# Partial Schauder estimates for unbounded solutions of the Kolmogorov
equation

Andrzej Palczewski111Faculty of Mathematics, Informatics and Mechanics,
University of Warsaw, Banacha 2, 02-097 Warszawa, Poland (e-mail:
<EMAIL_ADDRESS>
###### Abstract

The goal of the paper is to show, under possibly weak assumptions, that the
function given by the Feynman-Kac formula is a classical solution of the
associated Kolmogorov equation. We also show that although this solution is
unbounded it fulfills Schauder’s estimates. The paper collects mostly known
results on the regularity of solutions of linear parabolic equations of second
order and the regularity of solutions of Ito stochastic differential
equations. We aim to reproduce these and related results in a unified fashion.

Keywords: Schauder’s estimates, Kolmogorov equation, unbounded coefficients,
Cauchy problem, regularity of solutions

Mathematics Subject Classification (2010): 35K15, 35B65, 47D07, 60H30.

##  1 Introduction

The problem of the existence and regularity of solutions for the Kolmogorov
equation is a particular case of a more general analysis of second-order
parabolic equations. We regard this equation as a diffusion equation
corresponding to a suitable stochastic differential equation. Our ultimate
goal is to show that a functional of a solution of this stochastic equation
given by the Feynman-Kac formula solves the Kolmogorov equation. We also
obtain Schauder’s estimates for polynomially growing solutions of this
equation.

Our starting point is ${X}(s)=(X^{1}(s),\dots,X^{d}(s))$ a $d$-dimensional Ito
stochastic process driven by a Wiener process $W(s)$ in a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ with a filtration
$\mathbb{F}=(\mathcal{F}^{t}_{s})_{t\leq s\leq T}$ generated by $W(s)$. The
notation $\mathcal{F}^{t}_{s}$ means that we are considering the Wiener
process $W(s)$ starting at $s=t$ and that the $\sigma$-field
$\mathcal{F}^{t}_{t}$ is trivial. The process $X(s)$ is defined for
$s\in[t,T]$ by the system of stochastic differential equations

$d{X}(s)={b}(s,{X}(s))ds+{\sigma}(s,{X}(s))d{W}(s),\quad{X}(t)={x}.$ (1)

Here $b$ is a $d$-dimensional vector, $\sigma$ a matrix of dimension $d\times
m$, and ${W}(s)$ is an $m$-dimensional Wiener process. To indicate that
${X}(s)$ is a solution of (1) with initial condition ${X}(t)={x}$ we shall
write $X^{t,x}(s)$.

Assuming that the coefficients ${b}(s,{y})$ and ${\sigma}(s,{y})$ are
deterministic functions we define the family of elliptic operators

${\mathcal{A}}^{t}u(t,{x})=\sum_{i,j=1}^{d}a_{ij}(t,{x})\frac{\partial^{2}u}{\partial
x_{i}\partial x_{j}}+\sum_{i=1}^{d}b_{i}(t,{x})\frac{\partial u}{\partial
x_{i}},$ (2)

where coefficients $a_{ij}$ are given by the formula
$a_{ij}\\!=\\!\frac{1}{2}\sum_{k=1}^{m}\sigma_{ik}\sigma_{jk}$. The Kolmogorov
equation, which will be the object of our investigation, is then given by the
expression

$D_{t}u(t,x)+{\mathcal{A}}^{t}u(t,x)-c(t,x)u(t,x)+f(t,x)=0,$ (3)

where $x\in\mathbb{R}^{d}$ and $t\in[0,T]$.

It is well known (cf. Theorem 5.7.6 in [18]) that if equation (3) with
continuous coefficients and continuous terminal condition possesses a smooth
solution of class $C^{1,2}\bigl{(}[0,T]\times\mathbb{R}^{d}\bigr{)}$ then this
solution can be represented by the Feynman-Kac formula.

Equally important is the inverse problem: under which conditions the function
given by the Feynman-Kac formula is a solution of equation (3). A partial
answer to this question is given by Friedman (cf. Theorem 5.6.1 in [14]). A
more complete analysis can be found in the book by Krylov (Theorem V.7.4 in
[20]). In both mentioned theorems, the authors require a high smoothness of
the coefficients ($C^{2}$ in $x$ with $t$-uniform bounds on derivatives) and
the terminal condition (also $C^{2}$). On the other hand, the classical theory
of parabolic equations in bounded domains guarantees the existence of smooth
solutions of equation (3) for Hölder continuous coefficients and continuous
initial data (cf. Theorem 3.9 in [13]). But the picture is more complicated as
the stochastic process $X^{t,x}(s)$ that appears in the Feynman-Kac formula is
a solution of equation (1) in which the coefficients can grow linearly with
$x$. Hence, the associated equation (3) possesses unbounded coefficients, and
the results of Theorem 3.9 in [13] cannot be straightforwardly applied.

In the last sixty years, parabolic equations with unbounded coefficients in
unbounded domains were studied in great detail. After the pioneering works by
Aronson and Besala [6], and Besala [8], the problem was studied by many
authors who proved the existence, uniqueness, and regularity of solutions
under different hypotheses on the coefficients in spaces of bounded continuous
functions but also in $L^{p}$ spaces on $\mathbb{R}^{d}$ or on unbounded
domains in $\mathbb{R}^{d}$ (cf. [1], [2],[4], [5], [9], [11], [21], [23],
[26], [27], [28], [29]).

An essential difficulty in simultaneous analysis of the stochastic equation
(1) and the parabolic equation (3) is caused by the assumed regularity of
coefficients. The standard assumption for parabolic problems is Hölder’s
continuity of coefficients in both variables $(t,x)$ (see the book by Friedman
[13]). On the other hand, a standard assumption for Ito stochastic equations
is Lipschitz continuity in $x$ and continuity (or only measurability) in $t$.
To be in accord with the assumptions for stochastic differential equations, we
should construct a solution for the parabolic equation (3) for coefficients
only continuous in $t$, and Hölder continuous in the space variables. Then one
can expect the Schauder estimates controlling only the Hölder $x$-norms of the
derivatives of solutions. Such estimates for the mentioned weaker assumptions
on the coefficients were introduced under the name of partial (or
intermediate) Schauder estimates (cf. [10], [19], [24]). Then, to obtain the
existence and uniqueness of solutions for linear parabolic problems in a
bounded domain, it is sufficient to assume the continuity of coefficients in
$t$ and Hölder’s continuity in $x$, and the continuity of initial and boundary
data (cf. [17]). These assumptions are weaker than the standard assumptions
for stochastic equations where Lipschitz continuity in $x$ is required, but we
are not going to discuss this problem further (for the existence of unique
solutions of equation (1) with local Lipschitz continuity of coefficients see
Chapter 1 of the book by Cerrai [12]). Successfully, partial Schauder’s
estimates were extended to bounded solutions of parabolic problems on the
whole space (with lower order unbounded terms in [21] and all unbounded
coefficients in [26]). Partial Schauder’s estimates for unbounded solutions
(in weighted Hölder spaces) were obtained in [25] but only for bounded
coefficients. Our goal in this paper is to obtain these estimates in the whole
space for unbounded solutions and unbounded coefficients. These results are an
extension of results in [25] and [26].

Time regularity of the coefficients in the Kolmogorov equation requires an
additional comment. There are papers assuming only $(t,x)$ measurability of
the coefficients and Hölder’s regularity in the spatial variables (cf. [7],
[26]). Under these assumptions it can be proved that the Kolmogorov equation
is fulfilled in a generalized sense, i.e., the solution $u(t,x)$ is a
Lipschitz continuous function in $t$ and fulfills the equation outside a
negligible set. We are not going to explore this generalization, as our goal
is the construction of classical solutions of equation (3), which requires the
assumption of $t$-continuity of the coefficients. Then under suitable
assumptions on the space regularity, we get strong solutions of equation (1)
and $C^{1,2}\bigl{(}(0,T)\times\mathbb{R}^{d}\bigr{)}$ solutions of equation
(3).

The rest of the paper is organized as follows. In Section 2 we prove under
relatively weak assumptions that the function given by the Feynman-Kac formula
is a classical solution of the Kolmogorov equation (Theorem 2.4). We also
obtain estimates of the spatial derivatives of this solution (Theorem 2.7 and
Corollary 2.8). In Section 3 we extend the estimates of Section 2 to partial
Schauder’s estimates in spaces of polynomially growing functions (Theorem
3.11).

##  2 Equations of Kolmogorov type

We will now concentrate on the probabilistic representation of the solutions
to the Cauchy problem of Kolmogorov’s equation

$\begin{split}&D_{t}u(t,x)+{\mathcal{A}}^{t}u(t,x)-c(t,x)u(t,x)+f(t,x)=0,\
(t,x)\in[0,T)\times\mathbb{R}^{d},\\\ &u(T,x)=h(x),\quad
x\in\mathbb{R}^{d},\end{split}$ (4)

where ${\mathcal{A}}^{t}$ given by equation (2) is associated with the
stochastic differential equation

$d{X}(s)={b}(s,{X}(s))ds+{\sigma}(s,{X}(s))d{W}(s),\quad{X}(t)={x}.$ (5)

About the coefficients of equation (5) we make the following assumptions.

######  Assumption 2.1.

Let the coefficients in equation (5) be deterministic, continuous functions in
$(s,{y})$ and such that:

  1. (A1)

$|{b}(s,{y})-{b}(s,{y}^{\prime})|+|{\sigma}(s,{y})-{\sigma}(s,{y}^{\prime})|\leq
C|{y}-{y}^{\prime}|$;

  2. (A2)

$|{b}(s,{y})|+|{\sigma}(s,{y})|\leq C(1+|{y}|)$;

for ${y},{y}^{\prime}\in\mathbb{R}^{d}$ and $s\in[t,T]$, where $|\cdot|$
denotes the Euclidean norm in $\mathbb{R}^{d}$ and $C$ is a finite, positive
constant depending on $T$.

We also assume that $x$ is a deterministic vector in $\mathbb{R}^{d}$.

Under the above assumptions, equation (5) possesses a unique strong solution
${X}^{t,x}(s)$.

Our goal is to prove that a solution of the Cauchy problem (4) is given by the
Feynman-Kac formula

$\begin{split}u(t,x)=\,&\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}\\\
&\biggl{.}+\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)},\end{split}$
(6)

where ${X}^{t,x}(s)$ is a solution of equation (5).

In the rest of the paper, we will use the following notation.

######  Notation .

  1. 1.

The space of functions $f\kern-3.00003pt:\,\mathcal{B}\to\mathbb{R}$ which are
bounded and uniformly continuous on $\mathcal{B}$ will be denoted
$BC(\mathcal{B})$. This is a Banach space with the norm

$\|f\|_{BC(\mathcal{B})}=\sup_{z\in\mathcal{B}}|f(z)|.$

Here $\mathcal{B}$ can be a subset of $[0,T]\times\mathbb{R}^{d}$ or a subset
of $\mathbb{R}^{d}$.

  2. 2.

The space $BC^{p}(U)$, $p\geq 1$, where $U\subset\mathbb{R}^{d}$, is a space
of functions $f\kern-3.00003pt:\,U\to\mathbb{R}$ with continuous, bounded
derivatives $D^{\alpha}f(x)\in BC(U)$, for $1\leq|\alpha|\leq p$. This space
is a Banach space with the norm

$\|f\|_{BC^{p}(U)}=\sum_{|\alpha|\leq p}\sup_{x\in U}|D^{\alpha}f(x)|.$

  3. 3.

Let $U$ an open, bounded domain in $\mathbb{R}^{d}$. We will always assume
that an open bounded set $U$ is compactly embedded in $B(R)$, an open ball in
$\mathbb{R}^{d}$ of radius $R$, i.e., $\bar{U}\subset B(R)$. For an open,
bounded set $U$ we define the Hölder space $H^{\beta}(U)$, a space of
functions $f\kern-3.00003pt:\,U\to\mathbb{R}$ which are in $BC(U)$ and are
$\beta$-Hölder continuous, that is

$[f]_{\beta}=\sup_{\genfrac{}{}{0.0pt}{}{x,x^{\prime}\in U}{x\neq
x^{\prime}}}\biggl{(}\frac{|f(x)-f(x^{\prime})|}{|x-x^{\prime}|^{\beta}}\biggl{)}<+\infty.$

The Hölder space $H^{p+\beta}(U)$ is a space of functions $f(x)$ with
continuous derivatives $D^{\alpha}f(x)\in BC(U)$, for $0\leq|\alpha|\leq p$,
and $\beta$-Hölder continuous derivatives $D^{\alpha}f(x)$ for $|\alpha|=p$.

The space $H^{p+\beta}(U)$ is a Banach space for all $p\geq 0$ and
$\beta\in(0,1)$ with the norm

$\|f\|_{H^{p+\beta}(U)}=\sum_{|\alpha|\leq p}\sup_{x\in
U}|D^{\alpha}f(x)|+\sum_{|\alpha|=p}\bigl{[}D^{\alpha}f(x)\bigr{]}_{\beta}.$

Analogously, we define the space $H^{p+\beta}(\mathbb{R}^{d})$.

  4. 4.

The Hölder space $H^{p+\beta}_{\mathit{loc}}(\mathbb{R}^{d})$ is a space of
functions $f\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$, which for any
open, bounded set $U\subset\mathbb{R}^{d}$ belong to $H^{p+\beta}(U)$.

  5. 5.

Let $P(x)=1+|x|^{2q}$ denote for an integer $q\geq 0$ the polynomial weight in
$\mathbb{R}^{d}$. The space of functions
$f\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ such that $fP^{-1}\in
BC^{p}(\mathbb{R}^{d})$, $p\geq 0$, will be denoted
$BC^{p}_{P}(\mathbb{R}^{d})$. This is a Banach space with the norm

$\|f\|_{BC^{p}_{P}(\mathbb{R}^{d})}=\sum_{|\alpha|\leq
p}\sup_{x\in\mathbb{R}^{d}}\bigg{|}D^{\alpha}\frac{f(x)}{P(x)}\bigg{|}.$

  6. 6.

$H^{p+\beta}_{P}(\mathbb{R}^{d})$ denotes the space of functions
$f\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ such that $fP^{-1}\in
H^{p+\beta}(\mathbb{R}^{d})$. This is a Banach space with the norm

$\|f\|_{H^{p+\beta}_{P}(\mathbb{R}^{d})}=\sum_{|\alpha|\leq
p}\sup_{x\in\mathbb{R}^{d}}\bigg{|}D^{\alpha}\frac{f(x)}{P(x)}\bigg{|}+\sum_{|\alpha|=p}\biggl{[}D^{\alpha}\frac{f(x)}{P(x)}\biggr{]}_{\beta}.$

By simple computations, we obtain the following lemma that gives an equivalent
definition of the norm in $H^{p+\beta}_{P}(\mathbb{R}^{d})$ (this is Lemma 2.5
of [25]).

######  Lemma 2.2.

For $f\in H^{p+\beta}_{P}(\mathbb{R}^{d})$ the norm
$\|f\|_{H^{p+\beta}_{P}(\mathbb{R}^{d})}$ is equivalent to the norm

${|\kern-1.00006pt|\kern-1.00006pt|}f{|\kern-1.00006pt|\kern-1.00006pt|}_{H^{p+\beta}_{P}(\mathbb{R}^{d})}=\sum_{|\alpha|\leq
p}\sup_{x\in\mathbb{R}^{d}}\bigg{|}\frac{D^{\alpha}f(x)}{P(x)}\bigg{|}+\sum_{|\alpha|=p}\biggl{[}\frac{D^{\alpha}f(x)}{P(x)}\biggr{]}_{\beta}.$

We begin with the proof that under suitable assumptions, function $u(t,x)$
given by equation (6) is a continuous function for
$(t,x)\in[0,T]\times\mathbb{R}^{d}$. Results of this type are proved in many
papers on the Kolmogorov equation with unbounded coefficients, but under very
restrictive assumptions on the regularity of the coefficients (cf. [3], [23],
[26]). In the paper by Rubio [30], the continuity of $u(t,x)$ was proved
assuming $t$-Hölder and $x$-Lipschitz continuity of the coefficients and some
regularity of initial data. The following theorem is much stronger as we
assume only the continuity of coefficients and initial data, and Hölder’s
continuity in the spatial variables of the coefficients.

######  Theorem 2.3.

Let the coefficients of equation (5) fulfill the conditions of Assumptions
2.1, i.e., ${b}(t,{x})$ and ${\sigma}(t,{x})$ be $(t,x)$-continuous for
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, and with respect to $x$ be Lipschitz
continuous with linear growth.

The coefficients of equation (4) fulfill the following assumptions:

  1. 1.

$c\kern-3.00003pt:\,[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ and
$f\kern-3.00003pt:\,[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ are
$(t,x)$-continuous and locally Hölder continuous with respect to $x$, i.e.,
for an open bounded set $U$ such that $\bar{U}\subset B(R)$, and $t\in(0,T)$,
$c(t,\cdot)$, and $f(t,\cdot)$ belong to $H^{\beta}(U)$, $0<\beta<1$, and

$\|c(t,\cdot)\|_{H^{\beta}(U)}+\|f(t,\cdot)\|_{H^{\beta}(U)}\leq C(T,R).$

  2. 2.

$h\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ is $x$-continuous.

  3. 3.

There is $c_{0}>0$ such that $c(t,x)\geq c_{0}$ for
$(t,x)\in[0,T]\times\mathbb{R}^{d}$.

  4. 4.

There is an integer exponent $q\geq 1$ such that:

    1. $|f(t,x)|\leq C(T)\bigl{(}1+|x|^{2q}\bigr{)}$ for$(t,x)\in[0,T]\times\mathbb{R}^{d}$.

    2. $|h(x)|\leq C\bigl{(}1+|x|^{2q}\bigr{)}$ for $x\in\mathbb{R}^{d}$.

Under the above assumptions the function $u(t,x)$ given by equation (6)
belongs to $C\bigl{(}[0,T]\times\mathbb{R}^{d}\bigr{)}$. Furthermore

$\sup_{0\leq t\leq T}|u(t,x)|\leq C(T)\bigl{(}1+|x|^{2q}\bigr{)},\quad
x\in\mathbb{R}^{d},$

where $q$ is the exponent defining the rate of growth of $f(t,x)$ and $h(x)$.

Proof. The proof explores many ideas of the proof of Theorem 3.1 of [30]. We
are repeating this proof not only for the reader’s convenience but to point
out the differences that make our proof possible under much weaker
assumptions.

The proof of continuity of $u(t,x)$ on the interval $[0,T]$ requires special
treatment of boundary points. To avoid these problems, we consider the
interval $(-\gamma,T]$ for $\gamma$ positive but small. This overcomes the
problem of the boundary point $t=0$. The point $t=T$ still requires special
consideration. Let $(t,x)\in(-\gamma,T]\times\mathbb{R}^{d}$ and
$\\{(t_{n},x_{n})\\}_{n\in\mathcal{N}}$ be a sequence of points
$(t_{n},x_{n})\in(-\gamma,T]\times\mathbb{R}^{d}$ converging to $(t,x)$. We
assume that $t_{n}\nearrow t$, then the point $t=T$ can be treated in the same
way as interior points. This approach simplifies computations and puts no
restriction on the generality of considerations.

The proof of convergence

$u(t_{n},x_{n})\to u(t,x)$

can be reduced by formula (6) to the following

$\displaystyle\mathbb{E}\biggl{(}$ $\displaystyle
h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
$\displaystyle\biggl{.}-h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle+\mathbb{E}\biggl{(}$
$\displaystyle\int_{t_{n}}^{T}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}ds\biggr{.}$
$\displaystyle\biggl{.}-\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}\to
0.$

The strategy of the proof is to show that the expectations of random variables

$\displaystyle v^{n}_{1}=\,$
$\displaystyle\biggl{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
$\displaystyle\biggl{.}-h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{|}$

and

$\displaystyle v^{n}_{2}=\,$
$\displaystyle\biggl{|}\int_{t_{n}}^{T}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}ds\biggr{.}$
$\displaystyle\biggl{.}-\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{|}$

converge to zero.

An essential step in the proof is to show that variables $v^{n}_{1}$ and
$v^{n}_{2}$ are uniformly integrable. We obtain the required uniform
integrability by proving the square integrability of each of these variables.

For $v^{n}_{1}$ we have

$\displaystyle\int_{\Omega}|v^{n}_{1}|^{2}d\mathbb{P}$ $\displaystyle\leq
2\int_{\Omega}\biggl{(}|h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}|\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}\biggr{)}^{2}d\mathbb{P}$
$\displaystyle+2\int_{\Omega}\biggl{(}|h\bigl{(}X^{t,x}(T)\bigr{)}|\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}^{2}d\mathbb{P}$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\Bigl{(}\bigl{(}1+|X^{t,x}(T)|^{4q}\bigr{)}+\bigl{(}1+|X^{t_{n},x_{n}}(T)|^{4q}\bigr{)}\Bigr{)}d\mathbb{P}\leq
C(1+|x|^{4q}).$

Thus $v^{n}_{1}$ is uniformly integrable.

To prove that $\mathbb{E}(v^{n}_{1})\to 0$ as $n\to\infty$ we define the set

$\displaystyle B_{M,\eta}=$
$\displaystyle\\{\omega\in\Omega\kern-3.00003pt:\,\sup_{s\in[t,T]}|X^{t,x}(\omega,s)|\leq
M,\sup_{s\in[t_{n},T]}|X^{t_{n},x_{n}}(\omega,s)|\leq M,$
$\displaystyle\sup_{s\in[t_{n},T]}|X^{t,x}(\omega,s)-X^{t_{n},x_{n}}(\omega,s)|\leq\eta\\}.$

To avoid problem with the comparison of processes $X^{t_{n},x_{n}}$ and
$X^{t,x}$ on the interval $(t_{n},t)$ we extend $X^{t,x}$ on this interval
putting $X^{t,x}(s)=x$ for $s\in[t_{n},t)$.

It follows from the uniform integrability of $v^{n}_{1}$ that taking $M$
sufficiently large and $\eta$ sufficiently small we obtain

$\int_{\Omega\setminus B_{M,\eta}}v^{n}_{1}d\mathbb{P}\leq\delta,$

where $\delta$ is a small number. To estimate the integral over $B_{M,\eta}$,
we split $v^{n}_{1}$

$\begin{split}v^{n}_{1}\leq&\,\Big{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}\\\
&+\Big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\\\
&\times\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}.\end{split}$
(7)

For the mean value of the first term in the right hand side of (7), we have

$\displaystyle\mathbb{E}\biggl{(}$
$\displaystyle\Big{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle\leq\mathbb{E}\Big{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}$
$\displaystyle\leq
C\Bigl{(}1+\mathbb{E}\bigl{|}X^{t_{n},x_{n}}(T)\bigr{|}^{2q}+\mathbb{E}\bigl{|}X^{t,x}(T)\bigr{|}^{2q}\Bigr{)}\leq
C\bigl{(}1+|x|^{2q}\bigr{)}.$

Since $X^{t_{n},x_{n}}(T)$ converges in probability to $X^{t,x}(T)$ (cf.
Theorem II.2.1 in [22] and Lemma III.6.13 in [20]) then by Lemma III.6.13 in
[20] we have

$\mathbb{E}\biggl{(}\Big{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\to
0.$

Choosing $n$ large enough, we obtain from the above

$\int_{B_{M,\eta}}\biggl{(}\Big{|}h\bigl{(}X^{t_{n},x_{n}}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}d\mathbb{P}\leq\delta/2.$

The estimation of the integral of the second term in the right hand side of
(7) we begin with the estimation of the difference of exponents

$\displaystyle\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}$
$\displaystyle\leq\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{T}\Bigl{(}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}-c\bigl{(}r,X^{t,x}(r)\bigr{)}\Bigr{)}dr\Bigr{)}-1\bigg{|}$
$\displaystyle\leq\biggl{(}\exp\Bigl{(}\int_{t_{n}}^{T}\Big{|}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}-c\bigl{(}r,X^{t,x}(r)\bigr{)}\Big{|}dr\Bigr{)}-1\biggr{)}$
$\displaystyle\leq\biggl{(}\exp\Bigl{(}\int_{t_{n}}^{T}C\big{|}X^{t_{n},x_{n}}(r)-X^{t,x}(r)\big{|}^{\beta}dr\Bigr{)}-1\biggr{)}$
$\displaystyle\leq\mathrm{e}^{CT\eta^{\beta}}-1\leq C\theta T\eta^{\beta},$

where we used Hölder’s continuity of $c(t,x)$ and the mean value theorem which
introduced $\theta\in[0,1]$. Then for a sufficiently small $\eta$, we obtain

$\displaystyle\int_{B_{M,\eta}}\biggl{(}\big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\big{|}\biggr{.}$
.
$\displaystyle\times\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{T}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}\biggr{)}d\mathbb{P}\leq\delta/2.$

Collecting the above estimates, we have

$\mathbb{E}(v^{n}_{1})=\int_{\Omega}v^{n}_{1}d\mathbb{P}\leq 2\delta.$

Similarly to $v^{n}_{1}$, we prove the uniform integrability of $v^{n}_{2}$.
To this end, we compute

$\displaystyle\int_{\Omega}|v^{n}_{2}|^{2}$ $\displaystyle d\mathbb{P}\leq
2\int_{\Omega}\biggl{(}\int_{t_{n}}^{T}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}^{2}d\mathbb{P}$
$\displaystyle+2\int_{\Omega}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}^{2}d\mathbb{P}$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\int_{t_{n}}^{T}\Bigl{(}\bigl{(}1+\sup_{s\in[t,T]}|X^{t,x}(s)|^{4q}\bigr{)}+\bigl{(}1+\sup_{s\in[t_{n},T]}|X^{t_{n},x_{n}}(s)|^{4q}\bigr{)}\Bigr{)}dsd\mathbb{P}$
$\displaystyle\leq$ $\displaystyle
CT\int_{\Omega}\Bigl{(}2+\sup_{s\in[t,T]}|X^{t,x}(s)|^{4q}+\sup_{s\in[t_{n},T]}|X^{t_{n},x_{n}}(s)|^{4q}\Bigr{)}d\mathbb{P}$
$\displaystyle\leq$ $\displaystyle
CT\bigl{(}2+|x|^{4q}+|x_{n}|^{4q}\bigr{)}\leq CT(1+|x|^{4q}).$

We define the set $B_{M,\eta}$ in a similar way as for $v^{n}_{1}$ and select
$M$ and $\eta$ such that

$\int_{\Omega\setminus B_{M,\eta}}v^{n}_{2}d\mathbb{P}\leq\delta.$

To estimate the integral over $B_{M,\eta}$ we split $v^{n}_{2}$ as follows

$\begin{split}v^{n}_{2}\leq&\,\int_{t_{n}}^{T}\Big{|}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}-f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}ds\\\
&+\int_{t_{n}}^{T}\Big{|}f\bigl{(}s,X^{t,x}(T)\bigr{)}\Big{|}\\\
&\times\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}ds\end{split}$
(8)

For the first integral in the right hand side of (8), we apply Hölder’s
continuity of $f(t,x)$ to obtain

$\displaystyle\int_{B_{M,\eta}}$
$\displaystyle\int_{t_{n}}^{T}\\!\\!\Big{|}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}-f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}\exp\Bigl{(}\\!-\\!\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}dsd\mathbb{P}$
$\displaystyle\leq\,C\int_{B_{M,\eta}}\int_{t_{n}}^{T}\Big{|}f\bigl{(}s,X^{t_{n},x_{n}}(s)\bigr{)}-f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}dsd\mathbb{P}$
$\displaystyle\leq\,C\int_{B_{M,\eta}}\int_{t_{n}}^{T}\big{|}X^{t_{n},x_{n}}(s)-X^{t,x}(s)\big{|}^{\beta}dsd\mathbb{P}\leq\,CT\eta^{\beta}.$

To estimate the second integral in the right hand side of (8) for $t_{n}\leq
s\leq T$, we evaluate, like for $v^{n}_{1}$, the difference of exponents

$\displaystyle\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}\leq
C\theta s\eta^{\beta},$

Then, for a sufficiently small $\eta$, we obtain for the second integral in
the right hand side of (8)

$\displaystyle\int_{B_{M,\eta}}$
$\displaystyle\int_{t_{n}}^{T}\Big{|}f\bigl{(}s,X^{t,x}(T)\bigr{)}\Big{|}$
$\displaystyle\times\bigg{|}\exp\Bigl{(}-\int_{t_{n}}^{s}c\bigl{(}r,X^{t_{n},x_{n}}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}dsd\mathbb{P}$
$\displaystyle\leq\,C\eta^{\beta}\,\int_{B_{M,\eta}}\int_{t_{n}}^{T}s\Big{|}f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}ds$
$\displaystyle\leq\,C\eta^{\beta}\,\int_{B_{M,\eta}}\int_{t_{n}}^{T}s\Bigl{(}1+\sup_{s\in[t,T]}\big{|}X^{t,x}(s)\big{|}^{2q}\Bigr{)}dsd\mathbb{P}\leq\,CT^{2}\eta^{\beta}\bigl{(}1+|x|^{2q}\bigr{)}.$

By the above estimates, we have for a sufficiently small $\eta$

$\displaystyle\mathbb{E}(v^{n}_{2})=\int_{\Omega}v^{n}_{2}d\mathbb{P}\leq\int_{B_{M,\eta}}v^{n}_{2}d\mathbb{P}+\delta\leq
C\eta^{\beta}(1+|x|^{2q})+\delta\leq 2\delta.$

This proves that $u(t,x)$ is continuous in $(-\gamma,T]\times\mathbb{R}^{d}$,
hence also in $[0,T]\times\mathbb{R}^{d}$.

The growth estimate of $u(t,x)$ can be easily obtained from equation (6). For
the first term in (6), we have

$\displaystyle\bigg{|}\mathbb{E}\biggl{(}$ $\displaystyle
h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\bigg{|}$
$\displaystyle\leq\mathbb{E}\Big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}\leq
C\Bigl{(}1+\mathbb{E}\bigl{|}X^{t,x}(T)\bigr{|}^{2q}\Bigr{)}\leq
C\bigl{(}1+|x|^{2q}\bigr{)},$

where the constant $C$ depends on $T$, $q$, and the constant in the growth
estimate of $h(x)$.

For the second term in (6) we obtain the estimate

$\displaystyle\bigg{|}$
$\displaystyle\mathbb{E}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}\bigg{|}$
$\displaystyle\leq\,C\int_{t}^{T}\mathbb{E}\Big{|}f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}ds\leq\,C\int_{t}^{T}\Bigl{(}1+\mathbb{E}\big{|}X^{t,x}(s)\big{|}^{2q}\Bigr{)}ds\leq\,C\bigl{(}1+|x|^{2q}\bigr{)},$

where the constant $C$ depends on $T$, $q$, and the constant in the growth
estimate of $f(t,x)$.

The above estimates give

$\sup_{0\leq t\leq T}|u(t,x)|\leq\,C\bigl{(}1+|x|^{2q}\bigr{)},$

where the constant $C$ depends on $T$, $q$, and the constants in the growth
estimates of $f(t,x)$ and $h(x)$. ∎

The continuous function $u(t,x)$ defined by equation (6) is not a solution of
the Kolmogorov equation (4). But under slightly stronger assumptions on the
coefficients of equation (4), this function is already in
$C^{1,2}\bigl{(}(0,T)\times\mathbb{R}^{d}\bigr{)}$ and is a solution of the
Cauchy problem (4).

######  Theorem 2.4.

Let the conditions of Assumption 2.1 be fulfilled and operator
${\mathcal{A}}^{t}$ be uniformly elliptic, i.e., there exists $\delta(T)>0$
such that for $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$

$\sum_{i,j=1}^{d}a_{ij}(t,x)\xi_{i}\xi_{j}\geq\delta(T)|\xi|^{2}.$

Let $h\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ belong to
$BC(\mathbb{R}^{d})$,
$c\kern-3.00003pt:\,[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ and
$f\kern-3.00003pt:\,[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ be
$(t,x)$-continuous and locally Hölder continuous with respect to $x$, i.e.,
for an open, bounded set $U$ such that $\bar{U}\subset B(R)$, and $t\in(0,T)$,
$c(t,\cdot)$, and $f(t,\cdot)$ belong to $H^{\beta}(U)$, $0<\beta<1$, and

$\|c(t,\cdot)\|_{H^{\beta}(U)}+\|f(t,\cdot)\|_{H^{\beta}(U)}\leq C(T,R).$

In addition, let $c(t,x)\geq c_{0}>0$ for $(t,x)\in[0,T]\times\mathbb{R}^{d}$.

Then $u(t,x)$ defined by equation (6) is such that $u(t,\cdot)\in
H^{2+\beta}(U)$ for $t\in(0,T)$. Furthermore, $u(t,x)$ belongs to
$C^{1,2}\bigl{(}(0,T)\times\mathbb{R}^{d}\bigr{)}$ and is a unique classical
solution of the Cauchy problem (4).

Proof. For $(t,x)\in(t_{1},t_{2})\times U$, where $(t_{1},t_{2})\subset[0,T]$
and $U$ is an open set with a regular boundary $\partial U$ of class $C^{2}$,
consider the following Cauchy problem with the coefficients fulfilling the
assumptions of the theorem

$\begin{split}&D_{t}v(t,x)+{\mathcal{A}}^{t}v(t,x)-c(t,x)v(t,x)+f(t,x)=0,\
(t,x)\in(t_{1},t_{2})\times U,\\\ &v(t_{2},x)=u(t_{2},x),\quad x\in U,\\\
&v(t,x)=u(t,x),\quad(t,x)\in(t_{1},t_{2})\times\partial U,\end{split}$ (9)

where $u(t,x)$ defined by equation (6) is a continuous function by Theorem
2.3.

To obtain a regular solution of problem (9), let us observe that due to
conditions (A1) and (A2) of Assumption 2.1 the coefficients $a_{ij}$ and
$b_{i}$ as functions of $x$ belong to $H^{\beta}(U)$ and

$\|a_{ij}(t,\cdot)\|_{H^{\beta}(U)}+\|b_{i}(t,\cdot)\|_{H^{\beta}(U)}\leq
C(T,R).$

Then, we approximate $u(t,x)$ by a sequence of continuous functions
$u_{n}(t,x)$ which are Hölder continuous in $x$ with exponent $\beta$. Under
these conditions, we can apply Theorem 16.1 of [24]. According to this
theorem, there exist unique solutions $v_{n}(t,x)$ of problem (9) with
initial/boundary data $u_{n}(t,x)$ such that $v_{n}(t,x)\in
C^{1,2}\bigl{(}(t_{1},t_{2})\times U\bigr{)}$. Moreover, $v_{n}(t,\cdot)\in
H^{2+\beta}(U)$ for all $t\in(t_{1},t_{2})$. Similarly like in the proof of
Lemma 2.6 in [17], we can pass to the limit with solutions $v_{n}(t,x)$ using
the maximum principle and the condition $c(t,x)\geq c_{0}>0$, and obtain a
unique solution $v(t,x)\in C^{1,2}\bigl{(}(t_{1},t_{2})\times U\bigr{)}$ of
problem (9) with continuous initial/boundary data $u(t,x)$.

To obtain a stochastic representation of solution $v(t,x)$ under the weak
assumption of the theorem, we apply Theorem 7.4.2 of [31] (see also Theorem
IV.4.5 of [16]). Then we get

$\begin{split}v(t,x)=\,&\mathbb{E}_{t,x}\biggl{(}u\bigl{(}(\tau,X^{t,x}(\tau)\bigr{)}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\\\
&+\mathbb{E}_{t,x}\biggl{(}\int_{t}^{\tau}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)},\end{split}$
(10)

where $X^{t,x}(s)$ is a solution of the stochastic equation (5) with initial
condition $X(t)=x$ and $\tau=T\wedge\tau_{\varepsilon}$ with
$\tau_{\varepsilon}$ the first exit time from $U$, i.e.,
$\tau_{\varepsilon}=\inf\\{s>0\kern-3.00003pt:\,X^{t,x}(s)\notin\bar{U}\\}$
($\mathbb{E}(\tau_{\varepsilon})<\infty$ due to the proof of Theorem 5.5.1 in
[14]).

Let $w(t,x)$ denote the restriction of $u(t,x)$ defined by equation (6) to
domain $U$. It is clear that
$w\bigl{(}s,X^{t,x}(s)\bigr{)}=u\bigl{(}s,X^{t,x}(s)\bigr{)}$ for $0\leq
s\leq\tau$ and $X(t)=x$ with $x\in U$. Hence

$w\bigl{(}s,X^{t,x}(s)\bigr{)}=\mathbb{E}\bigl{(}u\bigl{(}s,X^{t,x}(s)\bigr{)}\big{|}\mathcal{F}^{t}_{\tau}\bigr{)}$

and

$\begin{split}w&(t,x)=\,\mathbb{E}_{t,x}\Biggl{(}\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}\\\
&+\mathbb{E}_{t,x}\Biggl{(}\mathbb{E}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}.\end{split}$
(11)

Using the Markov property of process $X(s)$, we obtain for the first integral

$\displaystyle\mathbb{E}_{t,x}\Biggl{(}$
$\displaystyle\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\Biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
.
$\displaystyle\times\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{\tau}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\Biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
.
$\displaystyle\times\mathbb{E}_{\tau,X(\tau)}\biggl{(}h\bigl{(}X^{\tau,X(\tau)}(T)\bigr{)}\exp\Bigl{(}-\int_{\tau}^{T}c\bigl{(}r,X^{\tau,X(\tau)}(r)\bigr{)}dr\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}u_{1}\bigl{(}\tau,X^{t,x}(\tau)\bigr{)}\biggr{)},$

where $u_{1}(t,x)$ corresponds to the first term in formula (6)

$u_{1}(t,x)=\mathbb{E}_{t,x}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}.$

For the second integral in (11), we obtain

$\displaystyle\mathbb{E}_{t,x}\Biggl{(}$
$\displaystyle\mathbb{E}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\Biggl{(}\mathbb{E}\biggl{(}\int_{t}^{\tau}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle+\,\mathbb{E}_{t,x}\Biggl{(}\mathbb{E}\biggl{(}\int_{\tau}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\biggl{(}\int_{t}^{\tau}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}$
$\displaystyle+\,\mathbb{E}_{t,x}\Biggl{(}\mathbb{E}\biggl{(}\int_{\tau}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}.$

Making a change of variables in the last integral and using the Markov
property of $X(s)$, we get

$\displaystyle\mathbb{E}_{t,x}\Biggl{(}$
$\displaystyle\mathbb{E}\biggl{(}\int_{\tau}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\Biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
.
$\displaystyle\times\mathbb{E}\biggl{(}\int_{\tau}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{\tau}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\bigg{|}\mathcal{F}^{t}_{\tau}\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\Biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
.
$\displaystyle\times\mathbb{E}_{\tau,X(\tau)}\biggl{(}\int_{\tau}^{T}f\bigl{(}s,X^{\tau,X(\tau)}(s)\bigr{)}\exp\Bigl{(}-\int_{\tau}^{s}c\bigl{(}r,X^{\tau,X(\tau)}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}\Biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}_{t,x}\biggl{(}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}u_{2}\bigl{(}\tau,X^{t,x}(\tau)\bigr{)}\biggr{)},$

where $u_{2}(t,x)$ corresponds to the second term in formula (6)

$u_{2}(t,x)=\mathbb{E}_{t,x}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}.$

Collecting all terms we have

$\displaystyle w(t,x)=\,$
$\displaystyle\mathbb{E}_{t,x}\biggl{(}u\bigl{(}(\tau,X^{t,x}(\tau)\bigr{)}\exp\Bigl{(}-\int_{t}^{\tau}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle+\mathbb{E}_{t,x}\biggl{(}\int_{t}^{\tau}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}$

which gives the equality $w(t,x)=v(t,x)$.

Therefore, $w(t,\cdot)\in H^{2+\beta}(U)$ and $u(t,\cdot)\in
H^{2+\beta}_{\mathit{loc}}(\mathbb{R}^{d})$ for $t\in(0,T)$ which by Theorem
16.1 of [24] implies $u\in C^{1,2}\bigl{(}(0,T)\times\mathbb{R}^{d}\bigr{)}$
and shows that $u(t,x)$ is a unique classical solution of the Cauchy problem
(4). ∎

The result of Theorem 2.4 can be easily extended to a theorem showing that a
solution of problem (4) is as smooth as the coefficients of the equation. The
theorem below can be obtained from Theorem 16.1 of [24] after applying the
approximation of a continuous initial condition by a smooth initial condition
in the spirit of Lemma 2.6 in [17].

######  Theorem 2.5.

Let $U$ be an arbitrary open bounded set $U$ in $\mathbb{R}^{d}$ with
$\bar{U}\subset B(R)$. If $h\in BC(U)$, $c(t,x)\geq c_{0}>0$ for
$(t,x)\in(0,T)\times U$, and the coefficients $a_{ij}$, $b_{i}$, $c$, and $f$
belong to $H^{p+\beta}(U)$, $\beta\in(0,1)$, with

$\displaystyle\|a_{ij}(t,\cdot)\|_{H^{p+\beta}(U)}+\|b_{i}(t,\cdot)$
$\displaystyle\|_{H^{p+\beta}(U)}+\|c(t,\cdot)\|_{H^{p+\beta}(U)}+\|f(t,\cdot)\|_{H^{p+\beta}(U)}$
$\displaystyle\leq C(T,R),\quad t\in(0,T),$

and $u(t,x)$ is a solution of equation (4) in $(0,T)\times U$, then
$u(t,\cdot)\in H^{p+2+\beta}(U)$.

######  Remark 2.6.

The condition $c_{0}>0$ in the growth assumption of Theorem 2.3, important for
the maximum principle used in the proofs of Theorems 2.4 and 2.5, can be
relaxed to $c_{0}\in\mathbb{R}$ by a change of variables. If $u(t,x)$ is a
solution of the Kolmogorov equation (4) with $c(t,x)\geq c_{0}>0$, then
$v(t,x)=\mathrm{e}^{-\gamma t}u(t,x)$ solves problem (4) with
$c^{\prime}(t,x)=c(t,x)-\gamma$, $f^{\prime}(t,x)=\mathrm{e}^{\gamma
t}f(t,x)$. Hence, we can replace $c_{0}$ with $c^{\prime}_{0}=c_{0}-\gamma$
where $c^{\prime}_{0}$ can be an arbitrary real number.

######  Theorem 2.7.

Let for an integer $p\geq 2$ the coefficients of equation (5) fulfill the
conditions:

  1. ( S1)

$|{b}(t,{x})|+|{\sigma}(t,{x})|\leq C(T)(1+|{x}|)$ and ${b}(t,{x})$,
${\sigma}(t,{x})$ are $(t,{x})$-continuous for
$(t,x)\in(0,T)\times\mathbb{R}^{d}$.

  2. (S2)

The derivatives $D_{x}^{\alpha}{b}(t,{x})$, $D_{x}^{\alpha}{\sigma}(t,{x})$
exist for $|\alpha|\leq p$ and are continuous with respect to ${x}$ for all
$t\in(0,T)$, ${x}\in\mathbb{R}^{d}$.

  3. (S3)

$\big{|}D_{x}^{\alpha}{b}(t,{x})\big{|}+\big{|}D_{x}^{\alpha}{\sigma}(t,{x})\big{|}\leq
C$ for $0<|\alpha|\leq p$ and $(t,x)\in(0,T)\times\mathbb{R}^{d}$.

Assume that for the same $p\geq 2$ the coefficients of the Cauchy problem (4)
are such that:

  1. ( K1)

There exists $\delta(T)>0$ such that for $(t,x)\in[0,T]\times\mathbb{R}^{d}$
and $\xi\in\mathbb{R}^{d}\setminus\\{0\\}$

$\sum_{i,j=1}^{d}a_{ij}(t,x)\xi_{i}\xi_{j}\geq\delta(T)|\xi|^{2}.$

  2. ( K2)

Functions $c\kern-3.00003pt:\,[0,\infty)\times\mathbb{R}^{d}\to[0,\infty)$ and
$f\kern-3.00003pt:\,[0,\infty)\times\mathbb{R}^{d}\to\mathbb{R}$ are
$(t,x)$-continuous and $c(t,{x})\geq c_{0}>0$. Function
$h\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ belongs to
$BC(\mathbb{R}^{d})$.

  3. ( K3)

The derivatives $D_{x}^{\alpha}c(t,{x})$, $D_{x}^{\alpha}f(t,{x})$,
$D_{x}^{\alpha}h({x})$ exist for $|\alpha|\leq p$ and are continuous with
respect to ${x}$ for all $t\in(0,T)$.

  4. (K4)

For all $t\in(0,T)$ and $|\alpha|\leq p$ there exist $C>0$, $q\geq 1$ such
that

$\big{|}D_{x}^{\alpha}c(t,{x})\big{|}+\big{|}D_{x}^{\alpha}f(t,{x})\big{|}+\big{|}D_{x}^{\alpha}h({x})\big{|}\leq
C(1+|{x}|^{2q}),\quad x\in\mathbb{R}^{d}.$

Then $u(t,x)$ given by equation (6) is a well defined function in
$[0,T]\times\mathbb{R}^{d}$ which is continuously differentiable with respect
to $t$ in $(0,T)\times\mathbb{R}^{d}$ and has $(t,{x})$-continuous derivatives
$D_{x}^{\alpha}u(t,{x})$, $|\alpha|\leq p$, with the growth estimate

$\sup_{0<t<T}\big{|}D_{x}^{\alpha}u(t,{x})\big{|}\leq
C(1+|{x}|^{2q(1+|\alpha|)}),\quad|\alpha|\leq p,\ {x}\in\mathbb{R}^{d},$ (12)

where $C$ is a positive constant, and $q$ is the same as in the growth
estimates of the derivatives of $c$, $f$, and $h$.

Proof. The assertion that $u(t,x)$ is continuously differentiable with respect
to $t$ and possesses continuous derivatives $D_{x}^{\alpha}u(t,{x})$,
$|\alpha|\leq p$, follows straightforwardly from Theorem 2.5. What we want to
prove is the possibility of differentiation under the integral sign. This
differentiation is also crucial to show the estimates of the derivatives.

To simplify the proof, we take $d=1$. First, we will show that under the
assumption of the theorem $u(t,x)$ given by formula (6) can be continuously
differentiated with respect to ${x}$ under the integral sign and
$D^{p}_{x}u(t,x)\leq C(1+|{x}|^{2q(1+p)})$.

Consider the difference $u(t,x+\Delta x)-u(t,x)$. We can write this difference
as the sum of terms:

$\displaystyle u($ $\displaystyle t,x+\Delta x)-u(t,x)$ $\displaystyle=$
$\displaystyle\,\mathbb{E}\biggl{(}\Bigl{(}h\bigl{(}X^{t,x+\Delta
x}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle+\mathbb{E}\Biggl{(}h\bigl{(}X^{t,x+\Delta
x}(T)\bigr{)}\Biggr{.}$ .
$\displaystyle\times\biggl{(}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x+\Delta
x}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\Biggr{)}$
$\displaystyle+\mathbb{E}\biggl{(}\int_{t}^{T}\Bigl{(}f\bigl{(}s,X^{t,x+\Delta
x}(s)\bigr{)}-f\bigl{(}s,X^{t,x}(s)\bigr{)}\Bigr{)}\biggr{.}$ .
$\displaystyle\phantom{f\bigl{(}s,X^{t,x}(s)\bigr{)}f\bigl{(}s,(s)\bigr{)}}\times\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}$
$\displaystyle+\mathbb{E}\Biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x+\Delta
x}(s)\bigr{)}\Biggr{.}$ .
$\displaystyle\times\biggl{(}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x+\Delta
x}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}ds\Biggr{)}.$

Our goal is to show that there exists a limit $\bigl{(}u(t,x+\Delta
x)-u(t,x)\bigr{)}/\Delta x$ for $\Delta x\to 0$. We will consider the
convergence for each term of the difference $\bigl{(}u(t,x+\Delta
x)-u(t,x)\bigr{)}$ separately. For the first term, we have by the mean value
theorem

$\displaystyle\mathbb{E}\biggl{(}\Bigl{(}$ $\displaystyle
h\bigl{(}X^{t,x+\Delta
x}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}\biggl{(}\int_{0}^{1}\frac{d}{d\mu}h\Bigl{(}X^{t,x}(T)+\mu\bigl{(}X^{t,x+\Delta
x}(T)-X^{t,x}(T)\bigr{)}\Bigr{)}d\mu\biggr{.}$ .
$\displaystyle\times\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle=\,$
$\displaystyle\mathbb{E}\biggl{(}\int_{0}^{1}D_{x}h\Bigl{(}X^{t,x}(T)+\mu\bigl{(}X^{t,x+\Delta
x}(T)-X^{t,x}(T)\bigr{)}\Bigr{)}d\mu\biggr{.}$ .
$\displaystyle\times\bigl{(}X^{t,x+\Delta
x}(T)-X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}.$

Dividing the above expression by $\Delta x$, we get for $\Delta x\to 0$ the
convergence

$\frac{X^{t,x+\Delta x}(T)-(X^{t,x}(T)}{\Delta x}\to\frac{\partial}{\partial
x}X^{t,x}(T)$

in $L^{2}$ due to Theorem 8.5.1 in [15]. We have also for $\Delta x\to 0$

$\int_{0}^{1}D_{x}h\Bigl{(}X^{t,x}(T)+\mu\bigl{(}X^{t,x+\Delta
x}(T)-X^{t,x}(T)\bigr{)}\Bigr{)}d\mu\to D_{x}h\bigl{(}X^{t,x}(T)\bigr{)}$

in $L^{2}$. This last convergence follows from the $x$-continuity of
$X^{t,x}(s)$, Lemma III.6.13 in [20], and the estimate

$\mathbb{E}\biggl{(}\int_{0}^{1}\\!\\!D_{x}h\Bigl{(}X^{t,x}(T)+\mu\bigl{(}X^{t,x+\Delta
x}(T)-X^{t,x}(T)\bigr{)}\Bigr{)}d\mu-
D_{x}h\bigl{(}(X^{t,x}(T)\bigr{)}\biggr{)}^{4}\leq C,$

which holds for any $|x|\leq C$ due to Theorem II.2.1 in [22] and the growth
condition for $D_{x}h(x)$.

Hence, due to the continuity of $c(t,x)$, the expression

$\frac{1}{\Delta x}\mathbb{E}\biggl{(}\Bigl{(}h\bigl{(}X^{t,x+\Delta
x}(T)\bigr{)}-h\bigl{(}X^{t,x}(T)\bigr{)}\Bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$

converges to

$\mathbb{E}\biggl{(}D_{x}h\bigl{(}X^{t,x}(T)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(T)\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}.$

Similar considerations show that the term

$\displaystyle\frac{1}{\Delta x}\mathbb{E}$
$\displaystyle\biggl{(}\int_{t}^{T}\Bigl{(}f\bigl{(}s,X^{t,x+\Delta
x}(s)\bigr{)}-f\bigl{(}s,X^{t,x}(s)\bigr{)}\Bigr{)}\biggr{.}$ .
$\displaystyle\phantom{f\bigl{(}s,X^{t,x}(s)\bigr{)}f\bigl{(}s,X^{t,x}(s)\bigr{)}}\times\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}$

converges in $L^{2}$ to

$\mathbb{E}\biggl{(}\int_{t}^{T}D_{x}f\bigl{(}s,X^{t,x}(s)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(s)\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}.$

To obtain convergence for the two other terms let us observe the equality

$\displaystyle\exp\Bigl{(}$
$\displaystyle-\int_{t}^{s}c\bigl{(}r,X^{t,x+\Delta
x}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}$
$\displaystyle=\,$
$\displaystyle\int_{0}^{1}\frac{d}{d\mu}\exp\biggl{(}-\int_{t}^{s}c\Bigl{(}r,X^{t,x}(r)+\mu\bigl{(}X^{t,x+\Delta
x}(r)-X^{t,x}(r)\bigr{)}\Bigr{)}dr\biggr{)}d\mu$ $\displaystyle=\,$
$\displaystyle-\int_{0}^{1}\Biggl{(}\exp\biggl{(}-\int_{t}^{s}c\Bigl{(}r,X^{t,x}(r)+\mu\bigl{(}X^{t,x+\Delta
x}(r)-X^{t,x}(r)\bigr{)}\Bigr{)}dr\biggr{)}\Biggr{.}$
$\displaystyle\phantom{X^{t,x+\Delta
x}(r)}\times\int_{t}^{s}\biggl{(}D_{x}c\Bigl{(}r,X^{t,x}(r)+\mu\bigl{(}X^{t,x+\Delta
x}(r)-X^{t,x}(r)\bigr{)}\Bigr{)}\biggr{.}$
$\displaystyle\Biggl{.}\biggl{.}\phantom{X^{t,x+\Delta
x}(r)-X^{t,x}(r)}\times\bigl{(}X^{t,x+\Delta
x}(r)-X^{t,x}(r)\bigr{)}\biggr{)}dr\Biggr{)}d\mu.$

By Lemma III.6.13 in [20], Theorem II.2.1 in [22], the continuity of $h(x)$,
$f(t,x)$ and $c(t,x)$, and the growth conditions for $h(x)$, $f(t,x)$,
$c(t,x)$, and $D_{x}c(t,x)$, we prove that

$\displaystyle\frac{1}{\Delta x}\mathbb{E}$
$\displaystyle\Biggl{(}h\bigl{(}X^{t,x+\Delta x}(T)\bigr{)}\biggr{.}$ .
$\displaystyle\times\biggl{(}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x+\Delta
x}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\Biggr{)}$

converges to

$\displaystyle-\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}\\!\\!\\!c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\\!\\!\int_{t}^{T}\\!\\!\\!D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(r)dr\biggr{)},$

and

$\displaystyle\frac{1}{\Delta x}$
$\displaystyle\mathbb{E}\Biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x+\Delta
x}(s)\bigr{)}\biggr{.}$ .
$\displaystyle\times\biggl{(}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x+\Delta
x}(r)\bigr{)}dr\Bigr{)}-\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}ds\Biggr{)}$

converges to

$\displaystyle-\mathbb{E}\Biggl{(}\int_{t}^{T}\biggl{(}f\bigl{(}X^{t,x}(s)\bigr{)}$
$\displaystyle\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
$\displaystyle\biggl{.}\times\int_{t}^{s}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(r)dr\biggr{)}ds\Biggr{)}.$

Together the above expressions prove that we can differentiate (6) under the
integral sign and obtain the following formula for the $x$-derivative of
$u(t,x)$

$\displaystyle D_{x}$ $\displaystyle
u(t,x)=\,\mathbb{E}\biggl{(}D_{x}h\bigl{(}X^{t,x}(T)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(T)\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
$\displaystyle-\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
$\displaystyle\biggl{.}\phantom{c\bigl{(}r,X^{t,x}(r)\bigr{)}}\times\int_{t}^{T}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(r)dr\biggr{)}$
$\displaystyle+\mathbb{E}\biggl{(}\int_{t}^{T}D_{x}f\bigl{(}s,X^{t,x}(s)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(s)\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}$
$\displaystyle-\mathbb{E}\Biggl{(}\int_{t}^{T}\biggl{(}f\bigl{(}X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}$
$\displaystyle\biggl{.}\phantom{c\bigl{(}r,X^{t,x}(r)\bigr{)}}\times\int_{t}^{s}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\frac{\partial}{\partial
x}X^{t,x}(r)dr\biggr{)}ds\Biggr{)}.$

The estimate of $D_{x}u(t,x)$ can be obtain similarly like in the proof of
Theorem 2.3 taking into account the above expression for $D_{x}u(t,x)$ and the
boundedness of $|D_{x}X^{t,x}(s)|$

$\displaystyle|D_{x}u(t,x)|\leq$
$\displaystyle\,C\,\mathbb{E}\Big{|}D_{x}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}$
$\displaystyle+C\,\mathbb{E}\biggl{(}\big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\big{|}\int_{t}^{T}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}dr\biggr{)}$
$\displaystyle+C\,\mathbb{E}\biggl{(}\int_{t}^{T}\big{|}D_{x}f\bigl{(}s,X^{t,x}(s)\bigr{)}\big{|}ds\biggr{)}$
$\displaystyle+C\,\mathbb{E}\Biggl{(}\int_{t}^{T}\biggl{(}\big{|}f\bigl{(}X^{t,x}(s)\bigr{)}\big{|}\int_{t}^{s}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}dr\biggr{)}ds\Biggr{)}.$

Straightforwardly we obtain

$\displaystyle\mathbb{E}\Big{|}D_{x}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}+\mathbb{E}\biggl{(}\int_{t}^{T}\big{|}D_{x}f\bigl{(}s,X^{t,x}(s)\bigr{)}\big{|}ds\biggr{)}\leq
C\,\bigl{(}1+|x|^{2q}\bigr{)}.$

To estimate the other two terms, we apply Hölder’s inequality and get

$\displaystyle\mathbb{E}\biggl{(}\big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\big{|}\int_{t}^{T}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}dr\biggr{)}$
$\displaystyle\phantom{h\bigl{(}X^{t,x}(T)\bigr{)}}\leq
C\,\Bigl{(}\mathbb{E}\big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\big{|}^{2}\Bigr{)}^{1/2}\biggl{(}\int_{t}^{T}\mathbb{E}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}^{2}dr\biggr{)}^{1/2}$
$\displaystyle\phantom{h\bigl{(}X^{t,x}(T)\bigr{)}}\leq
C\,\bigl{(}1+|x|^{4q}\bigr{)},$
$\displaystyle\mathbb{E}\Biggl{(}\int_{t}^{T}\biggl{(}\big{|}f\bigl{(}X^{t,x}(s)\bigr{)}\big{|}\int_{t}^{s}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}dr\biggr{)}ds\Biggr{)}$
$\displaystyle\phantom{h\bigl{(}X^{t,x}(T)\bigr{)}}\leq
C\,\int_{t}^{T}\Bigl{(}\mathbb{E}\big{|}f\bigl{(}X^{t,x}(s)\bigr{)}\big{|}^{2}\Bigr{)}^{1/2}\biggl{(}\int_{t}^{s}\mathbb{E}\big{|}D_{x}c\bigl{(}r,X^{t,x}(r)\bigr{)}\big{|}^{2}dr\biggr{)}^{1/2}ds$
$\displaystyle\phantom{h\bigl{(}X^{t,x}(T)\bigr{)}}\leq
C\,\bigl{(}1+|x|^{4q}\bigr{)}.$

For higher $x$-derivatives of $u(t,x)$ we can proceed analogously. ∎

The following corollary reformulates the results of Theorem 2.7 in terms of
the partial Schauder estimates.

######  Corollary 2.8.

Let the assumptions of Theorem 2.7 hold for $p\geq 2$. We assume additionally
that there exists a constant $C>0$ such that for all $t\in[0,T]$

$|D^{\alpha}_{x}c(t,x)|\leq C,\ 0<|\alpha|\leq p,\quad x\in\mathbb{R}^{d}.$

Then for the function $u(t,x)$ defined by equation (6) the following partial
Schauder estimates in the space of differentiable functions with the
polynomial weight $P(x)=1+|x|^{2q}$ are valid

$\sup_{0<t<T}\|u(t,\cdot)\|_{BC^{p}_{P}(\mathbb{R}^{d})}\leq
C\Bigl{(}\|h\|_{BC^{p}_{P}(\mathbb{R}^{d})}+\sup_{0<t<T}\|f(t,\cdot)\|_{BC^{p}_{P}(\mathbb{R}^{d})}\Bigr{)}.$

Proof. The estimate in $BC_{P}(\mathbb{R}^{d})$ can be obtained by repeating
the computations in the proof of Theorem 2.3. Let us recall that $u(t,x)$ is
given by the Feynman-Kac formula

$\begin{split}u(t,x)=\,&\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{.}\\\
&\biggl{.}+\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}.\end{split}$

For the first term of this formula, we have the estimate

$\displaystyle\sup_{0<t<T}$
$\displaystyle\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\bigg{|}\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}\bigg{|}$
$\displaystyle\leq\sup_{0<t<T}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\mathbb{E}\Big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}$
$\displaystyle=\sup_{0<t<T}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\mathbb{E}\Bigg{|}\frac{h\bigl{(}X^{t,x}(T)\bigr{)}}{1+\bigl{|}X^{t,x}(T)\bigr{|}^{2q}}\Bigl{(}1+\bigl{|}X^{t,x}(T)\bigr{|}^{2q}\Bigr{)}\Bigg{|}$
$\displaystyle\leq
C\|h\|_{BC(\mathbb{R}^{d})}\sup_{0<t<T}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\Bigl{(}1+\mathbb{E}\bigl{|}X^{t,x}(T)\bigr{|}^{2q}\Bigr{)}\leq
C\|h\|_{BC(\mathbb{R}^{d})}.$

For the second term in the Feynman-Kac formula we obtain

$\displaystyle\sup_{0<t<T}$
$\displaystyle\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\bigg{|}\mathbb{E}\biggl{(}\int_{t}^{T}f\bigl{(}s,X^{t,x}(s)\bigr{)}\exp\Bigl{(}-\int_{t}^{s}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}ds\biggr{)}\bigg{|}$
$\displaystyle\leq\,C\sup_{0<t<T}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\mathbb{E}\biggl{(}\int_{t}^{T}\Big{|}f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}ds\biggr{)}$
$\displaystyle=C\sup_{0<t<T}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\mathbb{E}\biggl{(}\int_{t}^{T}\frac{\Big{|}f\bigl{(}s,X^{t,x}(s)\bigr{)}\Big{|}}{1+\bigl{|}X^{t,x}(s)\bigr{|}^{2q}}\Bigl{(}1+\bigl{|}X^{t,x}(s)\bigr{|}^{2q}\Bigr{)}ds\biggr{)}$
$\displaystyle\leq\,C\
\sup_{0<t<T}\|f(t,\cdot)\|_{BC(\mathbb{R}^{d})}\sup_{x\in\mathbb{R}^{d}}\frac{1}{P(x)}\mathbb{E}\biggl{(}\int_{t}^{T}\Bigl{(}1+\big{|}X^{t,x}(s)\big{|}^{2q}\Bigr{)}ds\biggr{)}$
$\displaystyle\leq\,C\sup_{0<t<T}\|f(t,\cdot)\|_{BC(\mathbb{R}^{d})}.$

Together these estimates give the desired Schauder estimate in
$BC_{P}(\mathbb{R}^{d})$.

The estimates in $BC_{P}^{p}(\mathbb{R}^{d})$ for $p>0$ can be obtained
similarly. By assumption, the derivatives $D^{\alpha}_{x}c(t,x)$ and
$D^{\alpha}_{x}X^{t,x}(s)$ for $0<|\alpha|\leq p$ are bounded, then the
derivative of $u(t,x)$ with respect to $x_{j}$ has the estimate

$\displaystyle|D^{j}_{x}u(t,x)|\leq$
$\displaystyle\,C\,\mathbb{E}\Big{|}D^{j}_{x}h\bigl{(}X^{t,x}(T)\bigr{)}\Big{|}+C\,\mathbb{E}\big{|}h\bigl{(}X^{t,x}(T)\bigr{)}\big{|}$
$\displaystyle+C\,\mathbb{E}\biggl{(}\int_{t}^{T}\\!\\!\big{|}D^{j}_{x}f\bigl{(}s,X^{t,x}(s)\bigr{)}\big{|}ds\biggr{)}+C\,\mathbb{E}\biggl{(}\int_{t}^{T}\\!\\!\big{|}f\bigl{(}X^{t,x}(s)\bigr{)}\big{|}ds\biggr{)}.$

Performing similar computations as for $u(t,x)$ and applying Lemma 8.4.1 from
[15], we obtain

$\|D^{j}_{x}u(t,\cdot)\|_{BC_{P}(\mathbb{R}^{d})}\leq
C\|h\|_{BC^{1}_{P}(\mathbb{R}^{d})}+C\sup_{0<t<T}\|f(t,\cdot)\|_{BC^{1}_{P}(\mathbb{R}^{d})}.$

Computations for higher derivatives are analogous. ∎

##  3 Schauder’s estimates

This section aims to give an analytic proof of the partial Schauder estimates
similar to the estimates of Corollary 2.8 omitting the stochastic
representation of the solution given by the Feynman-Kac formula. The partial
Schauder estimates for the Kolmogorov equation in weighted Hölder’s spaces
have been obtained by Lorenzi in [25] under the assumption of bounded
coefficients. Here, we prove the extension of his result to unbounded
coefficients. The first step in this approach is to obtain known Schauder’s
estimates in spaces of bounded Hölder’s functions. We follow in this
presentation the approach from [26] (see also [7] where similar methods were
used). There is a non obvious choice in proving the partial Schauder estimates
concerning the assumed time regularity of the coefficients. In already
mentioned papers [25] and [26], the author assumes only time measurability of
the coefficients (at least for a part of the results). We assume that the
coefficients are continuous for $t\in[0,T]$ ($T$ finite). This assumption is
motivated by our consideration of classical solutions of the Kolmogorov
equation, i.e., solutions that are of class $C^{1,2}$. Thus, we get strong
stochastic solutions by standard assumptions of $(t,x)$-continuity of the
coefficients in the stochastic equation and classical solutions of the
Kolmogorov equation assuming $t$-continuity and $x$-Hölder continuity of its
coefficients.

We begin by proving the existence of solutions for the homogeneous version of
the Kolmogorov equation

$\begin{split}&D_{t}u(t,x)+{\mathcal{A}}^{t}u(t,x)-c(t,x)u(t,x)=0,\
(t,x)\in[0,T)\times\mathbb{R}^{d},\\\ &u(T,x)=h(x),\quad
x\in\mathbb{R}^{d},\end{split}$ (13)

where operator ${\mathcal{A}}^{t}$ and functions $c(t,x)$ and $h(x)$ are the
same as in equation (4).

From Theorem 2.4 we can easily obtain the following result.

######  Theorem 3.1.

Let the coefficients of equation (5) fulfill the conditions of Assumption 2.1
and operator ${\mathcal{A}}^{t}$ be uniformly elliptic, i.e., there exists
$\delta(T)>0$ such that for $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and
$\xi\in\mathbb{R}^{d}\setminus\\{0\\}$

$\sum_{i,j=1}^{d}a_{ij}(t,x)\xi_{i}\xi_{j}\geq\delta(T)|\xi|^{2}.$

On the coefficients of equation (13), we assume:

  1. 1.

Function $c\kern-3.00003pt:\,[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ is
$(t,x)$-continuous with $c(t,x)\geq c_{0}>0$. In addition, $c$ as a function
of $x$ is locally Hölder continuous, i.e., for an open bounded set $U$ in
$\mathbb{R}^{d}$ such that $\bar{U}\subset B(R)$, $c(t,\cdot)\in
H^{\beta}(U)$, $0<\beta<1$, and $\|c(t,\cdot)\|_{H^{\beta}(U)}\leq C(T,R)$.

  2. 2.

Function $h\kern-3.00003pt:\,\mathbb{R}^{d}\to\mathbb{R}$ belongs to
$BC(\mathbb{R}^{d})$.

Under the above assumptions the function $u(t,x)$ given by the equation

$u(t,x)=\,\mathbb{E}\biggl{(}h\bigl{(}X^{t,x}(T)\bigr{)}\exp\Bigl{(}-\int_{t}^{T}c\bigl{(}r,X^{t,x}(r)\bigr{)}dr\Bigr{)}\biggr{)}$
(14)

is such that $u(t,\cdot)\in H^{2+\beta}(U)$ and is a bounded function in
$[0,T]\times\mathbb{R}^{d}$ with

$\sup_{[0,T]\times\mathbb{R}^{d}}|u(t,x|\leq C(T)\|h\|_{BC(\mathbb{R}^{d})}.$

Furthermore, $u(t,x)$ belongs to
$C^{1,2}\bigl{(}(0,T)\times\mathbb{R}^{d}\bigr{)}$ and is a unique classical
solution of the Cauchy problem (13).

To obtain Schauder’s estimates for a solution of equation (13) we need a
maximum principle. The following maximum principle is a simple corollary from
Proposition 2.1 in [28].

######  Theorem 3.2.

Let $u\in BC([0,T]\times\mathbb{R}^{d})\cap
C^{1,2}([0,T)\times\mathbb{R}^{d})$ be a solution of the Cauchy problem

$\begin{split}&D_{t}u(t,x)+{\mathcal{A}}^{t}u(t,x)-c(t,x)u(t,x)+f(t,x)=0,\\\
&u(T,x)=h(x),\end{split}$

with $c(t,x)\geq c_{0}$.

If $f(t,x)\leq 0$ for $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and
$\sup_{[0,T]\times\mathbb{R}^{d}}u(t,x)>0$, then

$\sup_{x\in\mathbb{R}^{d}}u(t,x)\leq\mathrm{e}^{-c_{0}(T-t)}\sup_{x\in\mathbb{R}^{d}}h(x),\quad
t\in[0,T].$

If $f(t,x)\geq 0$ for $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and
$\inf_{[0,T]\times\mathbb{R}^{d}}u(t,x)<0$, then

$\inf_{x\in\mathbb{R}^{d}}u(t,x)\geq\mathrm{e}^{-c_{0}(T-t)}\inf_{x\in\mathbb{R}^{d}}h(x),\quad
t\in[0,T].$

In particular, if $f(t,x)=0$ then

$\|u(t,\cdot)\|_{BC(\mathbb{R}^{d})}\leq\mathrm{e}^{-c_{0}(T-t)}\|h\|_{BC(\mathbb{R}^{d})},\quad
t\in[0,T].$

Proof. Observe that in the proof of Proposition 2.1 in [28] no regularity of
coefficients is necessary. The proof requires only the uniform ellipticity of
${\mathcal{A}}^{t}$, the lower bound on $c(t,x)$, and the existence of the
Lyapunov function $\phi\in C^{2}(\mathbb{R}^{d})$ such that for a constant
$\lambda>0$

$\lim_{|x|\to+\infty}\phi(x)=+\infty,\quad{\mathcal{A}}^{t}\phi-c(t,x)\phi\leq\lambda\phi,\quad(t,x)\in[0,T]\times\mathbb{R}^{d}.$

Taking into account the growth conditions for ${b}(t,x)$ and ${\sigma}(t,x)$,
we can take $\phi(x)=(1+|x|^{2})$. Then we get

${\mathcal{A}}^{t}\phi-c(t,x)\phi\leq\big{|}{\mathcal{A}}^{t}(1+|x|^{2})\big{|}-c_{0}(1+|x|^{2})\leq
C(1+|x|)^{2}+|c_{0}|(1+|x|^{2})\leq\lambda\phi,$

where $\lambda>|c_{0}|+C$. ∎

######  Definition 3.3.

Let $u(t,x)$ be a solution of problem (13) with initial data $u(s,x)=h(x)$,
$s>t$. $G(\cdot,s)$ is defined as an operator on $BC(\mathbb{R}^{d})$ which
associates with any $h(x)\in BC(\mathbb{R}^{d})$ a unique classical solution
of problem (13).

The following corollary is a simple conclusion from Theorems 3.1 and 3.2.

######  Corollary 3.4.

For any $s>t$ and any $h\in BC(\mathbb{R}^{d})$, define $G(t,s)h=u(t,\cdot)$,
where $u(t,\cdot)$ is a unique solution of problem (13) such that
$u(s,\cdot)=h(\cdot)$. Then, $G(t,s)$ is a bounded linear operator in
$BC(\mathbb{R}^{d})$ and the following estimate holds

$\|G(t,s)\|_{\mathcal{L}\bigl{(}BC(\mathbb{R}^{d}),BC(\mathbb{R}^{d})\bigr{)}}\leq\mathrm{e}^{-c_{0}(s-t)}.$

To obtain the Schauder estimates for solutions of equation (13), we need
additional assumptions that include conditions on the coefficients of the
stochastic equation (5).

######  Assumption 3.5.

  1. 1.

Let the coefficients ${b}_{i}(t,{x})$ and ${\sigma}_{ij}(t,{x})$ of equation
(5) be $(t,x)$-continuous and with linear growth

$|{b}(t,{x})|+|{\sigma}(t,{x})|\leq C(1+|{x}|),\ t\in[0,T],$

and three times continuously differentiable with respect to ${x}$ with
derivatives
$D^{\alpha}_{x}{b}_{i}(t,{x}),D^{\alpha}_{x}{\sigma}_{ij}(t,{x})\in
BC(\mathbb{R}^{d})$ for $|\alpha|=1,2,3$, $t\in[0,T]$, and
$D^{\alpha}_{x}{b}_{i}(t,{x}),D^{\alpha}_{x}{\sigma}_{ij}(t,{x})\in
H^{\beta}_{\mathit{loc}}(\mathbb{R}^{d})$ for $|\alpha|=3$.

  2. 2.

Let the growth of ${\sigma}_{ij}(s,y)$ be limited by the inequalities

$C_{1}(1+|{x}|)^{1/2}\leq{\sigma}_{ij}(t,{x})\leq C_{2}(1+|{x}|),\quad
t\in[0,T],$

where $C_{1}$ and $C_{2}$ are positive constants.

  3. 3.

$c(t,x)\geq c_{0}$ for $(t,x)\in[0,T]\times\mathbb{R}^{d}$ with $c_{0}>0$.
Function $c(t,x)$ is three times continuously differentiable with respect to
$x$ and

$|D^{\alpha}_{x}c(t,x)|\leq C,\quad 1\leq|\alpha|\leq 3.$

As we mentioned earlier, we prove Schauder’s estimates in Hölder’s spaces of
bounded functions following the approach from [26]. An important part of this
proof is Theorem 2.4 of [26]. The main difference with the original
assumptions of this theorem is in the regularity of the coefficients. In
Theorem 2.4 of [26] the author assumes the Hölder regularity of the
coefficients with respect to variables $(t,x)$. In our approach, Hölder’s
regularity is required only with respect to the spatial variables due to the
partial Schauder estimates. With respect to variables $(t,x)$, we need only
uniform continuity on $[0,T]\times\mathbb{R}^{d}$. The following lemma shows
that Assumption 3.5 implies conditions required in [26] for the proof of
Theorem 2.4, and after some natural modifications, we can use that theorem.

######  Lemma 3.6.

Under Assumption 3.5 the following reformulation of Hypotheses 2.1 of [26] is
valid for a universal positive constant $C$:

  1. 1.

The coefficients $a_{ij}$, $b_{i}$, $i,j=1,\dots,d$, and $c$ belong to
$H^{3+\beta}_{\mathit{loc}}(\mathbb{R}^{d})$ for some $\beta\in(0,1)$.

  2. 2.

Operator ${\mathcal{A}}^{t}$ is uniformly elliptic

$\sum_{i,j=1}^{d}a_{ij}(t,x)\xi_{i}\xi_{j}\geq\nu(t,x)|\xi|^{2},\
(t,x)\in[0,T]\times\mathbb{R}^{d},\ \xi\in\mathbb{R}^{d}\setminus\\{0\\},$

for some function $\nu(t,x)\geq\nu_{0}>0$,
$(t,x)\in[0,T]\times\mathbb{R}^{d}$.

  3. 3.

The following estimates hold

$\displaystyle\Big{|}\sum_{j=1}^{d}a_{ij}(t,x)x_{j}\Big{|}\leq
C(1+|x|^{2})\nu(t,x),$ $\displaystyle\sum_{j=1}^{d}a_{jj}(t,x)\leq
C(1+|x|^{2})\nu(t,x),$ $\displaystyle\sum_{j=1}^{d}b_{j}(t,x)x_{j}\leq
C(1+|x|^{2})\nu(t,x).$

  4. 4.

$|D_{x}^{\alpha}a_{ij}(t,x)|\leq C\nu(t,x)$ for $|\alpha|=1,3$ and

$\sum_{i,j,k,l=1}^{d}D_{x}^{kl}a_{ij}(t,x)m_{ij}m_{kl}\leq
C\nu(t,x)\sum_{k,l=1}^{d}|m_{kl}|^{2},$

for any $d\times d$ symmetric matrix $M=\\{m_{kl}\\}$ and
$(t,x)\in[0,T]\times\mathbb{R}^{d}$.

  5. 5.

For the entries of $b$ we have

$\displaystyle\sum_{i,j=1}^{d}D_{x}^{j}b_{i}(t,x)\xi_{i}\xi_{j}\leq
C\nu(t,x)|\xi|^{2},$ $\displaystyle|D_{x}^{\alpha}b_{i}(t,x)|\leq C\nu(t,x),\
|\alpha|=2,3,\ i=1,\dots,d.$

  6. 6.

There exist a positive function $\phi\in C^{2}(\mathbb{R}^{d})$ and a constant
$\lambda>0$ such that

$\lim_{|x|\to+\infty}\phi(x)=+\infty,\quad\sup_{x\in\mathbb{R}^{d}}\bigl{(}{\mathcal{A}}^{t}\phi(x)-\lambda\phi(x)\bigr{)}<+\infty.$

Proof. Since $a_{ij}=\frac{1}{2}\sum_{k=1}^{m}{\sigma}_{ik}{\sigma}_{jk}$ then
point 2 of Assumption 3.5 gives $\nu(t,x)=C(1+|x|)$ with $C>0$. Therefore
points 1 and 2 of the lemma follow straightforwardly from Assumption 3.5.
Point 3 of the lemma holds true due to the linear growth of ${b}_{i}$ and
${\sigma}_{ij}$. To obtain point 4, let us observe that
$|D_{x}^{\alpha}a_{ij}(t,x)|\leq C(1+|{x}|)$, for $|\alpha|\leq 3$, due to the
linear growth of ${\sigma}_{ij}$ and the boundedness of derivatives
$D_{x}^{\alpha}{\sigma}_{ij}$. The boundedness of $D_{x}^{\alpha}{b}_{i}$ for
$|\alpha|\leq 3$ implies point 5 of the lemma. Point 6 can be proved similarly
like in Theorem 3.2 taking $\phi(x)=(1+|x|^{2})$. ∎

The following theorem is a reformulation of Theorem 2.4 from [26] (see also
Theorem 3.3 in [7]). We present an outline of the proof pointing to the
differences with the original proof implied by the modifications in estimates
due to Assumption 3.5. As we pointed out before, the main difference with
paper [26] is in the regularity of the coefficients. We will show that the
Hölder regularity with respect to variables $(t,x)$ can be replaced by a
weaker regularity only in the spatial variables and $t$-continuity.

######  Theorem 3.7.

Let Assumption 3.5 be satisfied. Then, for any
$p_{1}+\beta_{1},p_{2}+\beta_{2}\in[0,3]$, with $p_{1}+\beta_{1}\leq
p_{2}+\beta_{2}$, $\beta_{1},\beta_{2}\in[0,1)$, there exists a positive
constant $C$ such that

$\|G(t,T)h\|_{H^{p_{2}+\beta_{2}}(\mathbb{R}^{d})}\leq
C(T-t)^{-\frac{p_{2}+\beta_{2}-p_{1}-\beta_{1}}{2}}\|h\|_{H^{p_{1}+\beta_{1}}(\mathbb{R}^{d})},$

where $H^{p}(\mathbb{R}^{d})$ for an integer $p$ denotes the space
$BC^{p}(\mathbb{R}^{d})$.

Proof. We present only the part of the proof that corresponds to
$\beta_{1}=\beta_{2}=0$ and $p_{1}=0$, $p_{2}=3$. The idea of the proof in
[26] is to apply Bernstein’s method to equation (4) restricted to an open,
bounded ball $B(R)$ with radius $R$. Due to Theorem 2.4 there is a function
$u_{R}(t,x)$ that is a classical solution for the problem

$\begin{split}&D_{t}u_{R}(t,x)+{\mathcal{A}}^{t}u_{R}(t,x)-c(t,x)u_{R}(t,x)=0,\
(t,x)\in[0,T)\times B(R),\\\ &u_{R}(t,x)=0,\quad t\in[0,T],\ x\in\partial
B(R),\\\ &u_{R}(T,x)=\eta(x)h(x),\quad x\in B(R),\end{split}$

where $\eta(x)=\eta(|x|)\in C^{\infty}_{0}(\mathbb{R}^{d})$ such that
$\eta(x)=1$ for $x\in B(\frac{1}{2}R)$ and $\eta(x)=0$ outside $B(R)$.

By Assumption 3.5 and Theorem 2.5 $u_{R}(t,\cdot)$ is in $H^{4+\beta}(B(R))$
for any $\beta\in(0,1)$ and $t\in(0,T)$. Then, we define for $a>0$

$\displaystyle v_{R}(t,x)=$
$\displaystyle|u_{R}(t,x)|^{2}+a(T-t)\eta^{2}\sum_{i=1}^{d}\big{|}D^{i}u_{R}(t,x)\big{|}^{2}$
$\displaystyle+a^{2}(T-t)^{2}\eta^{4}\sum_{i,j=1}^{d}\big{|}D^{ij}u_{R}(t,x)\big{|}^{2}$
$\displaystyle+a^{3}(T-t)^{3}\eta^{6}\sum_{i,j,k=1}^{d}\big{|}D^{ijk}u_{R}(t,x)\big{|}^{2}.$

Straightforward computations show that the function $v_{R}$ solves the Cauchy
problem

$\begin{split}&D_{t}v_{R}(t,x)+{\mathcal{A}}^{t}v_{R}(t,x)-c(t,x)v_{R}(t,x)=f_{R}(t,x),\,(t,x)\in[0,T)\times
B(R),\\\ &v_{R}(t,x)=0,\quad t\in[0,T],\ x\in\partial B(R),\\\
&v_{R}(T,x)=\eta^{2}(x)h^{2}(x),\quad x\in B(R),\end{split}$

where

$f_{R}(t,x)=\sum_{i=1}^{9}f_{i,R}(t,x).$

For $i=1,\dots,8$, functions $f_{i,R}(t,x)$ are essentially equal $-g_{i,n}$,
where $g_{i,n}$ are functions in the proof of Theorem 2.4 in [26]. The only
difference is that terms $(t-s)^{k}$ in the original formulas for $g_{i,n}$
are replaced by $(T-t)^{k}$ in $f_{i,R}$. Since Hypotheses 2.1 of [26]
concerning estimates of the coefficients of equation (13) are fulfilled by our
Assumption 3.5, then also the estimate (2.23) of [26] is valid (time
regularity of the coefficients is not important in these estimates, hence our
weaker assumptions do not play a role) and

$f^{\prime}_{R}=\sum_{i=1}^{8}f_{i,R}(t,x)\geq 0.$

For $f_{9,R}(t,x)$ tedious computations give

$f_{9,R}=c(t,x)v_{R}+g_{R}$

where

$\displaystyle g_{R}=$ $\displaystyle
2a(T-t)\eta^{2}u_{R}\sum_{i=1}^{d}D^{i}cD^{i}u_{R}$
$\displaystyle+4a^{2}(T-t)^{2}\eta^{4}\sum_{i,j=1}^{d}D^{ij}u_{R}D^{i}u_{R}D^{j}c$
$\displaystyle+2a^{2}(T-t)^{2}\eta^{4}u_{R}\sum_{i,j=1}^{d}D^{ij}u_{R}D^{ij}c$
$\displaystyle+2a^{3}(T-t)^{3}\eta^{6}u_{R}\sum_{i,j,k=1}^{d}D^{ijk}u_{R}D^{ijk}c$
$\displaystyle+6a^{3}(T-t)^{3}\eta^{6}\sum_{i,j,k=1}^{d}D^{i}cD^{jk}u_{R}D^{ijk}u_{R}$
$\displaystyle+6a^{3}(T-t)^{3}\eta^{6}\sum_{i,j,k=1}^{d}D^{jk}cD^{i}u_{R}D^{ijk}u_{R}.$

Applying Cauchy’s inequality, we get

$\displaystyle|g_{R}|\leq C\,\bigl{(}1+c(t,x)\bigr{)}$
$\displaystyle\biggl{(}2aT\eta^{2}\sum_{i=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{i}u_{R})^{2}+\frac{1}{2\varepsilon}u^{2}_{R}\Bigr{)}\biggr{.}$
$\displaystyle+4a^{2}T^{2}\eta^{4}\sum_{i,j=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{ij}u_{R})^{2}+\frac{1}{2\varepsilon}(D^{i}u_{R})^{2}\Bigr{)}$
$\displaystyle+2a^{2}T^{2}\eta^{4}\sum_{i,j=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{ij}u_{R})^{2}+\frac{1}{2\varepsilon}(u_{R})^{2}\Bigr{)}$
$\displaystyle+2a^{3}T^{3}\eta^{6}\sum_{i,j,k=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{ijk}u_{R})^{2}+\frac{1}{2\varepsilon}(u_{R})^{2}\Bigr{)}$
$\displaystyle+6a^{3}T^{3}\eta^{6}\sum_{i,j,k=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{ijk}u_{R})^{2}+\frac{1}{2\varepsilon}(D^{ij}u_{R})^{2}\Bigr{)}$
$\displaystyle\biggl{.}+6a^{3}T^{3}\eta^{6}\sum_{i,j,k=1}^{d}\Bigl{(}\frac{\varepsilon}{2}(D^{ijk}u_{R})^{2}+\frac{1}{2\varepsilon}(D^{i}u_{R})^{2}\Bigr{)}\biggr{)}.$

Taking $\varepsilon$ sufficiently small and $a\leq a_{0}$, we have

$\displaystyle
C\varepsilon\Bigl{(}a_{0}T\eta^{2}\sum_{i=1}^{d}(D^{i}u_{R})^{2}$
$\displaystyle+3a_{0}^{2}T^{2}\eta^{4}\sum_{i,j=1}^{d}(D^{ij}u_{R})^{2}\Bigr{.}$
.
$\displaystyle+7a_{0}^{3}T^{3}\eta^{6}\sum_{i,j,k=1}^{d}(D^{ijk}u_{R})^{2}\Bigr{)}\leq
v_{R}/4.$

Then we can select $a$ so small that

$\displaystyle\frac{CaT\eta^{2}}{\varepsilon}\Bigl{(}\\!(d+a_{0}Td^{2}\eta^{2}$
$\displaystyle+a_{0}^{2}T^{2}d^{3}\eta^{4})u_{R}^{2}+(2a_{0}Td\eta^{2}+3a_{0}^{2}T^{2}d^{2}\eta^{4})\sum_{i=1}^{d}(D^{i}u_{R})^{2}\Bigr{.}$
.
$\displaystyle+3a_{0}^{2}T^{2}d\eta^{4}\sum_{i,j=1}^{d}(D^{ij}u_{R})^{2}\Bigr{)}\leq
v_{R}/4.$

Collecting these estimates, we get

$|g_{R}(t,x)|\leq\bigl{(}1+c(t,x)\bigr{)}v_{R}(t,x)/2$

and

$f_{9,R}(t,x)\geq
c(t,x)v_{R}-\bigl{(}1+c(t,x)\bigr{)}v_{R}/2=c(t,x)v_{R}/2-v_{R}/2.$

Since

$c(t,x)v_{R}(t,x)+f_{R}(t,x)\geq(3c_{0}/2-1/2)v_{R}(t,x)=c_{1}v_{R}(t,x),$

then applying Theorem 3.2, we obtain

$\sup_{x\in\mathbb{R}^{d}}|v_{R}(t,x)|\leq\mathrm{e}^{-c_{1}(T-t)}\sup_{x\in\mathbb{R}^{d}}|\eta^{2}(x)h^{2}(x)|\leq
C\,\sup_{x\in\mathbb{R}^{d}}|h^{2}(x)|,\quad t\in[0,T].$

As the constant in the above estimate is independent of $R$, we can pass to
the limit $R\to\infty$ obtaining the estimate of the theorem. Let us observe
that under Assumption 3.5 $u(t,x)$, a solution of (13), has a continuous third
spatial derivative and $D^{3}_{x}u_{R}$ converges to $D^{3}_{x}u$ locally
uniformly on $[0,T)\times\mathbb{R}^{d}$. This convergence follows from the
observation that fourth derivatives of $u_{R}(t,x)$ are uniformly bounded on
any compact set $\mathcal{B}\subset[0,T)\times\mathbb{R}^{d}$. Then
$D^{3}_{x}u_{R}$ is a sequence of bounded and equicontinuous functions and the
Arzelá-Ascoli theorem gives the convergence. This observation is essential for
the interpolation argument used in [26] to extend the estimate to
$\beta_{1}\neq 0$ and/or $\beta_{2}\neq 0$. ∎

The following theorem is a part of Theorem 3.8 of [26] where the author
assumes the spatial Hölder continuity and only the continuity in time of the
coefficients of (4).

######  Theorem 3.8.

Let the conditions of Assumption 3.5 hold and $u(t,x)$ be a solution of the
Cauchy problem (4) with $h=0$ and $f(t,\cdot)\in H^{\beta}(\mathbb{R}^{d})$,
$\beta\in(0,1)$. Then

$\|u(t,\cdot)\|_{H^{2+\beta}(\mathbb{R}^{d})}\leq
C\sup_{t\in[0,T]}\|f(t,\cdot)\|_{H^{\beta}(\mathbb{R}^{d})},\quad t\in[0,T).$

The following corollary summarizes the results of the two previous theorems.

######  Corollary 3.9.

Let Assumption 3.5 hold and $f(t,\cdot)\in H^{\beta}(\mathbb{R}^{d})$ for
$t\in[0,T]$, $\beta\in(0,1)$. Then for a solution $u(t,x)$ of the Cauchy
problem (4) we have the estimate

$\displaystyle\|u(t,\cdot)\|_{BC(\mathbb{R}^{d})}$
$\displaystyle+(T-t)^{1+\beta/2}\|u(t,\cdot)\|_{H^{2+\beta}(\mathbb{R}^{d})}$
$\displaystyle\leq
C\Bigl{(}\|h\|_{BC(\mathbb{R}^{d})}+\sup_{t\in[0,T]}\|f(t,\cdot)\|_{H^{\beta}(\mathbb{R}^{d})}\Bigr{)},\quad
t\in[0,T).$

From Theorem 3.8 of [26], we obtain the optimal Schauder’s estimate.

######  Theorem 3.10.

Let the conditions of Assumption 3.5 hold and $u(t,x)$ be a solution of the
Cauchy problem (4) with $h\in H^{2+\beta}(\mathbb{R}^{d})$ and $f(t,\cdot)\in
H^{\beta}(\mathbb{R}^{d})$ for $t\in[0,T]$, $\beta\in(0,1)$. Then

$\sup_{t\in[0,T]}\|u(t,\cdot)\|_{H^{2+\beta}(\mathbb{R}^{d})}\leq
C\Bigl{(}\|h\|_{H^{2+\beta}(\mathbb{R}^{d})}+\sup_{t\in[0,T]}\|f(t,\cdot)\|_{H^{\beta}(\mathbb{R}^{d})}\Bigr{)}.$

We now prove the main result of this section: Schauder’s estimates in the
weighted Hölder spaces with polynomial weights for the Kolmogorov equation
(4).

######  Theorem 3.11.

Let Assumption 3.5 hold, $h\in BC_{P}(\mathbb{R}^{d})$, and $f(t,\cdot)\in
H_{P}^{\beta}(\mathbb{R}^{d})$ for $t\in[0,T]$, $\beta\in(0,1)$, where
$P(x)=1+|x|^{2q}$, $q\geq 0$. In addition, we assume that the coefficients
$a_{ij}(t,x)$ are four times continuously differentiable with respect to $x$
and $D^{\alpha}_{x}a_{ij}(t,x)\in H^{\beta}_{\mathit{loc}}(\mathbb{R}^{d})$
for $|\alpha|=4$. Then for a solution $u(t,x)$ of the Cauchy problem (4) we
have the estimate

$\displaystyle\|u(t,\cdot)\|_{BC_{P}(\mathbb{R}^{d})}$
$\displaystyle+(T-t)^{1+\beta/2}\|u(t,\cdot)\|_{H_{P}^{2+\beta}(\mathbb{R}^{d})}$
$\displaystyle\leq
C\Bigl{(}\|h\|_{BC_{P}(\mathbb{R}^{d})}+\sup_{t\in[0,T]}\|f(t,\cdot)\|_{H_{P}^{\beta}(\mathbb{R}^{d})}\Bigr{)},\quad
t\in[0,T).$

If in addition, $h\in H_{P}^{2+\beta}(\mathbb{R}^{d})$, then we have the
estimate

$\sup_{t\in[0,T]}\|u(t,\cdot)\|_{H_{P}^{2+\beta}(\mathbb{R}^{d})}\leq
C\Bigl{(}\|h\|_{H_{P}^{2+\beta}(\mathbb{R}^{d})}+\sup_{t\in[0,T]}\|f(t,\cdot)\|_{H_{P}^{\beta}(\mathbb{R}^{d})}\Bigr{)}.$

Proof. To prove the theorem, we transform the Cauchy problem (4) with
unbounded $h(x)$ and $f(t,x)$ into a problem with bounded $h(x)$ and $f(t,x)$
where Schauder’s estimates are given in Corollary 3.9 and Theorem 3.10.

Assume that $u(t,x)$ is a solution of (4) with unbounded $h(x)$ and $f(t,x)$.
Let $u(t,x)=v(t,x)P(x)$. Then $v(t,x)$ fulfills the equation

$\begin{split}&D_{t}v(t,x)+\tilde{\mathcal{A}}^{t}v(t,x)-\tilde{c}(t,x)v(t,x)+\tilde{f}(t,x)=0,\
(t,x)\in[0,T)\times\mathbb{R}^{d},\\\ &v(T,x)=\tilde{h}(x),\quad
x\in\mathbb{R}^{d},\end{split}$ (15)

where

$\tilde{\mathcal{A}}^{t}v(t,x)=\sum_{i,j=1}^{d}a_{ij}(t,{x})D^{ij}_{x}v(t,x)+\sum_{i=1}^{d}\tilde{b}_{i}(t,{x})D^{i}_{x}v(t,x),$

and

$\displaystyle\tilde{b}_{i}(t,x)=b_{i}(t,x)+\sum_{j=1}^{d}D^{j}_{x}a_{ij}(t,x)P^{-1}(x)D^{i}_{x}P(x),$
$\displaystyle\tilde{c}(t,x)=c(t,x)+\sum_{i,j=1}^{d}a_{ij}(t,{x})P^{-1}(x)D^{ij}_{x}P(x)+\sum_{i=1}^{d}b_{i}(t,{x})P^{-1}(x)D^{i}_{x}P(x),$
$\displaystyle\tilde{f}(t,x)=P^{-1}(x)f(t,x),$
$\displaystyle\tilde{h}(x)=P^{-1}(x)h(x).$

It is obvious that $\tilde{h}\in BC(\mathbb{R}^{d})$ and
$\tilde{f}(t,\cdot)\in H^{\beta}(\mathbb{R}^{d})$. To show that
$\tilde{c}(t,x)$ fulfills the conditions of Lemma 3.6 let us observe that

$|a_{ij}(t,{x})P^{-1}(x)D^{ij}_{x}P(x)|+|b_{i}(t,{x})P^{-1}(x)D^{i}_{x}P(x)|\leq
C,$

which follows from the estimate

$|P^{-1}(x)D_{x}^{\alpha}P(x)|\leq C(1+|x|^{2})^{-|\alpha|/2}$

and the growth conditions for $a_{ij}$ and $b_{i}$.

Then

$\displaystyle\tilde{c}(t,x)\geq c_{0}$
$\displaystyle-\sup_{x\in\mathbb{R}^{d}}\Big{|}\sum_{i,j=1}^{d}a_{ij}(t,{x})P^{-1}(x)D^{ij}_{x}P(x)\Big{|}$
$\displaystyle-\sup_{x\in\mathbb{R}^{d}}\Big{|}\sum_{i=1}^{d}b_{i}(t,{x})P^{-1}(x)D^{i}_{x}P(x)\Big{|}\geq\tilde{c}_{0}.$

Since from Assumption 3.5 we have

$|D^{\alpha}_{x}a_{ij}(t,x)|\leq C(1+|x|),\quad|D^{\alpha}_{x}b_{i}(t,x)|\leq
C,$

for $1\leq|\alpha|\leq 3$, and

$\big{|}D^{\alpha_{2}}_{x}\bigl{(}P^{-1}(x)D^{\alpha_{1}}_{x}P(x)\bigr{)}\big{|}\leq
C(1+|x|^{2})^{-|\alpha_{1}|/2-|\alpha_{2}|/2},$

then for $|\alpha|\leq 3$

$\displaystyle\Big{|}D^{\alpha}_{x}\Bigl{(}\sum_{i,j=1}^{d}a_{ij}(t,{x})P^{-1}(x)D^{ij}_{x}P(x)$
$\displaystyle+\sum_{i=1}^{d}b_{i}(t,{x})P^{-1}(x)D^{i}_{x}P(x)\Bigr{)}\Big{|}\leq
C.$

Therefore, $\tilde{c}(t,x)$ fulfills condition 3 of Assumption 3.5.

From the above estimates, we also obtain

$\big{|}D^{j}_{x}a_{ij}(t,x)P^{-1}(x)D^{i}_{x}P(x)\big{|}\leq C,$

and for $|\alpha|\leq 3$

$\Big{|}D^{\alpha}_{x}\Bigl{(}\sum_{j=1}^{d}D^{j}_{x}a_{ij}(t,x)P^{-1}(x)D^{i}_{x}P(x)\Bigr{)}\Big{|}\leq
C.$

The additional assumption that $a_{ij}(t,x)$ is four times differentiable with
respect to $x$ and $D^{\alpha}_{x}a_{ij}(t,x)\in H^{\beta}(\mathbb{R}^{d})$
for $|\alpha|=4$ guaranties that $D^{\alpha}_{x}\tilde{b}_{i}(t,x)\in
H^{\beta}(\mathbb{R}^{d})$ for $|\alpha|=3$. Under this assumption condition 1
of Assumption 3.5 is fulfilled by $\tilde{b}(t,x)$.

Thus, problem (15) fulfills the conditions of Corollary 3.9 and Theorem 3.10
and the relevant Schauder’s estimates for $v(t,x)$ hold. This proves the
estimates of the theorem for $u(t,x)=v(t,x)P(x)$. ∎

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