdot}\beta_{\cdot}}_{x} 
 |x_{t}-x_{\kappa_{n}(t)}|^{2}I_{\tau>
\kappa_{n}(t)}\,dt$$ $$\leq Nn^{-1} \int_{0}^{\infty}E^{\alpha_{\cdot}\beta_{\cdot}}_{x} 
 I_{\tau>
\kappa_{n}(t)}\,dt=
Nn^{-1}  E^{\alpha_{\cdot}\beta_{\cdot}}_{x} 
(\tau+1/n)\leq Nn^{-1}.$$ The lemma is proved. 

For a stopping time $\gamma$ we say that a process $\xi_{t}$ is a submartingale on $[0,\gamma]$ if $\xi_{t\wedge\gamma}$ is a submartingale.  Similar definition applies to supermartingales. 

The proof of the following lemma and Lemma \[lemma 1.17.2\] follows a version of Świ[ȩ]{}ch’s ([@Sw96]) idea as it is presented in [@Kr_13_1]. 

\[lemma 1.16.1\] Let $H[\hat{u}]\leq0$ in $G$. Then for any $x\in{\mathbb{R}}^{d}$, $\alpha_{\cdot}\in{\mathfrak{A}}$, and $\varepsilon>0$, there exist a sequence $\beta^{n}_{\cdot}(\alpha_{\cdot})=
\beta^{n}_{\cdot}(\alpha_{\cdot},x,\varepsilon)
\in{\mathfrak{B}}$, $n=1,2,...$, and a sequence of increasing continuous $\{{\mathcal{F}}_{t}\}$-adapted processes $\eta^{n\varepsilon}_{t}(\alpha_{\cdot})
=\eta^{n\varepsilon}_{t}(\alpha_{\cdot},x)$ with $\eta^{n\varepsilon}_{0}(\alpha_{\cdot})=0$ such that $$\label{2.29.1}
\sup_{n}E 
\eta^{n\varepsilon}_{\infty}(\alpha_{\cdot})<\infty,$$ the processes $$\kappa^{n\varepsilon}_{t  }(\alpha_{\cdot}):=
\hat{u}(x^{n}_{t})
e^{-\phi^{n}_{t}-\psi^{n}_{t}}
-\eta^{n\varepsilon}_{t} (\alpha_{\cdot})
+\int_{0}^{t}[r^{n}_{t}]^{2}
f^{n}_{s} ( x^{n}_{s}) 
e^{-\phi^{n}_{s}-\psi^{n}_{s}}\,ds,$$ where $$\label{5.6.1}
(x^{n}_{t},\phi^{n}_{t},\psi^{n}_{t})
=(x_{t},\phi_{t},\psi_{t})
^{\alpha_{\cdot}\beta^{n}_{\cdot}(\alpha_{\cdot})  x},\quad
f^{n}_{t}( x)=f^{\alpha_{t}\beta^{n}_{t}(\alpha_{\cdot})}( x),\quad
r^{n}_{t}=r^{\alpha_{\cdot}\beta^{n}_{\cdot}(\alpha_{\cdot})}_{t}$$ are supermartingales on $[0,\tau^{\alpha_{\cdot}\beta^{n}_{\cdot}(\alpha_{\cdot})
  x}]$, and $$\label{3.3.1}
{\operatornamewithlimits{\overline{lim}}}_{n\to\infty}
\sup_{\alpha_{\cdot}\in{\mathfrak{A}}}E 
 \eta^{n\varepsilon}_{\tau}(\alpha_{\cdot})\leq
 N \varepsilon  ,$$ where $N$ is independent of $x$ and $\varepsilon$. Finally, $$\label{2.29.3}
\sup_{\alpha_{\cdot}\in{\mathfrak{A}}}\sup_{n}
E 
\sup_{t\geq0}|
\kappa^{n\varepsilon}_{t\wedge\tau }(\alpha_{\cdot})|<\infty .$$

Proof.  Since $B$ is separable and $a^{\alpha\beta},b^{\alpha\beta},
c^{\alpha\beta}$, and $f^{\alpha\beta}$ are continuous with respect to $\beta$ one can replace $B$ in with an appropriate countable subset $B_{0}=\{\beta_{1},\beta_{2},...\}$. Then for each $\alpha\in A$ and $x\in G$ define $\beta(\alpha,x)$ as $\beta_{i}\in B_{0}$ with the least $i$ such that $$\label{1.17.2}
0
\geq L^{\alpha\beta_{i}} \hat{u}(x)+f^{\alpha\beta_{i}} 
(x)-\varepsilon.$$ For each $i$ the right-hand side of is Borel in $x$ and continuous in $\alpha$. Therefore, it is a Borel function of $(\alpha,x)$, implying that $\beta(\alpha,x)$ also is a Borel function of $(\alpha,x)$. For $x\not\in G$ set $\beta(\alpha,x)=\beta^{*}$, where $\beta^{*}$ is a fixed element of $B$. Then we have that in $G$ $$\label{1.17.3}
0
\geq L^{\alpha\beta(\alpha,x)} \hat{u}(x)
+f^{\alpha\beta(\alpha,x)} (x)-\varepsilon.$$

After that fix $x$, define $\beta^{n0}_{t}(\alpha_{\cdot})=\beta(\alpha_{t},x)$, $t\geq0$, and for $k\geq1$ introduce $\beta^{nk}_{t}(\alpha_{\cdot})$ recursively so that $$\label{4.5.2}
\beta^{nk}_{t}(\alpha_{\cdot})=\beta^{n(k-1)}_{t}(\alpha_{\cdot})
\quad\text{for}\quad t<k/n,$$ $$\beta^{nk}_{t}(\alpha_{\cdot})=\beta(\alpha_{t},x^{nk }_{k/n})
\quad\text{for}\quad t\geq k/n,$$ where $x^{nk}_{t}$, $k=1,2,...$, is a unique solution of $$x _{t}=x+\int_{0}^{t}
r^{\alpha_{\cdot}\beta^{n(k-1)}_{\cdot}(\alpha_{\cdot})}_{s}
\sigma(\alpha_{s},\beta^{n(k-1)}_{s}(\alpha_{\cdot}),
 x_{s})\,
dw^{\alpha_{\cdot}\beta^{n(k-1)}_{\cdot}(\alpha_{\cdot})}_{s}$$ $$\label{4.5.1}
+\int_{0}^{t}
[r^{\alpha_{\cdot}\beta^{n(k-1)}_{\cdot}} _{s}]^{2}
\big[b(\alpha_{s},\beta^{n(k-1)}_{s}(\alpha_{\cdot}),x_{s}
)+\sigma(\alpha_{s},\beta^{n(k-1)}_{s}(\alpha_{\cdot}),
 x_{s})\pi^{\alpha_{\cdot}\beta^{n(k-1)}_{\cdot}}_{s}
\big]\,
ds.$$

To show that the above definitions make sense, observe that, by Assumption \[assumption 5.23.2\], $x^{n1}_{t}$ is well defined for all $t$. Therefore, $\beta^{n1}_{t}(\alpha_{\cdot})$ is also well defined, and by induction we conclude that $x^{nk}_{t}$ and $\beta^{nk}_{t}(\alpha_{\cdot}) $ are well defined for all $k\geq 1$.

Furthermore, owing to it makes sense to define $$\beta^{n}_{t}(\alpha_{\cdot})=\beta^{nk}_{t}(\alpha_{\cdot})
\quad\text{for}\quad t<k/n.$$ Notice that by definition $x^{n}_{t}:=x^{\alpha_{\cdot}\beta^{n}_{\cdot}(\alpha_{\cdot})x}_{t}$ satisfies the equation $$x _{t}=x+\int_{0}^{t}
r^{\alpha_{\cdot}\beta^{n }_{\cdot}(\alpha_{\cdot})}_{s}
\sigma(\alpha_{s},\beta^{n }_{s}(\alpha_{\cdot}),
 x_{s})\,
dw^{\alpha_{\cdot}\beta^{n }_{\cdot}(\alpha_{\cdot})}_{s}$$ $$\label{4.6.1}
+\int_{0}^{t}
[r^{\alpha_{\cdot}\beta^{n }_{\cdot}} _{s}]^{2}
\big[b(\alpha_{s},\beta^{n }_{s}(\alpha_{\cdot}),x_{s}
)+\sigma(\alpha_{s},\beta^{n }_{s}(\alpha_{\cdot}),
 x_{s})\pi^{\alpha_{\cdot}\beta^{n }_{\cdot}(\alpha_{\cdot})}_{s}
\big]\,
ds.$$ For $t<k/n$ we have $\beta^{n}_{t}(\alpha_{\cdot})
=\beta^{n(k-1)}_{t}(\alpha_{\cdot})$, so that for $t\leq k/n$ equation coincides with owing to the fact that $r^{\alpha_{\cdot}
\beta_{\cdot}}_{t},\pi^{\alpha_{\cdot}
\beta_{\cdot}}_{t}$, and $w^{\alpha_{\cdot}
\beta_{\cdot}}_{t}$ are control adapted.  It follows that (a.s.)  $$x_{t}^{n}=x_{t}^{n}(\alpha_{\cdot})=
x^{nk}_{t}\quad\text{for all}\quad t\leq k/n,$$ so that (a.s.)  $$\beta^{nk}_{t}(\alpha_{\cdot})=\beta(\alpha_{t},x^{n}_{k/n})$$ for all $t\geq k/n$. Therefore, if $(k-1)/n\leq t<k/n$, then $$\beta^{n}_{t}(\alpha_{\cdot})=\beta^{n(k-1)}_{t}
(\alpha_{\cdot})=\beta(\alpha_{t},x^{n}_{(k-1)/n}),$$

$$\label{1.19.1}
\beta^{n}_{s}:=\beta^{n}_{s}(\alpha_{\cdot})=\beta(\alpha_{s},
x^{n}_{\kappa_{n}(s)}),$$

and $x_{t}^{n}$ satisfies $$x^{n}_{t}=x+\int_{0}^{t}
r^{n}_{s}
\sigma(\alpha_{s},\beta(\alpha_{s},
x^{n}_{\kappa_{n}(s)}),
 x^{n}_{s})\,
dw^{n          }_{s}$$ $$\label{1.17.1}
+\int_{0}^{t}
[r^{n} _{s}]^{2}
\big[b(\alpha_{s},\beta(\alpha_{s},
x^{n}_{\kappa_{n}(s)}),x^{n}_{s}
)+\sigma(\alpha_{s},\beta(\alpha_{s},
x^{n}_{\kappa_{n}(s)}),
 x^{n}_{s})\pi^{n}_{s}
\big]\,
ds.$$ with $(r,\pi,w)^{n}_{s}
=(r,\pi,w)^{\alpha_{\cdot}\beta^{n }_{\cdot} }_{s}$.

Introduce $\tau^{n}=\tau^{n}(\alpha_{\cdot})$ as the first exit time of $x^{n}_{t}=x^{n}_{t}(\alpha_{\cdot})$ from $G$ and set $$\phi^{n}_{t}=\phi^{\alpha_{\cdot}
\beta^{n}_{\cdot}x}_{t},\quad
\psi^{n}_{t}=\psi^{\alpha_{\cdot}
\beta^{n}_{\cdot}x}_{t} .$$

Observe that by Itô’s formula $$\label{2.10.1}
\hat{u}(x^{n}_{t\wedge\tau^{n}})
e^{-\phi^{n}_{t\wedge \tau^{n}}
-\psi^{n}_{t\wedge \tau^{n}}}
=\hat{u}(x)+\int_{0}^{t\wedge\tau^{n}}
[r^{n} _{s}]^{2}e^{-\phi^{n}_{s}-\psi^{n}_{s}}
 L^{\alpha_{s}\beta^{n}_{s}}
\hat{u}( x^{n}_{s})
\,ds+m^{n}_{t} ,$$ where $m^{n}_{s}$ is a martingale.  Here according to our assumptions on the uniform continuity in $x$ of the data and $D_{ij}\hat u(x)$ we have that for $s<\tau^{n}$ (notice the change of $x^{n}_{s}$ to $x^{n}_{\kappa_{n}(s)}$) $$L^{\alpha_{s}\beta^{n}_{s}}
\hat{u}(x^{n}_{s})\leq  
  a_{ij}(\alpha_{s},\beta(\alpha_{s},
x^{n}_{\kappa_{n}(s)}),x^{n}_{\kappa_{n}(s)})
D_{ij}\hat{u}(x^{n}_{\kappa_{n}(s)})$$ $$+  b_{i}(\alpha_{s},\beta(\alpha_{s},x^{n}_{\kappa_{n}(s)}),x^{n}_{\kappa_{n}(s)})
D_{i}\hat{u}(x^{n}_{\kappa_{n}(s)})$$ $$-  c(\alpha_{s},\beta(\alpha_{s},x^{n}_{\kappa_{n}(s)}),x^{n}_{\kappa_{n}(s)})
 \hat{u}(x^{n}_{\kappa_{n}(s)}) 
+\chi 
(x^{n}_{s}-x^{n}_{\kappa_{n}(s)}).$$ where $\chi (y)$ is a (nonrandom) bounded function on ${\mathbb{R}}^{d}$ such that $\chi (y)\to0$ as $y\to0$. All such functions will be denoted by $\chi $ even if they may change from one occurrence to another. 

Then shows that, for $s<\tau^{n}$, $$L^{\alpha_{s}\beta^{n}_{s}}
\hat{u}(x^{n}_{s})\leq \varepsilon+\chi 
(x^{n}_{s}-x^{n}_{\kappa_{n}(s)})
 - f(\alpha_{s},\beta(\alpha_{s}(x^{n}_{\kappa_{n}(s)}),
x^{n}_{\kappa_{n}(s)})$$ $$\leq \varepsilon+\chi 
(x^{n}_{s}-x^{n}_{\kappa_{n}(s)}) 
  -  f^{\alpha_{s}\beta^{n}_{s}}
( x^{n}_{s}),$$ which along with implies that, for $$\eta^{n\varepsilon}_{t}=\eta^{n\varepsilon}_{t}(\alpha_{\cdot})=
\delta_{1}^{-2}\int_{0}^{t\wedge\tau^{n}}
e^{-\phi^{n}_{s}-\psi^{n}_{s}}[
\varepsilon+
 \chi (x^{n}_{s}-x^{n}_{\kappa_{n}(s)})] \,ds ,$$ $$\label{1.17.5}
\kappa^{n\varepsilon}_{t\wedge\tau^{n}} 
 =\zeta^{n\varepsilon}_{t}+m^{n}_{t},$$ where $\zeta^{n\varepsilon}_{t}$ is a decreasing process. 

Hence $\kappa^{n\varepsilon}_{t\wedge\tau^{n}}$ is at least a local supermartingale.  Owing to Lemmas \[lemma 2.20.1\] and \[lemma 2.10.1\], and hold.  It follows that the local supermartingale $\kappa^{n\varepsilon}_{t\wedge\tau^{n}}$ is, actually, a supermartingale. 

Furthermore, Lemmas \[lemma 2.20.1\] and \[lemma 2.10.1\], the boundedness of $\chi$, its continuity, and the fact that $\chi(0)=0$ easily yield that $$\label{1.17.6}
 \sup_{\alpha_{\cdot}\in{\mathfrak{A}}}E\int_{0}^{\tau^{n}(\alpha_{\cdot})}
e^{-\phi^{n}_{s}-\psi^{n}_{s}}
\chi (x^{n}_{s}(\alpha_{\cdot})-x^{n}_{\kappa_{n}(s)}
(\alpha_{\cdot})) \,ds\to0$$ as $n\to\infty$, which proves .  The lemma is proved. 

For treating $\check{u}$ we use the following result. 

\[lemma 1.17.2\] Let $H[\check{u}]\geq0$ in $G$. Then for any $x\in{\mathbb{R}}^{d}$, ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}$, and $\varepsilon>0$, there exist a sequence $\alpha^{n}_{\cdot} 
\in{\mathfrak{A}}$, $n=1,2,...$, and a sequence of increasing continuous $\{{\mathcal{F}}_{t}\}$-adapted processes $\eta^{n\varepsilon}_{t}({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})$ with $\eta^{n\varepsilon}_{0}({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})=0$ such that the processes $$\kappa^{n\varepsilon}_{t} :=
\check{u}(x^{n}_{t})
e^{-\phi^{n}_{t}-\psi^{n}_{t}}
+\eta^{n\varepsilon}_{t} ({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})
+\int_{0}^{t}
[r^{n}_{s}]^{2}f^{n}_{s}
( x^{n}_{s})
e^{-\phi^{n}_{s}-\psi^{n}_{s}}\,ds,$$ where $$\label{5.6.4}
(x^{n}_{t},
\phi^{n}_{t},\psi^{n}_{t})
=(x_{t},\phi_{t},\psi_{t})
^{\alpha^{n}_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n}_{\cdot})x},\quad
f^{n}_{t}( x)=
f^{\alpha^{n}_{t}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(\alpha^{n}_{\cdot}) }( x),
\quad r^{n}_{t}=
r^{\alpha^{n}_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n}_{\cdot})}_{t},$$ are submartingales on $[0,\tau^{\alpha^{n}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n}_{\cdot})
  x}]$ and $$\label{5.5.6}
\sup_{n}E 
\eta^{n\varepsilon}_{\infty}( {\text{\raise-.2ex\hbox{$\bm{\beta}$}}})<\infty,$$ $$\label{5.5.7}
{\operatornamewithlimits{\overline{lim}}}_{n\to\infty}
 E 
\eta^{n\varepsilon}_{\tau}({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})\leq
  N\varepsilon ,$$ where $N$ is independent of $x$, ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}$, and $\varepsilon$.

Finally, $$\sup_{n}E 
\sup_{t\geq0}|
\kappa^{n\varepsilon}_{t\wedge\tau }|<\infty .$$

Proof.  Owing to Assumptions \[assumption 1.9.1\] the function $$h(\alpha,x):=\inf_{\beta\in B}\big[   L^{\alpha\beta}\check{u}(x)+
  f^{\alpha\beta}(x)\big]$$ is a finite Borel function of $x$ and is continuous with respect to $\alpha$. Its $\sup$ over $A$ can be replaced with the $\sup$ over an appropriate countable subset of $A$ and since $$\sup_{\alpha\in A}h(\alpha,x)\geq0,$$ similarly to how $\beta(\alpha,x)$ was defined in the proof of Lemma \[lemma 1.16.1\], one can find a Borel function $\bar{\alpha}(x)$ in such a way that $$\label{1.17.07}
\inf_{\beta\in B}\big[   L^{\bar{\alpha}(x)\beta}\check{u}(x)+
  f^{\bar{\alpha}(x)\beta}(x)\big]\geq  -\varepsilon$$ in $G$. If $x\not\in G$ we set $\bar{\alpha}(x)=\alpha^{*}$, where $\alpha^{*}$ is a fixed element of $A$. 

After that we need some processes which we introduce recursively.  Fix $x$ and set $\alpha^{n0}_{t}\equiv\bar{\alpha}(x )$.  Then define $x^{n0}_{t}$, $t\geq0$, as a unique solution of the equation $$x _{t}=x+\int_{0}^{t}r_{s}^{\alpha^{n0}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n0}_{\cdot })}
\sigma(\alpha^{n0}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n0}_{\cdot }),x _{s})\,dw_{s}^{\alpha^{n0}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n0}_{\cdot })}$$ $$+\int_{0}^{t}[r_{s}^{\alpha^{n0}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n0}_{\cdot })}]^{2}\big[
b(\alpha^{n0}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n0}_{\cdot }) ,x _{s})
+\sigma(\alpha^{n0}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n0}_{\cdot }),x _{s})
\pi_{s}^{\alpha^{n0}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n0}_{\cdot })}\big]\,ds.$$ For $k\geq1$ introduce $\alpha^{nk}_{t}$ so that $$\alpha^{nk}_{t}=\alpha^{n(k-1)}_{t}\quad\text{for}\quad t<k/n,$$ $$\alpha^{nk}_{t}=\bar{\alpha}(x^{n(k-1)}_{k/n})
\quad\text{for}\quad t\geq k/n,$$ where $x^{n(k-1)}_{t}$ is a unique solution of $$x _{t}=x+\int_{0}^{t}r_{s}^{\alpha^{n(k-1)}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n(k-1)}_{\cdot })}
\sigma(\alpha^{n(k-1)}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n(k-1)}_{\cdot }), x _{s})\,dw_{s}^{\alpha^{n(k-1)}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n(k-1)}_{\cdot })}$$ $$+\int_{0}^{t}[r_{s}^{\alpha^{n(k-1)}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n(k-1)}_{\cdot })}]^{2}\big[b(\alpha^{n(k-1)}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n(k-1)}_{\cdot }) ,x _{s})$$ $$\label{5.5.1}
+\sigma(\alpha^{n(k-1)}_{s},
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n(k-1)}_{\cdot }), x _{s})
\pi_{s}^{\alpha^{n(k-1)}_{\cdot}
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n(k-1)}_{\cdot })}\big]\,ds.$$

As in the proof of Lemma \[lemma 1.16.1\] one can show that the above definitions make sense as well as the definition $$\label{5.5.2}
\alpha^{n}_{t}=\alpha^{n(k-1)}_{t}\quad\text{for}\quad t<k/n.$$ Next, by definition $x^{n}_{t}=
 x_{t}^{\alpha_{\cdot}^{n}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n}_{\cdot})x}$ satisfies $$x_{t}=x+\int_{0}^{t}r_{s}^{\alpha^{n}_{\cdot} {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(
\alpha^{n}_{\cdot})}
\sigma(\alpha^{n}_{s},{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(
\alpha^{n}_{\cdot}),x_{s})\,dw_{s}^{\alpha^{n}_{\cdot} {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(
\alpha^{n}_{\cdot})}$$ $$+\int_{0}^{t}[r_{s}^{\alpha^{n}_{\cdot} {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(
\alpha^{n}_{\cdot})}]^{2}\big[b(\alpha^{n}_{s},{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(
\alpha^{n}_{\cdot}) ,x_{s})
+\sigma(\alpha^{n}_{s},{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(
\alpha^{n}_{\cdot}),x_{s})\pi_{s}^{\alpha^{n}_{\cdot} {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(
\alpha^{n}_{\cdot})}\big]\,ds.$$ Equation and the definitions of ${\mathbb{B}}$ and of control adapted processes show that $x^{n}_{t}$ satisfies for $t\leq k/n$. Hence, (a.s.)  $x^{n}_{t}=x^{n(k-1)}_{t}$ for all $t\leq k/n$ and (a.s.)  for all $t\geq0$, $\alpha^{n}_{t}
=\bar{\alpha}(x^{n}_{\kappa_{n}(t)})$ and $$x^{n}_{t}=x+\int_{0}^{t}r^{n}_{s}\sigma(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n}_{\cdot }),x^{n}_{s})\,dw^{n}_{s}$$ $$+\int_{0}^{t}[r^{n}_{s}]^{2}\big[b(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n}_{\cdot }), x^{n}_{s})
+\sigma(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),
{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n}_{\cdot }),x^{n}_{s})\pi^{n}_{s}\big]\,ds,$$ where $(r,\pi,w)^{n}_{s} =(r,\pi,w)_{s}^{\alpha^{n}_{\cdot} {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(
\alpha^{n}_{\cdot})}$.

Now, introduce $\tau^{n}$ as the first exit time of $x^{n}_{t}$ from $G$, set $$\beta^{n}_{s}={\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{s}(\alpha^{n}_{\cdot}),\quad
\phi^{n}_{t}=\phi^{\alpha^{n}_{\cdot}
\beta^{n}_{\cdot} x}_{t}
,\quad
\psi^{n}_{t}=\psi^{\alpha^{n}_{\cdot}
\beta^{n}_{\cdot} x}_{t},\quad r^{n}_{s}=
r^{\alpha^{n}_{\cdot}
\beta^{n}_{\cdot} } ,$$ and observe that by Itô’s formula

$$\check{u}(x^{n}_{t\wedge\tau^{n}})
e^{-\phi^{n}_{t\wedge \tau^{n}}
-\psi^{n}_{t\wedge \tau^{n}}}
=\check{u}(x)+\int_{0}^{t\wedge\tau}
[r^{n}_{s}]^{2}e^{-\phi^{n}_{s}-\psi^{n}_{s}}
  L^{\alpha^{n}_{s} 
\beta^{n}_{s}}
\check{u}(x^{n}_{s})
\,ds+m^{n}_{t},$$ where $m^{n}_{s}$ is a martingale and, for $s<\tau^{n}$, $$L^{\alpha^{n}_{s}
\beta^{n}_{s}}
\check{u}(x^{n}_{s})=  
a_{ij}(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),\beta^{n}_{s}
,x^{n}_{s})
D_{ij}\check{u}(x^{n}_{s})$$ $$+  b_{i}(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),\beta^{n}_{s}
,x^{n}_{s})
D_{i}\check{u}(x^{n}_{s})
-  c(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),\beta^{n}_{s}
,x^{n}_{s})
 \check{u}(x^{n}_{s}).$$ Similarly to the proof of Lemma \[lemma 1.16.1\] we derive from that, for $s<\tau^{n}$, $$\bar L^{\alpha^{n}_{s}\beta^{n}_{s}}
\check{u}(x^{n}_{s})\geq -\varepsilon-\chi(x^{n}_{s}-x^{n}_{\kappa_{n}(s)})  -
f(\bar{\alpha}(x^{n}_{\kappa_{n}(s)}),\beta^{n}_{s},x^{n}_{\kappa_{n}(s)})$$ $$= -\varepsilon-\chi(x^{n}_{s}-x^{n}_{\kappa_{n}(s)})  
 -  f^{\alpha^{n}_{s}\beta^{n}_{s}}
( x^{n}_{s}),$$ where $\chi (y)$ are (nonrandom) bounded functions on ${\mathbb{R}}^{d}$ such that $\chi (y)\to0$ as $y\to0$. It follows that $$\label{1.17.8}
\check{u}(x^{n}_{t\wedge\tau^{n}})
e^{-\phi^{n}_{t\wedge\tau^{n}}
-\psi^{n}_{t\wedge\tau^{n}}}
+\int_{0}^{t\wedge\tau^{n}}[r^{n}_{s}]^{2}
f^{\alpha^{n}_{s}\beta^{n}_{s}   }
( x^{n}_{s})
e^{-\phi^{n}_{s}-\psi^{n}_{s}}\,ds+\eta^{n}_{t}
 =\zeta_{t}+m^{n}_{t},$$ where $\zeta_{t}$ is an increasing process and $$\eta^{n}_{t}=\eta^{n}_{t}({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})=
\delta_{1}^{-2}\int_{0}^{t\wedge\tau^{n}}
e^{-\phi^{n}_{s}-\psi^{n}_{s}}[\varepsilon
 +
 \chi_{\varepsilon}(x^{n}_{s}-x^{n}_{\kappa_{n}(s)})] \,ds.$$ Hence the left-hand side of is a local submartingale and we finish the proof in the same way as the proof of Lemma \[lemma 1.16.1\].  The lemma is proved. 

[**Proof of Theorem \[theorem 2.11.1\]**]{}. ( i) First we fix $x\in{\mathbb{R}}^{d}$, $\alpha_{\cdot}\in{\mathfrak{A}}$, and $\varepsilon>0$, take $\beta^{n}_{\cdot}(\alpha_{\cdot})$ form Lemma \[lemma 1.16.1\] and prove that the ${\mathfrak{B}}$-valued functions defined on ${\mathfrak{A}}$ by ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}^{n}(\alpha_{\cdot})=\beta^{n}_{\cdot}(\alpha_{\cdot})$ belong to ${\mathbb{B}}$. To do that observe that if holds and $T\leq 1/n$, then (a.s.)  $\beta^{n0}_{t}(\alpha^{1}_{\cdot})=\beta^{n0}_{t}(\alpha^{2}_{\cdot})$ for almost all $t\leq T$. By definition also (a.s.)  $$(r,\pi,w)^{\alpha^{1}_{\cdot}\beta_{\cdot}^{n0}(\alpha^{1}_{\cdot})}_{s}
=(r,\pi,w)^{\alpha^{2}_{\cdot}\beta_{\cdot}^{n0}(\alpha^{2}_{\cdot})}_{s}
\quad\text{for almost all}\quad s\leq T.$$ By uniqueness of solutions of (see Assumption \[assumption 5.23.2\]), the processes $x_{t}^{n1}$ found from for $\alpha_{\cdot}=\alpha^{1}_{\cdot}$ and for $\alpha_{\cdot}=\alpha^{2}_{\cdot}$ coincide (a.s.)  for all $t\leq T$.

If holds and $1/n<T\leq 2/n$, then by the above solutions of for $\alpha_{\cdot}=\alpha^{1}_{\cdot}$ and for $\alpha_{\cdot}=\alpha^{2}_{\cdot}$ coincide (a.s.)  for $t=1/n$ and then (a.s.)  $\beta^{n1}_{t}(\alpha^{1}_{\cdot} )=
\beta^{n1}_{t}(\alpha^{2}_{\cdot} )$ not only for all $t<1/n$ but also for all $t\geq1/n$, which implies that (a.s.)  $$(r,\pi,w)^{\alpha^{1}_{\cdot}\beta_{\cdot}^{n1}(\alpha^{1}_{\cdot})}_{s}
=(r,\pi,w)^{\alpha^{2}_{\cdot}\beta_{\cdot}^{n1}(\alpha^{2}_{\cdot})}_{s}
\quad\text{for almost all}\quad s\leq T$$ and again the processes $x_{t}^{n }$ found from for $\alpha_{\cdot}=\alpha^{1}_{\cdot}$ and for $\alpha_{\cdot}=\alpha^{2}_{\cdot}$ coincide (a.s.)  for all $t\leq T$.

By induction we get that if holds for a $T\in(0,\infty)$ and we define $k $ as the integer such that $k/n<T\leq(k+1)/n$, then (a.s.)  $$\label{4.5.5}
\beta^{n}_{t}(\alpha^{1}_{\cdot})=\beta^{nk}_{t}(\alpha^{1}_{\cdot} )=
\beta^{nk}_{t}(\alpha^{2}_{\cdot} )=
\beta^{n}_{t}(\alpha^{2}_{\cdot})\quad\text{for almost all}\quad
t< (k+1)/n ,$$ $$(r,\pi,w)^{\alpha^{1}_{\cdot}\beta_{\cdot}^{nk}(\alpha^{1}_{\cdot})}_{s}
=(r,\pi,w)^{\alpha^{2}_{\cdot}\beta_{\cdot}^{nk}(\alpha^{2}_{\cdot})}_{s}
\quad\text{for almost all}\quad s\leq T$$ and the processes $x_{t}^{n }$ found from for $\alpha_{\cdot}=\alpha^{1}_{\cdot}$ and for $\alpha_{\cdot}=\alpha^{2}_{\cdot}$ coincide (a.s.)  for all $t\leq T$. This means that ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}^{n} \in{\mathbb{B}}$ indeed. 

Furthermore, by the supermartingale property of $\kappa^{n\varepsilon}_{t}(\alpha_{\cdot})$, we have $$\hat{u}(x)\geq
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}^{n}(\alpha_{\cdot})}\big[
g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau} }
+\int_{0}^{\tau}r^{2}_{t}
 f( x_{t})  e^{-\phi_{t}-\psi_{t} }\,dt\big]
-E 
\eta^{n\varepsilon}_{\tau}
(\alpha_{\cdot})  ,$$ which owing to yields $$\hat{u}(x)\geq{\operatornamewithlimits{\underline{lim}}}_{n\to\infty}\sup_{\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}^{n} (\alpha_{\cdot})}
\big[
 \int_{0}^{\tau}
r^{2}_{t}f(x_{t}) e^{-\phi_{t}-\psi_{t}}\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau}} \big]
-N\varepsilon .$$

In light of the arbitrariness of $\varepsilon$ we conclude $\hat u\geq v$ and assertion (i) is proved. 

\(ii) Similarly to the above argument, for any ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}$, $$\check{u}(x)\leq E_{x}^{\alpha^{n}_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha^{n}_{\cdot})}
\big[\int_{0}^{\tau}r^{2}_{t}f( x_{t})
e^{-\phi_{t}-\psi_{t} }\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau}}\big]
+E\eta^{n\varepsilon}_{\tau}({\text{\raise-.2ex\hbox{$\bm{\beta}$}}}) .$$

It follows that $$\check{u}(x)\leq \sup_{\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha _{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha _{\cdot})}
\big[\int_{0}^{\tau}r^{2}_{t}f( x_{t})
e^{-\phi_{t} -\psi_{t} }\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau} }\big]
+{\operatornamewithlimits{\overline{lim}}}_{n\to\infty}
E 
\eta^{n\varepsilon}_{\tau}
({\text{\raise-.2ex\hbox{$\bm{\beta}$}}})$$ $$\leq \sup_{\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha _{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha _{\cdot})}
\big[\int_{0}^{\tau}r^{2}_{t}f( x_{t})
e^{-\phi_{t}-\psi_{t}}\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau}}\big]+N\varepsilon,$$ which in light of the arbitrariness of $\varepsilon$ and ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}$ finally yields that $\check u\leq v$.

This proves assertion (ii).  Assertion (iii) is an obvious consequence of (i) and (ii).  The theorem is proved. 

The case of uniformly nondegenerate processes
=============================================

\[section 4.6.1\]

As in Section \[section 2.11.1\] we replace Assumption \[assumption 1.9.1\] (iii) with Assumptions \[assumption 2.16.1\] and \[assumption 5.23.2\], and we assume that we are given two functions $\hat{u},\check{u} \in W^{2}_{d,loc}(G)\cap C(\bar{G})$.

In that case we have the following. 

\[theorem 2.16.1\] (i) If $H[\hat{u}]\leq0$ (a.e.)  in $G$ and $\hat{u}\geq g$ on $\partial G$, then $v\leq\hat{u}$ in $\bar{G}$.

\(ii) If $H[\check{u}]\geq0$ (a.e.)  in $G$ and $\check{u}\leq g$ on $\partial G$, then $v\geq\check{u}$ in $\bar{G}$.

\(iii) If $\hat{u}$ and $\check{u}$ are as in (i) and (ii) and $\hat{u}=\check{u}$, then $v=\hat{u}$ and $v$ is independent of the choice of the probability space, filtration, $r,\pi$, and $w$.

Proof. ( i) We basically repeat the proof of Theorem 4.1 of [@Kr_13_1] with considerable simplifications made possible due to our assumptions.  It is well known that there exists a sequence $\hat{u}_{n}\in C^{2}(\bar{G})$ such that $\hat{u}_{n}\to\hat{u}$ in $C(\bar{D})$ and in $W^{2}_{d}(G')$ for any subdomain $G'\subset \bar{G}'\subset G$. Introduce $\hat{h}_{n}=H[\hat{u}_{n}]$, $$f^{\alpha\beta}_{n}(x)= f^{\alpha\beta} (x)
 - \hat{h}_{n}(x),$$ and observe that owing to our continuity assumptions on $\sigma,b,c,f$, the functions $\hat{h}_{n}$ and $f^{\alpha\beta}_{n}(x)$ are continuous in $x$ uniformly with respect to $\alpha,\beta$ and $${\operatornamewithlimits{sup\,\,\,inf}}_{\alpha\in A\,\,\beta\in B}
\big[
L^{\alpha\beta}_{n}\hat{u}_{n}( x)+
f^{\alpha\beta}_{n}( x)\big]=0$$ in $G$. By Theorem \[theorem 2.11.1\], for any subdomain $G_{1}\subset \bar{G}_{1}\subset G$ we have in $G_{1}$ that $$\label{2.16.3}
\hat{u}_{n}(x)\geq{\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}\big[
\hat{u}_{n}(x_{\tau_{1}})e^{-\phi_{\tau_{1}}
-\psi_{\tau_{1}}}
+\int_{0}^{\tau_{1}}r^{2}_{t}
 f_{n}( x_{t}) e^{-\phi_{t}-\psi_{t}}\,dt \big],$$ where $$\tau_{1}^{\alpha_{\cdot}\beta_{\cdot}x}
=\inf\{t\geq0:x_{t}^{\alpha_{\cdot}\beta_{\cdot}x}\not\in G_{1}\}.$$

Notice that $$E^{\alpha_{\cdot}\beta_{\cdot}}_{ x}
 |\hat{u}_{n}(x_{\tau_{1}})- \hat{u} (x_{\tau_{1}})|
e^{-\phi_{\tau_{1}}
-\psi_{\tau_{1}}} \leq \sup_{G}|\hat{u}_{n}-\hat{u}|.$$ While estimating $$I_{n}(x):=E^{\alpha_{\cdot}\beta_{\cdot}}_{ x}\int_{0}^{\tau_{1}}
|f_{n}-f|(x_{t})e^{-\phi_{t}-\psi_{t}}\,dt
=E^{\alpha_{\cdot}\beta_{\cdot}}_{ x}e^{-\psi_{\tau_{1}}}\int_{0}^{\tau_{1}}
|f_{n}-f|(x_{t})e^{-\phi_{t}}\,dt.$$ Girsanow’s theorem allows us to concentrate on $\pi\equiv0$ and then the Alexandrov estimate guarantees that $$I_{n}(x)\leq N\|\hat{h}_{n}\|_{{\mathcal{L}}_{d}(G_{1})}
=N\| H[\hat{u}_{n}]-H[\hat{u} ]\|_{{\mathcal{L}}_{d}(G_{1})}
\leq N\|\hat{u}_{n} - \hat{u}\|_{W^{2}_{d}(G_{1})},$$ where the constants $N$ are independent of $n$ (and $x$). 

Hence by letting $n\to\infty$ in we obtain that for $k=1$ $$\label{2.16.4}
\hat{u} (x)\geq{\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}\,\,\alpha_{\cdot}\in{\mathfrak{A}}}
E_{x}^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}\big[
\hat{u} (x_{\tau_{k}})e^{-\phi_{\tau_{k}} 
-\psi_{\tau_{k}}}
+\int_{0}^{\tau_{k}}
 f ( x_{t}) e^{-\phi_{t}-\psi_{t}}\,dt \big],$$ Where $\tau_{k}^{\alpha_{\cdot}\beta_{\cdot}x}$ are defined as the first exit times of the processes $x_{t}^{\alpha_{\cdot}\beta_{\cdot}x}$ from an expanding sequence of subdomains $G_{k}\subset\bar{G}_{k}\subset G$ such that $\cup_{k}G_{k}= G$.

By letting $k\to\infty$ in and repeating the proof of Theorem 2.2 of [@Kr_13_1] given there in Section 6 we get that $\hat{u}\geq v$ in $G$ as stated.  Observe that in our situation in the proof of Theorem 2.2 of [@Kr_13_1] we need not mollify $f^{\alpha\beta}(x)$ because by assumption it is uniformly continuous in $x$.

The proof of assertion (ii) is quite similar and as usual assertion (iii) is obtained by simply combining assertions (i) and (ii).  The theorem is proved. 

A general approximation result from above
=========================================

\[section 2.25.1\]

In this section we suppose that all assumptions in Section \[section 2.26.3\] are satisfied.  Set $$A_{1}=A$$ and let $A_{2}$ be a separable metric space having no common points with $A_{1}$.

\[assumption 4.29.2\] The functions $ 
\sigma^{\alpha\beta}( x)$, $ b^{\alpha\beta}( x)$, $ 
c^{\alpha\beta}( x)$, and $ f^{\alpha\beta}( x)$ are also defined on $A_{2}\times B \times{\mathbb{R}}^{d}$ in such a way that they are [*independent*]{} of $\beta$ (on $A_{2}\times B \times{\mathbb{R}}^{d}$) and the assumptions in Section \[section 2.26.3\] are satisfied, of course, with $A_{2}$ in place of $A$. 

Define $$\hat{A}=A_{1}\cup A_{2}.$$

Then we introduce $\hat{{\mathfrak{A}}}$ as the set of progressively measurable $\hat{A}$-valued processes and $\hat{{\mathbb{B}}}$ as the set of ${\mathfrak{B}}$-valued functions $ {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})$ on $\hat{{\mathfrak{A}}}$ such that, for any $T\in[0,\infty)$ and any $\alpha^{1}_{\cdot},
\alpha^{2}_{\cdot}\in\hat{{\mathfrak{A}}}$ satisfying $$P(  \alpha^{1}_{t}=\alpha^{2}_{t} 
 \quad\text{for almost all}\quad t\leq T)=1,$$ we have $$P(  
 {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(\alpha^{1}_{\cdot})={\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(\alpha^{2}_{\cdot}) 
 \quad\text{for almost all}\quad t\leq T)=1.$$

We fix an element $\alpha^{*}\in A_{1}$ and for $\alpha_{\cdot}\in\hat{{\mathfrak{A}}}$ define $$(p\alpha)_{t}=\alpha_{t}\quad\text{if}
\quad\alpha_{t}\in A_{1},\quad
(p\alpha)_{t}=\alpha^{*}\quad\text{if}
\quad\alpha_{t}\in A_{2}.$$ By using this projection operator we extend $(w,r,\pi)^{\alpha_{\cdot}\beta_{\cdot}}_{t}$ originally defined for $\alpha_{\cdot}\in{\mathfrak{A}}$ and $\beta_{\cdot}\in{\mathfrak{B}}$ as $$\label{2.22.1}
(w,r,\pi)^{\alpha_{\cdot}\beta_{\cdot}}_{t}
=(w,r,\pi)^{p\alpha_{\cdot}\beta_{\cdot}}_{t}$$ thereby now defined for $\alpha_{\cdot}\in\hat{{\mathfrak{A}}}$ and $\beta_{\cdot}\in{\mathfrak{B}}$.

Next, take a constant $K\geq0$ and set $$v_{K}(x)={\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in\hat{{\mathbb{B}}}\,\,\alpha_{\cdot}\in\hat{{\mathfrak{A}}}}
v^{\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})}_{K}(x),$$ where $$v^{\alpha_{\cdot}\beta _{\cdot} }_{K}(x)=
E_{x}^{\alpha_{\cdot}\beta _{\cdot} }\big[\int_{0}^{\tau}
r^{2}_{t} f_{K} ( x_{t})e^{-\phi_{t}-\psi_{t}}\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau}
 }\big]$$ $$= :v^{\alpha_{\cdot}\beta _{\cdot} }  (x)-K
E_{x}^{\alpha_{\cdot}\beta _{\cdot} }\int_{0}^{\tau}
r^{2}_{t}I_{\alpha_{t}\in A_{2}}e^{-\phi_{t} 
-\psi_{t}}\,dt,$$ $$f^{\alpha\beta}_{K}( x)=f^{\alpha\beta}( x)-KI_{\alpha\in A_{2}}.$$ Notice that, obviously, $$v (x)={\operatornamewithlimits{inf\,\,\,sup}}_{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in\hat{{\mathbb{B}}}\,\,\alpha_{\cdot}\in\hat{{\mathfrak{A}}}}
v^{p\alpha_{\cdot}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(p\alpha_{\cdot})} (x).$$

These definitions make sense owing to Remark \[remark 2.20.1\], which also implies that $v^{\alpha_{\cdot}\beta _{\cdot} }_{K}$ and $v^{\alpha_{\cdot}\beta _{\cdot} }$ and bounded in $\bar{G}$.

\[theorem 1.20.1\] We have $v_{K}\to v$ uniformly on $\bar{G}$ as $K\to\infty$.

\[lemma 3.24.1\] Assume that $\pi\equiv0$. Then there exists a constant $N$ such that for any $\alpha_{\cdot}\in\hat{{\mathfrak{A}}}$, $\beta_{\cdot}\in{\mathfrak{B}}$, $x\in{\mathbb{R}}^{d}$, $T\in[0,\infty)$, and stopping time $\gamma$ $$E^{\alpha_{\cdot}\beta_{\cdot}}_{x}
\sup_{t\leq T\wedge\gamma}|x_{t}-y_{t}|\leq
Ne^{NT} \big(E^{\alpha_{\cdot}\beta_{\cdot}}_{x}\int_{0}^{
T\wedge\gamma}I_{\alpha_{t}\in A_{2}}\,dt\big)^{1/2} ,$$ where $$y^{\alpha_{\cdot} \beta_{\cdot}x}_{t}
=x^{p\alpha_{\cdot} \beta_{\cdot}x}_{t}.$$

Proof.  For simplicity of notation we drop the superscripts $\alpha_{\cdot} ,\beta_{\cdot},x$. Observe that $x_{t}$ and $y_{t}$ satisfy $$x_{t}=x+\int_{0}^{t}r_{s}\sigma^{\alpha_{s}\beta_{s}}( x_{s})
\,dw_{s}+\int_{0}^{t}r^{2}_{s}
b^{\alpha_{s}\beta_{s}}( x_{s})\,ds,$$ $$y_{t}=x+\int_{0}^{t}r_{s}\sigma^{\alpha_{s}\beta_{s}}( y_{s})
\,dw_{s}+\int_{0}^{t}r_{s}^{2}b^{\alpha_{s}\beta_{s}}( y_{s})
\,ds+\eta_{t},$$ where $\eta_{t}=I_{t}+J_{t}$, $$I_{t}=\int_{0}^{t}r_{s}
[\sigma^{p\alpha_{s}\beta_{s}}( y_{s})
-\sigma^{\alpha_{s}\beta_{s}}( y_{s})]\,dw_{s},$$ $$J_{t}=\int_{0}^{t}r^{2}_{s}[b^{p\alpha_{s}\beta_{s}}( y_{s})
-b^{\alpha_{s}\beta_{s}}( y_{s})]\,ds.$$

By Theorem II.5.9 of [@Kr77] (where we replace the processes $x_{t}$ and $\tilde{x}_{t}$ with appropriately stopped ones) for any $T\in[0,\infty)$ and any stopping time $\gamma
 $ $$\label{1.22.04}
E\sup_{t\leq T\wedge \gamma}|x _{t}-y _{t}|^{2}\leq
Ne^{NT}E\sup_{t\leq T\wedge \gamma}|\eta _{t}|^{2},$$ where $N$ depends only on $K_{1}$ and $d$, which by Theorem III.6.8 of [@Kr95] leads to $$\label{1.23.01}
E\sup_{t\leq T\wedge\gamma }|x _{t}-y _{t}|
\leq Ne^{NT}E\sup_{t\leq T\wedge\gamma }|\eta_{t}|$$ with the constant $N$ being three times the one from . 

By using Davis’s inequality we see that for any $T\in[0,\infty)$ $$E\sup_{t\leq T\wedge\gamma }|I _{t}| 
\leq NE\big(\int_{0}^{T\wedge\gamma}
I_{\alpha _{s}\in A_{2}}\,ds\big)^{1/2}
\leq N\big(E\int_{0}^{T\wedge\gamma}
I_{\alpha _{s}\in A_{2}}\,ds\big)^{1/2}.$$

Furthermore, almost obviously $$E\sup_{t\leq T\wedge\gamma }|J_{t}| 
\leq N E\int_{0}^{T\wedge\gamma}
I_{\alpha _{s}\in A_{2}}\,ds 
\leq NT^{1/2}\big(E\int_{0}^{T\wedge\gamma}
I_{\alpha _{s}\in A_{2}}\,ds\big)^{1/2}$$ and this in combination with proves the lemma. 

[**Proof of Theorem \[theorem 1.20.1\]**]{}.  Without losing generality we may assume that $g\in C^{3}({\mathbb{R}}^{d})$ since the functions of this class uniformly approximate in $\bar{G}$ any $g$ which is continuous in ${\mathbb{R}}^{d}$. Then notice that by Itô’s formula for $g\in C^{3}({\mathbb{R}}^{d})$ we have $$E_{x}^{\alpha_{\cdot}\beta _{\cdot} }\big[\int_{0}^{\tau}
r_{t}^{2} f_{K} ( x_{t})e^{-\phi_{t}-\psi_{t} }\,dt
+g(x_{\tau})e^{-\phi_{\tau}-\psi_{\tau}
 }\big]$$ $$=g(x)+E_{x }^{\alpha _{\cdot} \beta 
 _{\cdot} } \int_{0}^{\tau}r_{t}^{2}[ 
 \hat{f} ( x_{t} )-KI_{\alpha_{t}\in A_{2}}] 
e^{-\phi_{t}-\psi_{t}}\,dt,$$ where $$\hat{f} ^{\alpha\beta}( x ):=f^{\alpha\beta} ( x ) 
+ L^{\alpha\beta}g( x),$$ which is bounded and, for $(\alpha,\beta)\in\hat A\times B$, is uniformly continuous in $x$ uniformly with respect to $\alpha,\beta$. This argument shows that without losing generality we may (and will) also assume that $g=0$.

Next, since ${\mathfrak{A}}\subset\hat{{\mathfrak{A}}}$ and for $\alpha_{\cdot}\in
\hat{{\mathfrak{A}}}$ and ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in\hat{{\mathbb{B}}}$ we have ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(\alpha_{\cdot})\in{\mathfrak{B}}$, it holds that $$v_{K}\geq v.$$

To estimate $v_{K}$ from above, take ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}$ and define $\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}} \in\hat{{\mathbb{B}}}$ by $$\label{2.22.2}
\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}}_{t}(\alpha_{\cdot})={\text{\raise-.2ex\hbox{$\bm{\beta}$}}}_{t}(p\alpha_{\cdot}).$$ Also take any sequence $x^{n}
 \in \bar{G}$, $n=1,2,...$, recall that $r_{t}^{\alpha_{\cdot}\beta_{\cdot}}\geq \delta_{1}$, and find a sequence $\alpha^{n}_{\cdot}\in\hat{{\mathfrak{A}}}$ such that $$v_{K}(x^{n}) \leq  \sup_{\alpha\in\hat{{\mathfrak{A}}}}
E_{x^{n}}^{\alpha_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}}(\alpha_{\cdot})}
 \int_{0}^{\tau}r^{2}_{t}
 f _{K} ( x_{t})e^{-\phi_{t}-\psi_{t}}\,dt$$ $$\label{2.22.6}
=1/n+v^{\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}}(\alpha^{n}_{\cdot})}(x^{n})
-K\delta_{1}^{2}E\int_{0}^{\tau^{n} }I_{\alpha^{n}_{t}\in A_{2}}
e^{-\phi^{n}_{t}-\psi^{n}_{t}}\,dt,$$ where $$(\tau^{n},\phi^{n}_{t},\psi^{n}_{t} )
=( \tau,\phi _{t},\psi_{t} )
^{\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}}(\alpha^{n}_{\cdot})x^{n}}.$$ It follows that there is a constant $N$ independent of $n$ and $K$ such that $$\label{2.22.4}
E\int_{0}^{\tau^{n} }I_{\alpha^{n}_{t}\in A_{2}}
e^{-\phi^{n}_{t}-\psi^{n}_{t}}\,dt\leq N/K.$$ Below by $N$ we denote generic constants independent of $n$ and $K$ (and $T$ once it appears). 

We want to estimate the difference $$\label{2.22.3}
v^{\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}}(\alpha^{n}_{\cdot})}(x^{n})
-v^{p\alpha_{\cdot}^{n}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(p\alpha_{\cdot}^{n})}(x^{n}).$$ Observe that in the expression of this difference by the definition through the mathematical expectations of certain quantities the processes $\psi_{t}$ involved are just the same, thanks to and .  This allows us to rewrite the mathematical expectations similarly to how it is done in Remark \[remark 2.20.1\] and then by using Girsanov’s theorem allows us to assume that $\pi_{t}\equiv0$, at the expense that the underlying probability measures will now depend on $n$. However, for simplicity of notation we keep the symbol $E$ for expectations with respect to the new probability measures depending on $n$. Thus, while estimating we assume that $\pi_{t}\equiv0$.

Introduce $$x^{n}_{t} =
x^{\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}} 
(\alpha^{n}_{\cdot}) x^{n}}_{t} ,\quad
 y^{n}_{t} 
=x^{p\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}} 
( \alpha^{n}_{\cdot}) x^{n} }_{t} ,$$ $$c^{n}_{t}=c^{\alpha^{n}_{t}\hat{\beta}_{t}(\alpha^{n}_{\cdot})}
(x^{n}_{t}),\quad
pc^{n}_{t}=c^{p\alpha^{n}_{t}\hat{\beta}_{t}(\alpha^{n}_{\cdot})}
(y^{n}_{t})$$ $$f^{n}_{t}=f^{\alpha^{n}_{t}\hat{\beta}_{t}(\alpha^{n}_{\cdot})}
(x^{n}_{t}),\quad
pf^{n}_{t}=f^{p\alpha^{n}_{t}\hat{\beta}_{t}(\alpha^{n}_{\cdot})}
(y^{n}_{t})$$ $$r^{n}_{t}=r^{\alpha^{n}_{\cdot}\hat{{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}} 
( \alpha^{n}_{\cdot})}_{t},\quad
p\phi^{n}_{t}=\int_{0}^{t}[r^{n}_{s}]^{2}pc^{n}_{s}  \,ds,$$ and define $\gamma^{n}$ as the first exit time of $y^{n}_{t}$ from $G$. Notice that, for any $T\in[0,\infty)$, equals $$I_{1n}(T)+I_{2n}(T)-I_{3n}(T),$$ where $$I_{1n}(T)=
E\int_{0}^{\tau^{n}\wedge\gamma^{n}\wedge T}
[r^{n}_{t}]^{2}\big[f^{n}_{t}\exp(-\phi^{n}_{t})
-pf^{n}_{t}\exp(-p\phi^{n}_{t})\big]\,dt,$$ $$I_{2n}(T)=E\int^{\tau^{n}}_{\tau^{n}\wedge\gamma^{n}\wedge T}[r^{n}_{t}]^{2}
f^{n}_{t}\exp(-\phi^{n}_{t})\,dt,$$ $$I_{3n}(T)=E\int^{\gamma^{n}}_{\tau^{n}\wedge\gamma^{n}\wedge T}[r^{n}_{t}]^{2}
pf^{n}_{t}\exp(-p\phi^{n}_{t})\,dt,$$

By using the inequalities $|e^{-a}-e^{-b}|\leq|a-b|$ valid for $a,b\geq0$ and $|ab-cd|\leq |b|\cdot|a-c|+|c|\cdot
|b-d|$ and also using the boundedness of $r^{\alpha_{\cdot}\beta_{\cdot}}$, $c^{\alpha\beta}$, and $f^{\alpha\beta}$ we easily conclude that $$|I_{1n}(T)|\leq N(1+T)E\int_{0}^{\tau^{n}\wedge T}\big[
|f^{n}_{t}-pf^{n}_{t}|+|c^{n}_{t}-pc^{n}_{t}|\big]\,dt .$$ Observe that, if $\alpha^{n}_{t}\in A_{1}$, then $$|f^{n}_{t}-pf^{n}_{t}|\leq W_{f}(|x^{n}-y^{n}_{t}|),$$ where $W_{f}$ is the modulus of continuity of $f^{\alpha\beta}(x)$ with respect to $x$ uniform with respect to $\alpha,\beta$. Similar estimate holds for $|c^{n}_{t}-pc^{n}_{t}|$ in which $W_{c}$ is the modulus of continuity of $c^{\alpha\beta}(x)$. Furthermore, $$E\int_{0}^{\tau^{n}\wedge T}I_{\alpha^{n}_{t}\in A_{2}}\,dt
\leq e^{T/\delta}E\int_{0}^{\tau^{n} }I_{\alpha^{n}_{t}\in A_{2}}
e^{-\phi^{n}_{t}}\,dt\leq Ne^{T/\delta}/K,$$ where the last inequality is due to .  Hence, $$|I_{1n}(T)|\leq N(1+T)^{2}E[W_{c}+W_{f}](
\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|)
+ Ne^{T/N}/K.$$ We may and will assume that $W_{c}(r)$ and $W_{f}(r)$ are concave functions on $[0,\infty)$, so that $$|I_{1n}(T)|\leq N(1+T)^{2}[W_{c}+W_{f}](E
\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|)
+ Ne^{T/N}/K.$$ Next use the fact that as follows from Lemma \[lemma 3.24.1\] $$E\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|
\leq Ne^{NT}/\sqrt{K}.$$ Then we conclude that $$\label{2.22.5}
|I_{1n}(T)|\leq N(1+T)^{2}[W_{c}+W_{f}](Ne^{NT}/\sqrt{K})
+ Ne^{T/\delta}/K.$$

While estimating $I_{2n}(T)$ we again use the boundedness of the data and use Remark \[remark 2.22.2\] and by Itô’s formula to obtain that $$|I_{2n}(T)|\leq NEI_{\tau^{n}\geq\gamma^{n}\wedge T}
\int_{\gamma^{n}\wedge T}^{\tau^{n} }
[r^{n}_{t}]^{2}\,dt\leq NEI_{\tau^{n}\geq\gamma^{n}\wedge T}
G(x^{n}_{\gamma^{n}\wedge T})$$ $$\leq NEI_{\tau^{n}\geq\gamma^{n}\wedge T}|
G(x^{n}_{\gamma^{n}\wedge T})-G(y^{n}_{\gamma^{n}\wedge T})|
+NEI_{\tau^{n}\geq\gamma^{n}\wedge T}|G(y^{n}_{\gamma^{n}\wedge T})|$$ $$\leq NE\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|
+NEI_{\gamma^{n}>T}G(y^{n}_{T}).$$ By Lemma 5.1 of [@Kr_13_1] $$EI_{\gamma^{n}>T}G(y^{n}_{T})\leq Ne^{-T/N}.$$

Next, $$|I_{3n}(T)|\leq NEI_{\gamma^{n}\geq\tau^{n}\wedge T}
\int_{\tau^{n}\wedge T}^{\gamma^{n} }
[r^{n}_{t}]^{2}\,dt\leq NEI_{\gamma^{n}\geq\tau^{n}\wedge T}
G(y^{n}_{\tau^{n}\wedge T})$$ $$\leq
NE\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|
+NEI_{\gamma^{n}\geq\tau^{n}\wedge T}
G(x^{n}_{\tau^{n}\wedge T})$$ $$\leq NE\sup_{t\leq \tau^{n}\wedge T} |x^{n}_{t}-y^{n}_{t}|
+NEI_{\tau^{n}>\  T}
G(x^{n}_{ T}).$$ We use again Lemma 5.1 of [@Kr_13_1] and conclude that, for $K\geq1$, is less than $$w(T,K):=N(1+T)^{2}[W_{c}+W_{f}](Ne^{N_{1}T}/\sqrt{K})
+  Ne^{N_{1}T}/\sqrt{K}+Ne^{-T/N_{2}}.$$

Thus, yields $$v_{K}(x^{n})\leq 1/n
+v^{p\alpha_{\cdot}^{n}{\text{\raise-.2ex\hbox{$\bm{\beta}$}}}(p\alpha_{\cdot}^{n})}(x^{n})
+w(T,K).$$ Hence $$v_{K}(x^{n})\leq \sup_{\alpha_{\cdot}\in{\mathfrak{A}}}
 v^{ \alpha _{\cdot}
 {\text{\raise-.2ex\hbox{$\bm{\beta}$}}}( \alpha_{\cdot} )}(x^{n})+w(T,K) +1/n.$$ Owing to the arbitrariness of ${\text{\raise-.2ex\hbox{$\bm{\beta}$}}}\in{\mathbb{B}}$ we have $$v_{K}(x^{n})\leq v(x^{n})+w(T,K) +1/n,$$ and the arbitrariness of $x^{n}$ yields that for $K\geq1$ $$\label{2.23.2}
\sup_{\bar{D}}(v_{K}-v) \leq  w(T,K)  ,$$ which leads to the desired result after first letting $K\to\infty$ and then $T\to\infty$. The theorem is proved. 

\[remark 2.23.1\] Assume that $c^{\alpha\beta}(x)$ and $f^{\alpha\beta}(x)$ are Hölder continuous with respect to $x$ with exponent $\kappa\in(0,1]$ and constant independent of $\alpha$ and $\beta$. Then by taking $T$ such that $e^{N_{1}T}=K^{1/4}$ we see that, for $K\geq1$, the left-hand side of is dominated by $$N(1+\ln K)^{2}K^{-\kappa/4}+NK^{-1/(4N_{1}N_{2})}.$$ Hence, there is a $\chi\in(0,1]$ such that the left-hand side of is dominated by $NK^{-\chi}$ for $K\geq 1$. Thus, we have justified a claim made in Section 5 of [@Kr_14_1]. 

Proof of Theorem \[theorem 1.14.1\]
===================================

\[section 4.6.2\]

The properties of $P$ listed before Theorem \[theorem 9.23.1\] or just the construction of $P$ in [@Kr_12] yield that there is a set $A_{2}$, having no common points with $A$, and bounded continuous functions $\sigma^{\alpha}=\sigma^{\alpha\beta}$, $b^{\alpha}=b^{\alpha\beta}$, $c^{\alpha}=c^{\alpha\beta}$ (independent of $x$ and $\beta$), and $f^{\alpha\beta}\equiv0$ defined on $A_{2}$ such that the assumptions in Section \[section 2.26.3\] are satisfied perhaps with different constants $\delta $ and $K_{0}$ and for $a^{\alpha}:=a^{\alpha\beta}
=(1/2)\sigma^{\alpha}(\sigma^{\alpha})^{*}$ we have $$\label{2.8.5}
P[u](x)=\sup_{\alpha\in A_{2}}
\big[a_{ij}^{\alpha} D_{ij}u(x)
+b^{\alpha}_{i} D_{i}u(x)-c^{\alpha} u(x) \big].$$

Use the notation from Section \[section 2.25.1\] and observe that $$\max(H[u](x),
P[u](x)-K)$$ $$=\max\big\{{\operatornamewithlimits{sup\,\,\,inf}}_{\alpha\in A_{1}\,\,\beta\in B}
[L^{\alpha\beta}u(x)+f^{\alpha\beta}(x)],
{\operatornamewithlimits{sup\,\,\,inf}}_{\alpha\in A_{2}\,\,\beta\in B}
[L^{\alpha\beta}u(x)+f^{\alpha\beta}(x)-K]\big\}$$ $$={\operatornamewithlimits{sup\,\,\,inf}}_{\alpha\in\hat{A}\,\,\beta\in B}
\big[ L^{\alpha\beta}u( x)+f^{\alpha\beta}_{K}( x)]
\quad (f^{\alpha\beta}_{K}( x)=f^{\alpha\beta} ( x)
I_{\alpha\in A_{1}}-KI_{\alpha\in A_{2}}),$$ where the first equality follows from the definition of $H[u]$, , and the fact that $L^{\alpha \beta}$ is independent of $\beta$ for $\alpha\in A_{2}$. It follows by Theorems \[theorem 9.23.1\] and \[theorem 2.16.1\] that $u_{K}=v_{K}$ and by Theorem \[theorem 1.20.1\] that in $\bar{G}$ $$v=\lim_{K\to\infty}u_{K},$$ where the right-hand side is indeed independent of the probability space, filtration, and the choice of $w,r,\pi$. Since the above convergence is uniform, $v$ is continuous in $\bar{G}$. The theorem is proved. 

[mm]{}

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[^1]: The author was partially supported by NSF Grant DMS-1160569
