# Sandwiched SDEs with unbounded drift driven by Hölder noises

Giulia Di Nunno1,2
<EMAIL_ADDRESS>Yuliya Mishura3
<EMAIL_ADDRESS>Anton Yurchenko-Tytarenko1
<EMAIL_ADDRESS>
(1Department of Mathematics, University of Oslo
2Department of Business and Management Science, NHH Norwegian School of
Economics, Bergen
3Department of Probability, Statistics and Actuarial Mathematics, Taras
Shevchenko National University of Kyiv
)

###### Abstract

We study a stochastic differential equation with an unbounded drift and
general Hölder continuous noise of an arbitrary order. The corresponding
equation turns out to have a unique solution that, depending on a particular
shape of the drift, either stays above some continuous function or has
continuous upper and lower bounds. Under some additional assumptions on the
noise, we prove that the solution has moments of all orders. We complete the
study providing a numerical scheme for the solution. As an illustration of our
results and motivation for applications, we suggest two stochastic volatility
models which we regard as generalizations of the CIR and CEV processes.
Keywords: sandwiched process, unbounded drift, Hölder continuous noise,
numerical scheme, stochastic volatility
MSC 2020: 60H10; 60H35; 60G22; 91G30

## Introduction

Stochastic differential equations (SDEs) the solutions of which take values in
a given bounded domain are largely applied in several fields. Just as an
illustration, we can consider the Tsallis–Stariolo–Borland (TSB) model
employed in biophysics defined as

$dY_{1}(t)=-\frac{\theta Y_{1}(t)}{1-Y^{2}_{1}(t)}dt+\sigma
dW(t),\quad\theta>0,~{}\sigma>0,$ (0.1)

with $W$ being a standard Wiener process. If
$\frac{\sigma^{2}}{\theta}\in(0,1]$, the TSB process is “sandwiched” between
$-1$ and $1$ (for more details, see e.g. [17, Subsection 2.3] or [18, Chapter
3 and Chapter 8]). Another example is the Cox–Ingersoll–Ross (CIR) process
[13, 14, 15] defined via the SDE of the form

$dX(t)=(\theta_{1}-\theta_{2}X(t))dt+\sigma\sqrt{X(t)}dW(t),\quad\theta_{1},~{}\theta_{2},~{}\sigma>0.$

Under the so-called Feller condition $2\theta_{1}>\sigma^{2}$, the CIR process
is lower bounded (more precisely, positive) a.s. which justifies its
popularity in interest rates and stochastic volatility modeling in finance.
Moreover, by [31, Theorem 2.3], the square root $Y_{2}(t):=\sqrt{X(t)}$ of the
CIR process satisfies the SDE of the form

$dY_{2}(t)=\frac{1}{2}\left(\frac{\theta_{1}-\sigma^{2}/4}{Y_{2}(t)}-\theta_{2}Y_{2}(t)\right)dt+\frac{\sigma}{2}dW(t)$
(0.2)

and SDEs (0.1) and (0.2) both have an additive noise term and an unbounded
drift with points of singularity at the bounds ($\pm 1$ for the TSB and $0$
for CIR) which have a “repelling” action so that the corresponding processes
never cross or even touch the bounds.

The goal of this paper is to study a family of SDEs of the type similar to
(0.1) and (0.2), namely

$Y_{t}=Y_{0}+\int_{0}^{t}b(s,Y_{s})ds+Z_{t},\quad t\in[0,T],$ (0.3)

where where the drift $b$ is unbounded. We consider separately two cases.

  * (A)

$b$ is a real function defined on the set
$\\{(t,y)\in[0,T]\times\mathbb{R}~{}|~{}y>\varphi(t)\\}$ such that $b(t,y)$
has an explosive growth of the type $(y-\varphi(t))^{-\gamma}$ as
$y\downarrow\varphi(t)$, where $\varphi$ is a given Hölder continuous function
and $\gamma>0$. The we will see that the process $Y$ satisfying (0.3) is lower
bounded by $\varphi$, i.e.

$Y_{t}>\varphi(t),\quad a.s.,\quad t\in[0,T],$ (0.4)

which can be called a one-sided sandwich.

  * (B)

$b$ is a real function defined on the set
$\\{(t,y)\in[0,T]\times\mathbb{R}~{}|~{}\varphi(t)<y<\psi(t)\\}$ such that
$b(t,y)$ has an explosive growth of the type $(y-\varphi(t))^{-\gamma}$ as
$y\downarrow\varphi(t)$ and an explosive decrease of the type
$-(\psi(t)-y)^{-\gamma}$ as $y\uparrow\psi(t)$, where $\varphi$ and $\psi$ are
given Hölder continuous functions such that $\varphi(t)<\psi(t)$, $t\in[0,T]$,
and $\gamma>0$. We will see that in this case the solution to (0.3) turns out
to be sandwiched, namely

$\varphi(t)<Y_{t}<\psi(t)\quad a.s.,\quad t\in[0,T],$ (0.5)

as a two-sided sandwich.

The noise term $Z$ in (0.3) is an arbitrary $\lambda$-Hölder continuous noise,
$\lambda\in(0,1)$. Our main motivation to consider $Z$ from such a general
class instead of the classical Wiener process lies in the desire to go beyond
Markovianity and include memory in the dynamics (0.3) via the noise term. It
should be noted that the presence of memory is a commonly observed empirical
phenomenon (in this regard, we refer the reader to [7, Chapter 1], where
examples of datasets with long memory are collected, and to [34] for more
details on stochastic processes with long memory) and the particular
application which we have in mind throughout this paper comes from finance
where the presence of market memory is well-known and extensively studied (see
e.g. [3, 16, 37] or [36] for a detailed historical overview on the subject).
In the current literature, processes with memory in the noise have been used
as stochastic volatilities allowing for the inclusion of empirically detected
features, such as volatility smiles and skews in long term options [11], see
also [9, 10] for more details on long memory models and [23] for short memory
coming from the microstructure of the market. Some studies (see e.g. [1])
indicate that the roughness of the volatility changes over time which
justifies the choice of multifractional Brownian motion [4] or even general
Gaussian Volterra processes [26] as drivers. Separately we mention the series
of papers [28, 29, 30] that study an SDE of the type (0.2) with memory
introduced via a fractional Brownian motion with $H>\frac{1}{2}$:

$dY_{t}=\left(\frac{\theta_{1}}{Y_{t}}-\theta_{2}Y_{t}\right)dt+\sigma
dB^{H}_{t},\quad\theta_{1},~{}\theta_{2},~{}\sigma>0,~{}t\in[0,T].$ (0.6)

Our model (0.3) can thus be regarded as a generalization of (0.6)
accommodating a much flexible choice of noise to deal with problems of local
roughness mentioned above.

In this paper, we first consider existence and uniqueness of solution to (0.3)
and then focus on the moments of both positive and negative orders. We
recognize that similar problems concerning equations of the type (0.3) with
lower bound $\varphi\equiv 0$ and the noise $Z$ being a fractional Brownian
motion with $H>\frac{1}{2}$ were addressed in [25]. There, authors used
pathwise arguments to prove existence and uniqueness of the solution whereas a
Malliavin calculus-based method was applied to obtain finiteness of the
inverse moments. Despite its elegance, the latter technique requires the noise
to be Gaussian and, moreover, is unable to ensure the finiteness of the
inverse moments on the entire time interval $[0,T]$. It should be stressed
that the inverse moments are crucial for numerical simulation of (0.3) since
it is necessary to control the explosive growth of the drift near bounds, and
hence any limitations embedded in the techniques used to prove results on the
inverse moments directly influence the convergence of the numerical schemes.
These disadvantages of the Malliavin method mentioned above resulted in
restrictive conditions involving all parameters of the model and $T$ in
numerical schemes in e.g. [24, Theorem 4.2] and [38, Theorem 4.1].

The approach we have is to use pathwise calculus together with stopping times
arguments also for the inverse moments. This allows us, on the one hand, to
choose from a much broader family of noises well beyond the Gaussian framework
and, on the other hand, to prove the existence of the inverse moments of the
solution on the entire $[0,T]$. The corresponding inverse moment bounds are
presented in Theorems 2.6 and 3.2 and can be regarded as the main results of
the paper.

This paper is organised as follows. In Section 1, the general framework is
described and the main assumptions are listed. Furthermore, some examples of
possible noises $Z$ are provided (including continuous martingales and
Gaussian Volterra processes). In Section 2, we provide existence and
uniqueness of the solution to (0.3) as well as discuss property (0.4), derive
upper and lower bounds for the solution in terms of the noise and study
finiteness of $\mathbb{E}\left[\sup_{t\in[0,T]}|Y_{t}|^{r}\right]$ and
$\mathbb{E}\left[\sup_{t\in[0,T]}(Y_{t}-\varphi(t))^{-r}\right]$, $r\geq 1$.
Full details of the proof of the existence are provided in the Appendix A.
Section 3 is devoted to the sandwiched case (0.5): existence, uniqueness and
properties of the solution are discussed. Our approach is readily put in
application introducing the generalized CIR and CEV (see [2, 12]) processes in
Section 4. Finally, in Section 5 we provide a modification of the standard
Euler numerical scheme which we call _semi-heuristic_ in view of the type of
convergence and dependence on a random variable that cannot be computed
explicitly and has to be estimated from the discretised data. The results are
illustrated by simulations.

##  1 Preliminaries and assumptions

In this section, we present the framework and collect all the assumptions
regarding both the noise $Z$ and the drift functional $b$ from equation (0.3).

###  1.1 The noise

Fix $T>0$ and consider two $\lambda$-Hölder continuous functions $\varphi$,
$\psi$: $[0,T]\to\mathbb{R}$, $\lambda\in(0,1)$, such that
$\varphi(t)<\psi(t)$ for all $t\in[0,T]$. Throughout this paper, the noise
term $Z=\\{Z_{t},~{}t\in[0,T]\\}$ in equation (0.3) is an arbitrary stochastic
process such that:

  * (Z1)

$Z_{0}=0$ a.s.;

  * (Z2)

$Z$ has Hölder continuous paths of the same order $\lambda$ as functions
$\varphi$ and $\psi$, i.e. there exists a random variable
$\Lambda=\Lambda_{\lambda}(\omega)\in(0,\infty)$ such that

$|Z_{t}-Z_{s}|\leq\Lambda|t-s|^{\lambda},\quad t,s\in[0,T].$ (1.1)

Note that we do not require any particular assumptions on distribution of the
noise (e.g. Gaussianity), but, for some results, we will need the random
variable $\Lambda$ from (1.1) to have moments of sufficiently high orders. In
what follows, we list several examples of admissible noises as well as
properties of the corresponding random variable $\Lambda$. In order to discuss
the latter, we will use a corollary from the well-known Garsia-Rodemich-Rumsey
inequality (see [22] for more details).

######  Lemma 1.1.

Let $f$: $[0,T]\to\mathbb{R}$ be a continuous function, $p\geq 1$ and
$\alpha>\frac{1}{p}$. Then for all $t,s\in[0,T]$ one has

$|f(t)-f(s)|\leq
A_{\alpha,p}|t-s|^{\alpha-\frac{1}{p}}\left(\int_{0}^{T}\int_{0}^{T}\frac{|f(x)-f(y)|^{p}}{|x-y|^{\alpha
p+1}}dxdy\right)^{\frac{1}{p}},$

with the convention $0/0=0$, where

$A_{\alpha,p}=2^{3+\frac{2}{p}}\left(\frac{\alpha p+1}{\alpha p-1}\right).$
(1.2)

Note that this lemma was stated, for example, in [33] and [5] without
computing the constant $A_{\alpha,p}$ explicitly, but we will need the latter
for the approximation scheme in section 5.

###### Proof.

The proof can be easily obtained from [22, Lemma 1.1] by putting in the
notation of [22] $\Psi(u):=|u|^{\beta}$ and
$p(u):=|u|^{\alpha+\frac{1}{\beta}}$, where $\beta=p\geq 1$ in our statement.
∎

######  Example 1.2 (_Hölder continuous Gaussian processes_).

Let $Z=\\{Z_{t},~{}t\in 0\\}$ be a centered Gaussian process with $Z_{0}=0$
and $H\in(0,1)$ be a given constant. Then, by [5], $Z$ has a modification with
Hölder continuous paths of any order $\lambda\in(0,H)$ if and only if for any
$\lambda\in(0,H)$ there exists a constant $C_{\lambda}>0$ such that

$\left(\mathbb{E}|Z_{t}-Z_{s}|^{2}\right)^{\frac{1}{2}}\leq
C_{\lambda}|t-s|^{\lambda},\quad s,t\in[0,T].$ (1.3)

Furthermore, according to [5, Corollary 3], the class of all Gaussian
processes on $[0,T]$, $T\in(0,\infty)$, with Hölder modifications of any order
$\lambda\in(0,H)$ consists exclusively of Gaussian Fredholm processes

$Z_{t}=\int_{0}^{T}\mathcal{K}(t,s)dB_{s},\quad t\in[0,T],$

with $B=\\{B_{t},~{}t\in[0,T]\\}$ being some Brownian motion and
$\mathcal{K}\in L^{2}([0,T]^{2})$ satisfying, for all $\lambda\in(0,H)$,

$\int_{0}^{T}|\mathcal{K}(t,u)-\mathcal{K}(s,u)|^{2}du\leq
C_{\lambda}|t-s|^{2\lambda},\quad s,t\in[0,T],$

where $C_{\lambda}>0$ is some constant depending on $\lambda$.

Finally, using Lemma 1.1, one can prove that the corresponding random variable
$\Lambda$ can be chosen to have moments of all positive orders. Namely, assume
that $\lambda\in(0,H)$ and take $p\geq 1$ such that $\frac{1}{p}<H-\lambda$.
If we take

$\Lambda=A_{\lambda+\frac{1}{p},p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z(x)-Z(y)|^{p}}{|x-y|^{\lambda
p+2}}dxdy\right)^{\frac{1}{p}},$ (1.4)

then, for any $r\geq 1$

$\mathbb{E}\Lambda^{r}<\infty$

and for all $s,t\in[0,T]$:

$|Z_{t}-Z_{s}|\leq\Lambda|t-s|^{\lambda},$

see e.g. [33, Lemma 7.4] for fractional Brownian motion or [5, Theorem 1] for
the general Gaussian case.

In particular, the condition (1.3) presented in Example 1.2 is satisfied by
the following stochastic processes.

######  Example 1.3 (_fractional Brownian motion_).

Fractional Brownian motion $B^{H}=\\{B^{H}_{t},~{}t\geq 0\\}$ with $H\in(0,1)$
(see e.g. [32]) since

$\left(\mathbb{E}|B^{H}_{t}-B^{H}_{s}|^{2}\right)^{\frac{1}{2}}=|t-s|^{H}\leq
T^{H-\lambda}|t-s|^{\lambda},$

i.e. $B^{H}$ has a modification with Hölder continuous paths of any order
$\lambda\in(0,H)$.

######  Example 1.4 (_Gaussian Volterra processes with fBm-type kernel_).

Gaussian Volterra processes

$Z_{t}=\int_{0}^{t}\mathcal{K}(t,s)dB_{s},\quad t\in[0,T],$

with the kernel of the form

$\mathcal{K}(t,s)=a(s)\int_{s}^{t}b(u)c(u-s)du\mathbbm{1}_{s<t},$

where $a\in L^{p}[0,T]$, $b\in L^{q}[0,T]$ and $c\in L^{r}[0,T]$ with $p$,
$q$, $r$ such that

  * 1)

$p\in[2,\infty]$, $q\in(1,\infty]$, $r\in[1,\infty]$,

  * 2)

$\frac{1}{p}+\frac{1}{r}\geq\frac{1}{2}$,

  * 3)

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<\frac{3}{2}$.

Under the conditions specified above, the process $Z$ satisfies (see [27,
Lemma 1])

$\left(\mathbb{E}|Z_{t}-Z_{s}|^{2}\right)^{\frac{1}{2}}\leq\lVert
a\rVert_{p}\lVert b\rVert_{q}\lVert
c\rVert_{r}|t-s|^{\frac{3}{2}-\frac{1}{p}-\frac{1}{q}-\frac{1}{r}},\quad
t,s\in[0,T],$

and therefore has a modification with Hölder continuous paths of all orders
$\lambda\in\left(0,\frac{3}{2}-\frac{1}{p}-\frac{1}{q}-\frac{1}{r}\right)$.

######  Example 1.5 (_multifractional Brownian motion_).

The harmonizable multifractional Brownian motion $Z=\\{Z_{t},~{}t\in[0,T]\\}$
with functional parameter $H$: $[0,T]\to(0,1)$ (for more detail on this
process, see e.g. [6], [35], [19] and references therein). Namely,

$Z_{t}:=\int_{\mathbb{R}}\frac{e^{itu}-1}{|u|^{H_{t}+\frac{1}{2}}}\widetilde{W}(du),\quad
t\in[0,T],$

where $\widetilde{W}(du)$ is a unique Gaussian complex-valued random measure
such that for all $f\in L^{2}(\mathbb{R})$

$\int_{\mathbb{R}}f(u)W(du)=\int_{\mathbb{R}}\widehat{f}(u)\widetilde{W}(du)\quad\text{a.s.}$

Also let $H$ satisfy the following assumptions:

  * 1)

there exist constants $0<h_{1}<h_{2}<1$ such that for any $t\in[0,T]$

$h_{1}<H_{t}<h_{2},$

  * 2)

there exist constants $D>0$ and $\alpha\in(0,1]$ such that

$|H_{t}-H_{s}|\leq D|t-s|^{\alpha},\quad t,s\in[0,T].$

Then, according to Lemma 3.1 from [19], there is a constant $C>0$ such that
for all $s,t\in[0,T]$:

$\left(\mathbb{E}(Z_{t}-Z_{s})^{2}\right)^{\frac{1}{2}}\leq
C|t-s|^{h_{1}\wedge\alpha}$

and, since $Z$ is clearly Gaussian, it has a Hölder continuous modification of
any order $\lambda\in(0,h_{1}\wedge\alpha)$.

######  Example 1.6 (_non-Gaussian continuous martingales_).

Denote $B=\\{B_{t},~{}t\in[0,T]\\}$ a standard Brownian motion and
$\sigma=\\{\sigma_{t},~{}t\in[0,T]\\}$ an Itô integrable process such that,
for all $\beta>0$,

$\sup_{u\in[0,T]}\mathbb{E}\sigma^{2+2\beta}_{u}<\infty.$ (1.5)

Define

$Z_{t}:=\int_{0}^{t}\sigma_{u}dB_{u},\quad t\in[0,T].$

Then, by the Burkholder-Davis-Gundy inequality, for any $0\leq s<t\leq T$ and
any $\beta>0$:

$\displaystyle\mathbb{E}|Z_{t}-Z_{s}|^{2+2\beta}$ $\displaystyle\leq
C_{\beta}\mathbb{E}\left[\left(\int_{s}^{t}\sigma^{2}_{u}du\right)^{1+\beta}\right]\leq
C_{\beta}(t-s)^{\beta}\int_{s}^{t}\mathbb{E}\sigma^{2+2\beta}_{u}du$ (1.6)
$\displaystyle\leq
C_{\beta}\sup_{u\in[0,T]}\mathbb{E}\sigma^{2+2\beta}_{u}(t-s)^{1+\beta}.$

Therefore, by the Kolmogorov continuity theorem and an arbitrary choice of
$\beta$, $Z$ has a modification that is $\lambda$-Hölder continuous of any
order $\lambda\in\left(0,\frac{1}{2}\right)$.

Next, for an arbitrary $\lambda\in\left(0,\frac{1}{2}\right)$, choose $p\geq
1$ such that $\lambda+\frac{1}{p}<\frac{1}{2}$ and put

$\Lambda:=A_{\lambda+\frac{1}{p},p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z(x)-Z(y)|^{p}}{|x-y|^{\lambda
p+2}}dxdy\right)^{\frac{1}{p}},$

where $A_{\lambda+\frac{1}{p},p}$ is defined by (1.2). By the the Burkholder-
Davis-Gundy inequality, for any $r>p$, we obtain

$\mathbb{E}|Z_{t}-Z_{s}|^{r}\leq|t-s|^{\frac{r}{2}}C_{r}\sup_{u\in[0,T]}\mathbb{E}\sigma^{r}_{u},\quad
s,t\in[0,T].$

Hence, using Lemma (1.1) and the Minkowski integral inequality, we have:

$\displaystyle\left(\mathbb{E}\Lambda^{r}\right)^{\frac{p}{r}}$
$\displaystyle=A^{p}_{\lambda+\frac{1}{p},p}\left(\mathbb{E}\left[\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z_{u}-Z_{v}|^{p}}{|u-v|^{\lambda
p+2}}dudv\right)^{\frac{r}{p}}\right]\right)^{\frac{p}{r}}$ $\displaystyle\leq
A^{p}_{\lambda+\frac{1}{p},p}\int_{0}^{T}\int_{0}^{T}\frac{\left(\mathbb{E}[|Z_{u}-Z_{v}|^{r}]\right)^{\frac{p}{r}}}{|u-v|^{\lambda
p+2}}dudv$ $\displaystyle\leq
A^{p}_{\lambda+\frac{1}{p},p}C_{r}^{\frac{p}{r}}\left(\sup_{t\in[0,T]}\mathbb{E}\sigma^{r}_{t}\right)^{\frac{p}{r}}\int_{0}^{T}\int_{0}^{T}|u-v|^{\frac{p}{2}-\lambda
p-2}dudv<\infty,$

since $\frac{p}{2}-\lambda p-2>-1$, i.e. $\mathbb{E}\Lambda^{r}<\infty$ for
all $r>0$. Note that condition (1.5) can actually be relaxed (see e.g. [8,
Lemma 14.2]).

###  1.2 The drift

Set $T\in(0,\infty)$. Let and $\varphi$, $\psi$: $[0,T]\to\mathbb{R}$,
$\varphi(t)<\psi(t)$, $t\in[0,T]$, be $\lambda$-Hölder continuous functions,
with $\lambda\in(0,1)$ being the same as in assumption (Z2), i.e. there exists
a constant $K=K_{\lambda}$ such that

$|\varphi(t)-\varphi(s)|+|\psi(t)-\psi(s)|\leq K|t-s|^{\lambda},\quad
t,s\in[0,T].$

For an arbitrary pair $a_{1}$, $a_{2}\in[-\infty,\infty]$ denote

$\mathcal{D}_{a_{1},a_{2}}:=\\{(t,y)~{}|~{}t\in[0,T],y\in(\varphi(t)+a_{1},\psi(t)-a_{2})\\}.$
(1.7)

By $\mathcal{D}_{a_{1}}$ we shall mean the set
$\mathcal{D}_{a_{1},-\infty}=\\{(t,y)~{}|~{}t\in[0,T],y\in(\varphi(t)+a_{1},\infty)\\}$.

Consider the stochastic differential equation of the form (0.3), where
$Y(0)>\varphi(0)$ and the drift $b$ is a function satisfying the following
assumptions:

  * (A1)

$b$: $\mathcal{D}_{0}\to\mathbb{R}$ is continuous;

  * (A2)

for any $\varepsilon>0$ there is a constant $c_{\varepsilon}>0$ such that for
any $(t,y_{1}),(t,y_{2})\in\mathcal{D}_{\varepsilon}$:

$|b(t,y_{1})-b(t,y_{2})|\leq c_{\varepsilon}|y_{1}-y_{2}|;$

  * (A3)

there are such positive constants $y_{*}$, $c$ and $\gamma$ that for all
$(t,y)\in\mathcal{D}_{0}\setminus\mathcal{D}_{y_{*}}$:

$b(t,y)\geq\frac{c}{\left(y-\varphi(t)\right)^{\gamma}}.$

  * (A4)

the constant $\gamma$ from assumption (A3) satisfies condition

$\gamma>\frac{1-\lambda}{\lambda}$

with $\lambda$ being the order of Hölder continuity of $\varphi$, $\psi$ and
paths of $Z$.

######  Example 1.7.

Let $\alpha_{1}$: $[0,T]\to(0,\infty)$ be an arbitrary continuous function and
$\alpha_{2}$: $\mathcal{D}_{0}\to\mathbb{R}$ be such that

$|\alpha_{2}(t,y_{1})-\alpha_{2}(t,y_{2})|\leq
C|y_{1}-y_{2}|,\quad(t,y_{1}),(t,y_{2})\in\mathcal{D}_{0},$

for some constant $C>0$. Then

$b(t,y):=\frac{\alpha_{1}(t)}{(y-\varphi(t))^{\gamma}}-\alpha_{2}(t,y),\quad(t,y)\in\mathcal{D}_{0},$

satisfies assumptions (A1)–(A4) (provided that
$\gamma>\frac{1-\lambda}{\lambda}$).

In section 3, we discuss the equation (0.3) in a slightly different setting.
Namely, we consider $\varphi(0)<Y_{0}<\psi(0)$ as well as an alternative list
of assumptions on $b$:

  * (B1)

$b$: $\mathcal{D}_{0,0}\to\mathbb{R}$ is continuous;

  * (B2)

for any pair $\varepsilon_{1}$, $\varepsilon_{2}>0$ such that
$\varepsilon_{1}+\varepsilon_{2}<\lVert\varphi-\psi\rVert_{\infty}$ there is a
constant $c_{\varepsilon_{1},\varepsilon_{2}}>0$ such that for any
$(t,y_{1}),(t,y_{2})\in\mathcal{D}_{\varepsilon_{1},\varepsilon_{2}}$:

$|b(t,y_{1})-b(t,y_{2})|\leq
c_{\varepsilon_{1},\varepsilon_{2}}|y_{1}-y_{2}|;$

  * (B3)

there are constants $\gamma$, $y_{*}>0$,
$y_{*}<\frac{1}{2}\lVert\varphi-\psi\rVert_{\infty}$ and $c>0$ such that for
all $(t,y)\in{\mathcal{D}}_{0,0}\setminus{\mathcal{D}}_{y_{*},0}$:

$b(t,y)\geq\frac{c}{\left(y-\varphi(t)\right)^{\gamma}},$

and for all $(t,y)\in{\mathcal{D}}_{0,0}\setminus{\mathcal{D}}_{0,y_{*}}$:

$b(t,y)\leq-\frac{c}{\left(\psi(t)-y\right)^{\gamma}}.$

  * (B4)

the constant $\gamma$ from assumption (B3) satisfies condition

$\gamma>\frac{1-\lambda}{\lambda}$

with $\lambda$ being the order of Hölder continuity of $\varphi$, $\psi$ and
paths of $Z$.

######  Remark 1.8.

We always assume that the initial value $Y_{0}$ is a deterministic constant
such that $Y_{0}>\varphi(0)$, if the drift $b$ satisfies Assumptions
(A1)–(A4), and $\varphi(0)<Y_{0}<\psi(0)$, if the drift $b$ satisfies
Assumptions (B1)–(B4).

######  Example 1.9.

Let $\alpha_{1}$: $[0,T]\to(0,\infty)$, $\alpha_{2}$: $[0,T]\to(0,\infty)$ and
$\alpha_{3}$: $\mathcal{D}_{0,0}\to\mathbb{R}$ be continuous and

$|\alpha_{3}(t,y_{1})-\alpha_{3}(t,y_{2})|\leq
C|y_{1}-y_{2}|,\quad(t,y_{1}),(t,y_{2})\in\mathcal{D}_{0,0},$

for some constant $C>0$. Then

$b(t,y):=\frac{\alpha_{1}(t)}{(y-\varphi(t))^{\gamma}}-\frac{\alpha_{2}(t)}{(\psi(t)-y)^{\gamma}}-\alpha_{3}(t,y),\quad
t\in[0,T],y\in\mathcal{D}_{0,0},$

satisfies assumptions (B1)–(B4) provided that
$\gamma>\frac{1-\lambda}{\lambda}$.

##  2 SDE with lower-sandwiched solution case

In this section, we discuss existence, uniqueness and properties of the
solution of (0.3) under assumptions (A1)–(A4). First, we demonstrate that
(A1)–(A3) ensure the existence and uniqueness of the solution to (0.3) until
the first moment of hitting the lower bound $\\{\varphi(t),~{}t\in[0,T]\\}$
and then we prove that (A4) guarantees that the solution exists on the entire
$[0,T]$, since it always stays above $\varphi(t)$. The latter property
justifies the name _lower-sandwiched_ in the section title.

Finally, we derive additional properties of the solution, still in terms of
some form of bounds.

######  Remark 2.1.

Throughout this paper, the pathwise approach will be used, i.e. we fix a
Hölder continuous trajectory of $Z$ in most proofs. For simplicity, we omit
$\omega$ in brackets in what follows.

###  2.1 Existence and uniqueness result

As mentioned before, we shall start from the existence and uniqueness of the
local solution.

######  Theorem 2.2.

Let assumptions (A1)–(A3) hold. Then SDE (0.3) has a unique local solution in
the following sense: there exists a continuous process
$Y=\\{Y_{t},~{}t\in[0,T]\\}$ such that

$Y_{t}=Y_{0}+\int_{0}^{t}b(s,Y_{s})ds+Z_{t},\quad\forall t\in[0,\tau_{0}],$

with

$\displaystyle\tau_{0}:$ $\displaystyle=\sup\\{t\in[0,T]~{}|~{}\forall
s\in[0,t):Y_{s}>\varphi(s)\\}$
$\displaystyle=\inf\\{t\in[0,T]~{}|~{}Y_{t}=\varphi(t)\\}\wedge T.$

Furthermore, if $\tilde{Y}$ is another process satisfying equation (0.3) on
any interval $[0,t]\subset[0,\tilde{\tau}_{0})$, where

$\tilde{\tau}_{0}:=\sup\\{s\in[0,T]~{}|~{}\forall
u\in[0,s):\tilde{Y}_{u}>\varphi(s)\\},$

then $\tau_{0}=\tilde{\tau}_{0}$ and $\tilde{Y}_{t}=Y_{t}$ for all
$t\in[0,\tau_{0}]$.

###### Proof.

The proof is based on careful approximation of the non-Lipschitz drift by some
Lipschitz functions. The approximants are explicit and can be used for
numerical purposes. Nevertheless, the proof is quite technical and we have set
it in the Appendix A. ∎

Theorem 2.2 shows that equation (0.3) has a unique solution until the latter
stays above $\\{\varphi(t),t\in[0,T]\\}$. However, an additional condition
(A4) on the constant $\gamma$ from assumption (A3) allows to ensure that the
corresponding process $Y$ always stays above $\varphi$. More precisely, we
have the following result.

######  Theorem 2.3.

Let assumptions (A1)–(A4) hold. Then the process $Y$ introduced in (0.3)
satisfies

$Y_{t}>\varphi(t),\quad t\in[0,T],$

and therefore the equation (0.3) has a unique solution on the entire $[0,T]$.

###### Proof.

Assume that $\tau:=\inf\\{t\in[0,T]~{}|~{}Y_{t}=\varphi(t)\\}\in[0,T]$ (here
we assume that $\inf\emptyset=+\infty$). For any
$\varepsilon<\min\left\\{y_{*},Y_{0}-\varphi(0)\right\\}$, where $y_{*}$ is
from assumption (A3), consider

$\tau_{\varepsilon}:=\sup\\{t\in[0,\tau]~{}|~{}Y_{t}=\varphi(t)+\varepsilon\\}.$

Due to the definitions of $\tau$ and $\tau_{\varepsilon}$,

$\varphi(\tau)-\varphi(\tau_{\varepsilon})-\varepsilon=Y_{\tau}-Y_{\tau_{\varepsilon}}=\int^{\tau}_{\tau_{\varepsilon}}b(s,Y_{s})ds+Z_{\tau}-Z_{\tau_{\varepsilon}}.$

Moreover, for all $t\in[\tau_{\varepsilon},\tau)$:
$(t,Y_{t})\in\mathcal{D}_{0}\setminus\mathcal{D}_{\varepsilon}$, so, using the
fact that $\varepsilon<y_{*}$ and assumption (A3), we obtain that for
$t\in[\tau_{\varepsilon},\tau)$:

$b(t,Y_{t})\geq\frac{c}{(Y_{t}-\varphi(t))^{\gamma}}\geq\frac{c}{\varepsilon^{\gamma}}.$
(2.1)

Finally, due to the Hölder continuity of $\varphi$ and $Z$,

$-(Z_{\tau}-Z_{\tau_{\varepsilon}})+(\varphi(\tau)-\varphi(\tau_{\varepsilon}))\leq(\Lambda+K)(\tau-\tau_{\varepsilon})^{\lambda}=:\bar{\Lambda}(\tau-\tau_{\varepsilon})^{\lambda}.$

Therefore, taking into account all of the above, we get:

$\bar{\Lambda}(\tau-\tau_{\varepsilon})^{\lambda}\geq\int^{\tau}_{\tau_{\varepsilon}}\frac{c}{\varepsilon^{\gamma}}ds+\varepsilon=\frac{c(\tau-\tau_{\varepsilon})}{\varepsilon^{\gamma}}+\varepsilon,$

i.e.

$\frac{c(\tau-\tau_{\varepsilon})}{\varepsilon^{\gamma}}-\bar{\Lambda}(\tau-\tau_{\varepsilon})^{\lambda}+\varepsilon\leq
0.$ (2.2)

Now consider the function $F_{\varepsilon}$: $\mathbb{R}^{+}\to\mathbb{R}$
such that

$F_{\varepsilon}(t)=\frac{c}{\varepsilon^{\gamma}}t-\bar{\Lambda}t^{\lambda}+\varepsilon.$

According to (2.2), $F_{\varepsilon}(\tau-\tau_{\varepsilon})\leq 0$ for any
$0<\varepsilon<\min\left\\{y_{*},Y_{0}-\varphi(0)\right\\}$. It is easy to
verify that $F_{\varepsilon}$ attains its minimum at the point

$t^{*}=\left(\frac{\lambda\bar{\Lambda}}{c}\right)^{\frac{1}{1-\lambda}}\varepsilon^{\frac{\gamma}{1-\lambda}}$

and

$F_{\varepsilon}(t^{*})=\varepsilon-D\bar{\Lambda}^{\frac{1}{1-\lambda}}\varepsilon^{\frac{\gamma\lambda}{1-\lambda}},$

where
$D:=\left(\frac{1}{c}\right)^{\frac{\lambda}{1-\lambda}}\left(\lambda^{\frac{\lambda}{1-\alpha}}-\lambda^{\frac{1}{1-\lambda}}\right)>0$.
Note that, by (A4), we have $\frac{\gamma\lambda}{1-\lambda}>1$. Hence it is
easy to verify that there exists $\varepsilon^{*}$ such that for all
$\varepsilon<\varepsilon^{*}$ $F_{\varepsilon}(t^{*})>0$, which contradicts
(2.2). Therefore, $\tau$ cannot belong to $[0,T]$ and $Y$ exceeds $\varphi$. ∎

######  Remark 2.4.

  1. 1.

The result above can be generalized to the case of infinite time horizon in a
straightforward manner. For this, it is sufficient to assume that $\varphi$ is
locally $\lambda$-Hölder continuous, $Z$ has locally Hölder continuous paths,
i.e. for each $T>0$ there exist constant $K_{T}>0$ and random variable
$\Lambda=\Lambda_{T}(\omega)>0$ such that

$|\varphi(t)-\varphi(s)|\leq
K_{T}|t-s|^{\lambda},\quad|Z_{t}-Z_{s}|\leq\Lambda_{T}|t-s|^{\lambda},\quad
t,s\in[0,T],$

and assumptions (A1)–(A4) hold on $[0,T]$ for any $T>0$ (in such case,
constants $c_{\varepsilon}$, $y_{*}$ and $c$ from the corresponding
assumptions are allowed to depend on $T$).

  2. 2.

Since all the proofs above are based on pathwise calculus, it is possible to
extend the results to stochastic $\varphi$ and $Y_{0}$ (provided that
$Y_{0}>\varphi(0)$).

###  2.2 Upper and lower bounds for the solution

As we have seen in the previous subsection, each random variable $Y_{t}$,
$t\in[0,T]$, is _a priori_ lower sandwiched by the deterministic value
$\varphi(t)$ (under assumptions (A1)–(A4)). In this subsection, we derive
additional bounds from above and below for $Y_{t}$ in terms of the random
variable $\Lambda$ characterizing the noise from (1.1). Furthermore, such
bounds allow us to establish existence of moments of $Y$ of all orders
including the negative ones.

######  Theorem 2.5.

Let assumptions (A1)–(A4) hold and $\Lambda$ be the random variable such that

$|Z_{t}-Z_{s}|\leq\Lambda|t-s|^{\lambda},\quad t,s\in[0,T].$

Then, for any $r>0$,

  * 1)

there exist positive deterministic constants $M_{1}(r,T)$ and $M_{2}(r,T)$
such that

$|Y_{t}|^{r}\leq M_{1}(r,T)+M_{2}(r,T)\Lambda^{r},\quad t\in[0,T];$

  * 2)

additionally, if $\Lambda$ can be chosen in such a way that
$\mathbb{E}\Lambda^{r}<\infty$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y_{t}|^{r}\right]<\infty.$

###### Proof.

It is enough to prove item 1) for $r=1$ as the rest of the Theorem will then
become clear. Denote $\eta:=\frac{Y(0)-\varphi(0)}{2}$ and let

$\tau_{1}:=\sup\left\\{s\in[0,T]~{}|~{}\forall
u\in[0,s]:Y_{u}\geq\varphi(u)+\eta\right\\}.$

Our initial goal is to prove the inequality of the form

$\displaystyle\left|Y_{t}\right|\leq|Y_{0}|+TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda
T^{\lambda}+\max_{u\in[0,T]}|\varphi(u)|+\eta,$ (2.3)

where

$A_{T}:=c_{\eta}\left(1+\max_{u\in[0,T]}|\varphi(u)|+\eta\right)+\max_{u\in[0,T]}\left|b\left(u,\varphi(u)+\eta\right)\right|$

and $c_{\eta}$ is from assumption (A2).

Similarly to Proposition A.2 in Appendix A, we get (2.3) considering the cases
$t\leq\tau_{1}$ and $t>\tau_{1}$ separately.
Case  $\boldsymbol{t\leq\tau_{1}}$. For any $s\in[0,t]$:
$(s,Y_{s})\in\mathcal{D}_{\eta}$ and, therefore, by assumption (A2), for all
$s\in[0,t]$:

$\left|b(s,Y_{s})-b\left(s,\varphi(s)+\eta\right)\right|\leq
c_{\eta}\left|Y_{s}-\varphi(s)-\eta\right|,$

hence

$\displaystyle|b(s,Y_{s})|$ $\displaystyle\leq
c_{\eta}|Y_{s}|+c_{\eta}\left(\max_{u\in[0,T]}|\varphi(u)|+\eta\right)+\max_{u\in[0,T]}\left|b\left(u,\varphi(u)+\eta\right)\right|$
$\displaystyle\leq A_{T}(1+|Y_{s}|).$

Therefore, taking into account that $|Z_{t}|\leq\Lambda T^{\lambda}$, we have:

$\displaystyle\left|Y_{t}\right|$
$\displaystyle=\left|Y_{0}+\int_{0}^{t}b(s,Y_{s})ds+Z_{t}\right|$
$\displaystyle\leq|Y_{0}|+\int_{0}^{t}|b(s,Y_{s})|ds+|Z_{t}|$
$\displaystyle\leq|Y_{0}|+TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda
T^{\lambda}$
$\displaystyle\leq|Y_{0}|+TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda
T^{\lambda}+\max_{u\in[0,T]}|\varphi(u)|+\eta.$

Case $\boldsymbol{t>\tau_{1}}$. From the definition of $\tau_{1}$ and
continuity of $Y$, $Y_{\tau_{1}}=\eta$. Furthermore, since $Y_{s}>\varphi(s)$
for all $s\geq 0$, we can consider

$\tau_{2}(t):=\sup\left\\{s\in(\tau_{1},t]~{}|~{}Y_{s}<\varphi(s)+\eta\right\\}.$

Note that $\left|Y_{\tau_{2}(t)}\right|\leq\max_{u\in[0,T]}|\varphi(u)|+\eta$,
so

$\displaystyle\left|Y_{t}\right|$
$\displaystyle\leq\left|Y_{t}-Y_{\tau_{2}(t)}\right|+\left|Y_{\tau_{2}(t)}\right|$
(2.4)
$\displaystyle\leq\left|Y_{t}-Y_{\tau_{2}(t)}\right|+\max_{u\in[0,T]}|\varphi(u)|+\eta.$

If $\tau_{2}(t)<t$, we have that $(s,Y_{s})\in\mathcal{D}_{\eta}$ for all
$s\in[\tau_{2}(t),t]$, therefore, similarly to Step 1,

$|b(s,Y_{s})|\leq A_{T}(1+|Y_{s}|),$

so

$\displaystyle\left|Y_{t}-Y_{\tau_{2}(t)}\right|$
$\displaystyle=\left|\int_{\tau_{2}(t)}^{t}b(s,Y_{s})ds+(Z_{t}-Z_{\tau_{2}(t)})\right|$
$\displaystyle\leq\int_{\tau_{2}(t)}^{t}|b(s,Y_{s})|ds+|Z_{t}-Z_{\tau_{2}(t)}|$
$\displaystyle\leq TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda T^{\lambda},$

whence, taking into account (2.4), we have:

$\displaystyle\left|Y_{t}\right|$ $\displaystyle\leq
TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda
T^{\lambda}+\max_{u\in[0,T]}|\varphi(u)|+\eta$ (2.5)
$\displaystyle\leq|Y_{0}|+TA_{T}+A_{T}\int_{0}^{t}|Y_{s}|ds+\Lambda
T^{\lambda}+\max_{u\in[0,T]}|\varphi(u)|+\eta.$

Now, when we have seen that (2.3) holds for any $t\in[0,T]$, we apply the
Gronwall’s inequality to get

$\displaystyle|Y_{t}|$ $\displaystyle\leq\left(|Y_{0}|+TA_{T}+\Lambda
T^{\lambda}+\max_{u\in[0,T]}|\varphi(u)|+\eta\right)e^{TA_{T}}$
$\displaystyle=:M_{1}(1,T)+M_{2}(1,T)\Lambda,$

where

$\displaystyle M_{1}(1,T)$
$\displaystyle:=\left(|Y_{0}|+TA_{T}+\max_{u\in[0,T]}|\varphi(u)|+\frac{Y_{0}-\varphi(0)}{2}\right)e^{TA_{T}},$
$\displaystyle M_{2}(1,T)$ $\displaystyle:=T^{\lambda}e^{TA_{T}}.$

∎

Hereafter we provide further specifications on the solution bounds. For this,
recall that, for all $t,s\in[0,T]$, the lower sandwich function $\varphi$
satisfies

$|\varphi(t)-\varphi(s)|\leq K|t-s|^{\lambda}$

and the noise satisfies (Z2):

$|Z_{t}-Z_{s}|\leq\Lambda|t-s|^{\lambda},$

where $K>0$ is a deterministic constant and $\Lambda$ is a positive random
variable.

######  Theorem 2.6.

Let assumptions (A1)–(A4) hold and $\Lambda$ be the random variable such that

$|Z_{t}-Z_{s}|\leq\Lambda|t-s|^{\lambda},\quad t,s\in[0,T].$

Then, for any $r>0$,

  * 1)

there exists a constant $M_{3}(r,T)>0$ depending only on $r$, $T$, $\lambda$,
$\gamma$ and the constant $c$ from assumption (A3) such that for all
$t\in[0,T]$:

$(Y_{t}-\varphi(t))^{-r}\leq
M_{3}(r,T){\tilde{\Lambda}^{\frac{r}{\gamma\lambda+\lambda-1}}},$ (2.6)

where

$\tilde{\Lambda}:=\max\left\\{\Lambda,K,\left(2\beta\right)^{\lambda-1}\left(\frac{(Y_{0}-\varphi(0))\wedge
y_{*}}{2}\right)^{1-\lambda-\gamma\lambda}\right\\}$

with

$\beta:=\frac{\lambda^{\frac{\lambda}{1-\lambda}}-\lambda^{\frac{1}{1-\lambda}}}{c^{\frac{\lambda}{1-\lambda}}}>0;$

  * 2)

additionally, if $\Lambda$ can be chosen in such a way that
$\mathbb{E}\Lambda^{\frac{r}{\gamma\lambda+\lambda-1}}<\infty$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}(Y_{t}-\varphi(t))^{-r}\right]<\infty.$

###### Proof.

Just as in Theorem 2.5, it is enough to prove that there exists a constant
$L>0$ that depends only on $T$, $\lambda$, $\gamma$ and the constant $c$ from
assumption (A3) such that for all $t\in[0,T]$

$Y_{t}-\varphi(t)\geq\frac{L}{\tilde{\Lambda}^{\frac{1}{\gamma\lambda+\lambda-1}}}$

and then the rest of the Theorem will follow.

Put

$\varepsilon=\varepsilon(\omega):=\frac{1}{(2\beta)^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}\tilde{\Lambda}^{\frac{1}{\gamma\lambda+\lambda-1}}}.$

Note that $\tilde{\Lambda}$ is chosen is such a way that

$|\varphi(t)-\varphi(s)|+|Z_{t}-Z_{s}|\leq\tilde{\Lambda}|t-s|^{\lambda},\quad
t,s\in[0,T],$

and, furthermore, $\varepsilon<Y_{0}-\varphi(0)$ and $\varepsilon<y_{*}$. Fix
an arbitrary $t\in[0,T]$. If $Y_{t}-\varphi(t)>\varepsilon$, then, by
definition of $\varepsilon$, estimate of the type (2.6) holds automatically.
If $Y_{t}-\varphi(t)<\varepsilon$, then, since $Y_{0}-\varphi(0)>\varepsilon$,
one can define

$\tau(t):=\sup\\{s\in[0,t]~{}|~{}Y_{s}-\varphi(s)=\varepsilon\\}.$

Since for all $s\in[\tau(t),t]$ $Y_{s}-\varphi(s)\leq\varepsilon<y_{*}$, one
can apply Assumption (A3) and write

$\displaystyle Y_{t}-\varphi(t)$
$\displaystyle=Y_{\tau(t)}-\varphi(t)+\int_{\tau(t)}^{t}b(s,Y_{s})ds+Z_{t}-Z_{\tau(t)}$
$\displaystyle=\varepsilon+\varphi(\tau(t))-\varphi(t)+\int_{\tau(t)}^{t}b(s,Y_{s})ds+Z_{t}-Z_{\tau(t)}$
$\displaystyle\geq\varepsilon+\frac{c}{\varepsilon^{\gamma}}(t-\tau(t))-\tilde{\Lambda}(t-\tau(t))^{\lambda}.$

Consider the function $F_{\varepsilon}:\mathbb{R}_{+}\to\mathbb{R}$ such that

$F_{\varepsilon}(x)=\varepsilon+\frac{c}{\varepsilon^{\gamma}}x-\tilde{\Lambda}x^{\lambda}.$

It is straightforward to verify that $F_{\varepsilon}$ attains its minimum at

$x_{*}:=\left(\frac{\lambda}{c}\right)^{\frac{1}{1-\lambda}}\varepsilon^{\frac{\gamma}{1-\lambda}}\tilde{\Lambda}^{\frac{1}{1-\lambda}}$

and, taking into account the explicit form of $\varepsilon$,

$\displaystyle F_{\varepsilon}(x_{*})$
$\displaystyle=\varepsilon+\frac{\lambda^{\frac{1}{1-\lambda}}}{c^{\frac{\lambda}{1-\lambda}}}\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}\tilde{\Lambda}^{\frac{1}{1-\lambda}}-\frac{\lambda^{\frac{\lambda}{1-\lambda}}}{c^{\frac{\lambda}{1-\lambda}}}\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}\tilde{\Lambda}^{\frac{1}{1-\lambda}}$
$\displaystyle=\varepsilon-\beta\varepsilon^{\frac{\gamma\lambda}{1-\lambda}}\tilde{\Lambda}^{\frac{1}{1-\lambda}}$
$\displaystyle=\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}\tilde{\Lambda}^{\frac{1}{\gamma\lambda+\lambda-1}}}$
$\displaystyle=\frac{\varepsilon}{2},$

i.e., if $Y(t)<\varphi(t)+\varepsilon$, we have that

$Y(t)-\varphi(t)\geq F_{\varepsilon}(t-\tau(t))\geq
F_{\varepsilon}(x_{*})=\frac{\varepsilon}{2},$

and thus for any $t\in[0,T]$

$Y(t)\geq\varphi(t)+\frac{\varepsilon}{2}=\varphi(t)+\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}\tilde{\Lambda}^{\frac{1}{\gamma\lambda+\lambda-1}}}=:\frac{L}{\tilde{\Lambda}^{\frac{1}{\gamma\lambda+\lambda-1}}},$

where
$L:=\frac{1}{2^{\frac{\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{1-\lambda}{\gamma\lambda+\lambda-1}}}$.
By this, the proof is complete. ∎

######  Remark 2.7.

As one can see, the existence of moments for $Y$ comes down to existence of
moments for $\Lambda$. Note that the noises given in Examples 1.2 and 1.6 fit
into this framework.

######  Remark 2.8.

The constant $M_{3}(r,T)$ from Theorem 2.6 can be explicitly written as

$M_{3}(r,T)=2^{\frac{r\gamma\lambda}{\gamma\lambda+\lambda-1}}\beta^{\frac{r(1-\lambda)}{\gamma\lambda+\lambda-1}}.$

##  3 SDE with sandwiched solution case

The fact that, under assumptions (A1)–(A4), the solution $Y$ of (0.3) stays
above the function $\varphi$ is essentially based on the rapid growth to
infinity of $b(t,Y_{t})$ whenever $Y_{t}$ approaches $\varphi(t)$, $t\geq 0$.
The same effect is exploited in the case of assumptions (B1)–(B4) and the
corresponding solution turns out to be both upper and lower bounded, i.e.
_sandwiched_.

Recall that $\varphi$, $\psi$: $[0,T]\to\mathbb{R}$, $\varphi(t)<\psi(t)$,
$t\in[0,T]$, are $\lambda$-Hölder continuous functions, $\lambda\in(0,1)$.
Consider a stochastic differential equation of the form (0.3) with
$\varphi(0)<Y_{0}<\psi(0)$, $Z$ being, as before, a stochastic process with
$\lambda$-Hölder continuous trajectories and the drift $b$ satisfying
assumptions (B1)–(B4).

In line with the previous section, we show that the solution exists and it is
sandwiched.

######  Theorem 3.1.

Let assumptions (B1)–(B4) hold. Then the equation (0.3) has a unique solution
$Y=\\{Y_{t},~{}t\in[0,T]\\}$ such that

$\varphi(t)<Y_{t}<\psi(t),\quad t\in[0,T].$ (3.1)

###### Proof.

The proof uses the techniques presented in Appendix A and section 2 in a
straightforward manner, so full details will be omitted. Here we present only
the kernel points.

First, let $n_{0}>-\min_{t\in[0,T]}b(t,\psi(t)-y_{*})$, with $y_{*}$ being
from assumption (B3). For an arbitrary $n\geq n_{0}$ define the set

$\widehat{\mathcal{G}}_{n}:=\left\\{(t,y)\in{\mathcal{D}}_{0,0}\setminus{\mathcal{D}}_{0,y_{*}}~{}|~{}b(t,y)>-n\right\\},$

and consider the stochastic process $Y^{(n)}_{t}$ that is the solution to the
stochastic differential equation of the form

$dY^{(n)}_{t}=\hat{f}_{n}(t,Y^{(n)}_{t})dt+dZ_{t},\quad Y^{(n)}_{0}=Y_{0}>0,$

where

$\hat{f}_{n}(t,y)=\begin{cases}b(t,y)+\frac{1}{n},&\quad(t,y)\in\widehat{\mathcal{G}}_{n}\cup{\mathcal{D}}_{0,y_{*}},\\\
-n+\frac{1}{n},&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus\left(\widehat{\mathcal{G}}_{n}\cup{\mathcal{D}}_{0,y_{*}}\right).\end{cases}$

Observe that each $\hat{f}_{n}$ satisfies assumptions (A1)–(A4). Therefore, by
Theorem 2.3, each $\\{Y^{(n)}_{t},~{}t\in[0,T]\\}$, $n\geq n_{0}$, exists, is
unique and exceeds $\\{\varphi(t),~{}t\in[0,T]\\}$. Furthermore, by the virtue
of Theorem 2.6,

$Y^{(n)}_{t}\geq\varphi(t)+\xi,\quad t\in[0,T],$

where $\xi<y_{*}$ is a positive random variable that does not depend on $n$.
In other words, each $\\{Y^{(n)}_{t},~{}t\in[0,T]\\}$ is, in fact, a unique
solution to the equation

$dY^{(n)}_{t}=\hat{b}_{n}(t,Y^{(n)}_{t})dt+dZ_{t},\quad Y^{(n)}_{0}=Y_{0},$

with

$\hat{b}_{n}(t,y)=\begin{cases}\hat{f}_{n}(t,y),&\quad(t,y)\in{\mathcal{D}}_{\xi},\\\
b(t,\varphi(t)+\xi)+\frac{1}{n},&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus{\mathcal{D}}_{\xi}.\end{cases}$

Now, following Appendix A, it is easy to verify that
$Y_{t}=Y^{(\infty)}_{t}=\lim_{n\to\infty}Y^{(n)}_{t}$, $t\in[0,T]$, is
correctly defined and is a unique stochastic process that satisfies (0.3)
until the first moment of crossing $\psi(t)$, $t\in[0,T]$. The given claim
follows by the argument similar to the one in Theorem 2.3. ∎

Similarly to Theorem 2.6, the bounds (3.1) obtained in the
existence–uniqueness Theorem 3.1 can be refined.

######  Theorem 3.2.

Let $r>0$ be fixed.

  1. 1.

Under conditions (B1)–(B4), there exists a constant $L>0$ depending only on
$\lambda$, $\gamma$ and the constant $c$ from assumption (B3) such that the
solution $Y$ to the equation (0.3) has the property

$\varphi(t)+L\tilde{\Lambda}^{-\frac{1}{\gamma\lambda+\lambda-1}}~{}\leq~{}Y_{t}~{}\leq~{}\psi(t)-L\tilde{\Lambda}^{-\frac{1}{\gamma\lambda+\lambda-1}},\quad
t\in[0,T],$

where

$\tilde{\Lambda}:=\max\left\\{\Lambda,K,\left(4\beta\right)^{\lambda-1}\left(\frac{(Y_{0}-\varphi(0))\wedge
y_{*}\wedge(\psi(0)-Y_{0})}{2}\right)^{1-\lambda-\gamma\lambda}\right\\}$

with

$\beta:=\frac{\lambda^{\frac{\lambda}{1-\lambda}}-\lambda^{\frac{1}{1-\lambda}}}{\left(2^{\gamma}c\right)^{\frac{\lambda}{1-\lambda}}}>0$

and $K$ being such that

$|\varphi(t)-\varphi(s)|+|\psi(t)-\psi(s)|\leq K|t-s|^{\lambda},\quad
t,s\in[0,T].$

  2. 2.

If $\Lambda$ can be chosen in such a way that
$\mathbb{E}\Lambda^{\frac{r}{\gamma\lambda+\lambda-1}}<\infty$, then

$\mathbb{E}\left[\sup_{t\in[0,T]}(Y_{t}-\varphi(t))^{-r}\right]<\infty\quad\text{and}\quad\mathbb{E}\left[\sup_{t\in[0,T]}(\psi(t)-Y_{t})^{-r}\right]<\infty.$

###### Proof.

The proof is similar to the one of Theorem 2.6. ∎

##  4 Applications: generalized CIR and CEV processes

In this section, we show how two classical processes used in stochastic
volatility modeling can be generalized under our framework.

###  4.1 CIR and CEV processes driven by a Hölder continuous noise

Let $\varphi\equiv 0$. Consider

$b(y)=\frac{\kappa}{y^{\frac{\alpha}{1-\alpha}}}-\theta y,$

where $\kappa$, $\theta>0$ are positive constants,
$\alpha\in\left[\frac{1}{2},1\right)$, and the process $Z$ is a process with
$\lambda$-Hölder continuous paths with $\alpha+\lambda>1$. It is easy to
verify that for $\gamma=\frac{\alpha}{1-\alpha}$ assumptions (A1)–(A4) hold
and the prosess $Y$ satisfying the stochastic differential equation

$dY_{t}=\left(\frac{\kappa}{Y_{t}^{\frac{\alpha}{1-\alpha}}}-\theta
Y_{t}\right)dt+dZ_{t}$ (4.1)

exists, is unique and positive. Furthermore, as it is noted in Theorems 2.5
and 2.6, if the corresponding Hölder continuity constant $\Lambda$ can be
chosen to have all positive moments, $Y$ will have moments of all real orders,
including the negative ones.

The process $X=\\{X_{t},~{}t\in[0,T]\\}$ such that

$X_{t}=Y_{t}^{\frac{1}{1-\alpha}},\quad t\in[0,T],$

can be interpreted as a generalization of CIR (if $\alpha=\frac{1}{2}$) or CEV
(if $\alpha\in\left(\frac{1}{2},1\right)$) process in the following sense.
Assume that $\lambda>\frac{1}{2}$. Fix the partition
$0=t_{0}<t_{1}<t_{2}<...<t_{n}=t$ where $t\in[0,T]$, $\lvert\Delta
t\rvert:=\max_{k=1,...,n}(t_{k}-t_{k-1})$. It is clear that

$X_{t}=X_{0}+\sum_{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})=X_{0}+\sum_{k=1}^{n}(Y^{\frac{1}{1-\alpha}}_{t_{k}}-Y^{\frac{1}{1-\alpha}}_{t_{k-1}}),$

so, using the Taylor’s expansion, we obtain that

$X_{t}=X_{0}+\sum_{k=1}^{n}\left(\frac{1}{1-\alpha}Y^{\frac{\alpha}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k-1}})+\frac{\alpha\Theta_{k}^{\frac{2\alpha-1}{1-\alpha}}}{2(1-\alpha)^{2}}(Y_{t_{k}}-Y_{t_{k-1}})^{2}\right)$

with $\Theta_{k}$ being a real value between $Y_{t_{k}}$ and $Y_{t_{k-1}}$.

Using equation (4.1) and Theorem 2.6, it is easy to prove that $Y$ has
trajectories which are $\lambda$-Hölder continuous, therefore, since
$\lambda>\frac{1}{2}$,

$\sum_{k=1}^{n}\frac{\lambda\Theta_{k}^{\frac{2\alpha-1}{1-\alpha}}}{2(1-\alpha)^{2}}(Y_{t_{k}}-Y_{t_{k-1}})^{2}\to
0,\quad\lvert\Delta t\rvert\to 0,$ (4.2)

and

$\begin{gathered}\sum_{k=1}^{n}\frac{1}{1-\alpha}Y^{\frac{\alpha}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k-1}})=\frac{1}{1-\alpha}\sum_{k=1}^{n}X^{\alpha}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k-1}})\\\
=\frac{1}{1-\alpha}\sum_{k=1}^{n}X^{\alpha}_{t_{k-1}}\left(\int_{t_{k-1}}^{t_{k}}\left(\frac{\kappa}{Y_{s}^{\frac{\alpha}{1-\alpha}}}-\theta
Y_{s}\right)ds+(Z_{t_{k}}-Z_{t_{k-1}})\right)\\\
=\frac{1}{1-\alpha}\sum_{k=1}^{n}X^{\alpha}_{t_{k-1}}\int_{t_{k-1}}^{t_{k}}\left(\frac{\kappa}{X_{s}^{\alpha}}-\theta
X_{s}^{1-\alpha}\right)ds+\frac{1}{1-\alpha}\sum_{k=1}^{n}X^{\alpha}_{t_{k-1}}(Z_{t_{k}}-Z_{t_{k-1}})\\\
\to\frac{1}{1-\alpha}\int_{0}^{t}(\kappa-\theta
X_{s})ds+\frac{1}{1-\alpha}\int_{0}^{t}X_{s}^{\alpha}dZ_{s},\quad\lvert\Delta
t\rvert\to 0.\end{gathered}$ (4.3)

Note that the integral with respect to $Z$ in (4.3) exists as a pathwise limit
of Riemann-Stieltjes integral sums due to sufficient Hölder continuity of both
the integrator and integrand, see e.g. [39].

Taking into account all of the above, the $X$ satisfies (pathwisely) the
stochastic differential equation of the CIR (or CEV) type, namely

$dX_{t}=\left(\frac{\kappa}{1-\alpha}-\frac{\theta}{1-\alpha}X_{t}\right)dt+\frac{1}{1-\alpha}X_{t}^{\alpha}dZ_{t}=(\tilde{\kappa}-\tilde{\theta}X_{t})dt+\tilde{\nu}X^{\alpha}_{t}dZ_{t},$
(4.4)

where the integral with respect to $Z$ is the pathwise Riemann-Stieltjes
integral.

######  Remark 4.1.

The integral $\int_{0}^{t}X_{s}^{\alpha}dZ_{s}$ arising above is a pathwise
Young integral, see e.g. [20, Section 4.1] and references therein.

######  Remark 4.2.

Note that the reasoning described above also implies that, for
$\alpha\in\left[\frac{1}{2},1\right)$ and $\lambda>\frac{1}{2}$, the SDE
(4.4), where the integral w.r.t. $Z$ is understood pathwisely, has a unique
strong solution in the class of non-negative stochastic processes with
$\lambda$-Hölder continuous trajectories. Indeed,
$\\{Y^{\frac{1}{1-\alpha}}_{t},~{}t\in[0,T]\\}$ with $Y$ defined by (4.1) is a
solution to (4.4). Moreover, if $X$ is another solution to (4.4), then by the
chain rule [39, Theorem 4.3.1], the process
$\\{X^{1-\alpha}_{t},~{}t\in[0,T]\\}$ must satisfy the SDE (4.1) until the
first moment of zero hitting. However, the SDE (4.1) has a unique solution
that never hits zero and thus $X^{1-\alpha}$ coincides with $Y$.

######  Remark 4.3.

Some of the properties of the process $Y$ given by (4.1) in the case of
$\lambda=\frac{1}{2}$ and $Z$ being a fractional Brownian motion with
$H>\frac{1}{2}$ were discussed in [28].

###  4.2 Mixed-fractional CEV-process

Assume that $\kappa$, $\theta$, $\nu_{1}$, $\nu_{2}$ are positive constants,
$B=\\{B_{t},~{}t\in[0,T]\\}$ is a standard Wiener process,
$B^{H}=\\{B^{H}_{t},~{}t\in[0,T]\\}$ is a fractional Brownian motion
independent of $B$ with $H\in\left(0,1\right)$, $Z=\nu_{1}B+\nu_{2}B^{H}$,
$\alpha\in\left(\frac{1}{2},1\right)$ is such that
$H\wedge\frac{1}{2}+\alpha>1$ and the function $b$ has the form

$b(y)=\frac{\kappa}{y^{\frac{\alpha}{1-\alpha}}}-\frac{\alpha\nu_{1}^{2}}{2y}-\theta
y.$

Then the process $Y$ defined by the equation

$dY_{t}=\left(\frac{\kappa}{Y_{t}^{\frac{\alpha}{1-\alpha}}}-\frac{\alpha\nu_{1}^{2}}{2(1-\alpha)Y_{t}}-\theta
Y_{t}\right)dt+\nu_{1}dB_{t}+\nu_{2}dB^{H}_{t}$ (4.5)

exists, is unique, positive and has all the moments of real orders.

If $H>\frac{1}{2}$, just as in subsection 4.1, the process
$X_{t}:=Y_{t}^{\frac{1}{1-\alpha}}$, $t\in[0,T]$, can be interpreted as a
generalization of the CEV-process.

######  Proposition 4.4.

Let $H>\frac{1}{2}$. Then the process $X_{t}:=Y_{t}^{\frac{1}{1-\alpha}}$,
$t\in[0,T]$, a.s. satisfies the SDE of the form

$dX_{t}=\left(\frac{\kappa}{1-\alpha}-\frac{\theta}{1-\alpha}X_{t}\right)dt+\frac{\nu_{1}}{1-\alpha}X_{t}^{\alpha}dB_{t}+\frac{\nu_{2}}{1-\lambda}X_{t}^{\alpha}dB^{H}_{t},$
(4.6)

where the integral with respect to $B$ is the regular Itô integral (w.r.t.
filtration generated jointly by $(B,B^{H})$) and the integral with respect to
$B^{H}$ is understood as the $L^{2}$-limit of Riemann-Stieltjes integral sums.

###### Proof.

We will use the argument that is similar to the one presented in subsection
4.1 with one main difference: since we are going to treat the integral w.r.t.
the Brownian motion $B$ as a regular Itô integral, all the convergences
(including convergence of integral sums w.r.t. $B^{H}$) must be considered in
$L^{2}$-sense. For reader’s convenience, we split the proof into several
steps.

Step 1. First, we will prove that the integral
$\int_{0}^{t}X_{s}^{\alpha}dB^{H}_{s}$ is well defined as the $L^{2}$-limit of
Riemann-Stieltjes integral sums. Let $0=t_{0}<t_{1}<t_{2}<...<t_{n}=t$ be a
partition of $[0,t]$ with the mesh $|\Delta
t|:=\max_{k=0,...,n-1}(t_{k+1}-t_{k})$.

Choose $\lambda\in\left(\frac{1}{2},H\right)$,
$\lambda^{\prime}\in\left(0,\frac{1}{2}\right)$ and $\varepsilon>0$ such that
$\lambda+\lambda^{\prime}>1$ and $\lambda+\varepsilon<H$,
$\lambda^{\prime}+\varepsilon<\frac{1}{2}$. Using Theorem 2.6 and the fact
that for any $\lambda^{\prime}\in\left(0,\frac{1}{2}\right)$ the random
variable $\Lambda_{Z,\lambda^{\prime}+\varepsilon}$ which corresponds to the
noise $Z$ and Hölder order $\lambda^{\prime}+\varepsilon$ can be chosen to
have moments of all orders, it is easy to prove that there exists a random
variable $\Upsilon_{X}$ having moments of all orders such that

$|X^{\alpha}_{t}-X^{\alpha}_{s}|\leq\Upsilon_{X}|t-s|^{\lambda^{\prime}+\varepsilon},\quad
s,t\in[0,T],\quad a.s.$

By the Young-Lóeve inequality (see e.g. [21, Theorem 6.8]), it holds a.s. that

$\displaystyle\left|\int_{0}^{t}X^{\alpha}_{s}dB^{H}_{s}-\sum_{k=0}^{n-1}X^{\alpha}_{t_{k}}(B^{H}_{t_{k+1}}-B^{H}_{t_{k}})\right|$
$\displaystyle\leq\sum_{k=0}^{n-1}\left|\int_{t_{k}}^{t_{k+1}}X^{\alpha}_{s}dB^{H}_{s}-X^{\alpha}_{t_{k}}(B^{H}_{t_{k+1}}-B^{H}_{t_{k}})\right|$
$\displaystyle\leq\frac{1}{2^{1-(\lambda+\lambda^{\prime})}}\sum_{k=0}^{n-1}[X^{\alpha}]_{\lambda^{\prime};[t_{k},t_{k+1}]}[B^{H}]_{\lambda;[t_{k},t_{k+1}]},$

where

$[f]_{\lambda;[t,t^{\prime}]}:=\left(\sup_{\Pi[t,t^{\prime}]}\sum_{l=0}^{m-1}|f(s_{l+1})-f(s_{l}))|^{\frac{1}{\lambda}}\right)^{\lambda},$

with supremum taken over all partitions
$\Pi[t,t^{\prime}]=\\{t=s_{0}<...<s_{m}=t^{\prime}\\}$ of $[t,t^{\prime}]$.

It is clear that, a.s.,

$\displaystyle[X^{\alpha}]_{\lambda^{\prime};[t_{k},t_{k+1}]}$
$\displaystyle=\left(\sup_{\Pi[t_{k},t_{k+1}]}\sum_{l=0}^{m-1}|X^{\alpha}(s_{l+1})-X^{\alpha}(s_{l}))|^{\frac{1}{\lambda^{\prime}}}\right)^{\lambda^{\prime}}$
$\displaystyle\leq\Upsilon_{X}\left(\sup_{\Pi[t_{k},t_{k+1}]}\sum_{k=0}^{m-1}(s_{l+1}-s_{l})^{1+\frac{\varepsilon}{\lambda^{\prime}}}\right)^{\lambda^{\prime}}$
$\displaystyle\leq\Upsilon_{X}|\Delta t|^{\lambda^{\prime}+\varepsilon}$

and, similarly,

$\displaystyle[B^{H}]_{\lambda;[t_{k},t_{k+1}]}\leq\Lambda_{B^{H}}|\Delta
t|^{\lambda+\varepsilon},$

where $\Lambda_{B^{H}}$ has moments of all orders and

$|B^{H}_{t}-B^{H}_{s}|\leq\Lambda_{B^{H}}|t-s|^{\lambda+\varepsilon},$

whence

$\displaystyle\mathbb{E}\left|\int_{0}^{t}X^{\alpha}_{s}dB^{H}_{s}-\sum_{k=0}^{n-1}X^{\alpha}_{t_{k}}(B^{H}_{t_{k+1}}-B^{H}_{t_{k}})\right|^{2}$
$\displaystyle\leq\mathbb{E}\left[\left(\frac{1}{2^{1-(\lambda+\lambda^{\prime})}}\sum_{k=0}^{n-1}[X^{\alpha}]_{\lambda^{\prime};[t_{k},t_{k+1}]}[B^{H}]_{\lambda;[t_{k},t_{k+1}]}\right)^{2}\right]$
$\displaystyle\leq\mathbb{E}\left[\Lambda^{2}_{B^{H}}\Upsilon_{X}^{2}\frac{1}{2^{2-2(\lambda+\lambda^{\prime})}}\left(\sum_{k=0}^{n-1}|\Delta
t|^{\lambda+\lambda^{\prime}+2\varepsilon}\right)^{2}\right]\to 0,$

as $|\Delta t|\to 0$. It is now enough to note that each Riemann-Stieltjes sum
is in $L^{2}$ (thanks to the fact that
$\mathbb{E}[\sup_{t\in[0,T]}X^{r}_{t}]<\infty$ for all $r>0$), so the integral
$\int_{0}^{t}X^{\alpha}_{s}dB^{H}_{s}$ is indeed well-defined as the
$L^{2}$-limit of Riemann-Stieltjes integral sums.

Step 2. Now, we would like to get representation (4.6). In order to do that,
one should follow the proof of the Itô formula in a similar manner to
subsection 4.1. Namely, for a partition $0=t_{0}<t_{1}<t_{2}<...<t_{n}=t$ one
can write

$\displaystyle X_{t}$
$\displaystyle=X_{0}+\sum_{k=1}^{n}\left(Y^{\frac{1}{1-\alpha}}_{t_{k}}-Y^{\frac{1}{1-\alpha}}_{t_{k-1}}\right)$
$\displaystyle=X_{0}+\frac{1}{1-\alpha}\sum_{k=0}^{n-1}\left(Y^{\frac{\alpha}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k}-1})\right)+\frac{1}{2}\frac{\alpha}{(1-\alpha)^{2}}\sum_{k=0}^{n-1}\left(Y^{\frac{2\alpha-1}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k}-1})^{2}\right)$
$\displaystyle\quad+\frac{1}{6}\frac{\alpha(2\alpha-1)}{(1-\alpha)^{3}}\sum_{k=1}^{n}\left(\Theta_{k}^{\frac{3\alpha-2}{1-\alpha}}(Y_{t_{k}}-Y_{t_{k-1}})^{3}\right),$

where $\Theta_{k}$ is a value between $Y_{t_{k-1}}$ and $Y_{t_{k}}$.

Note that, using Theorems 2.5 and 2.6, it is easy to check that for any
$\lambda^{\prime}\in\left(\frac{1}{3},\frac{1}{2}\right)$ there exists a
random variable $\Upsilon_{Y}$ having moments of all orders such that

$|Y_{t}-Y_{s}|\leq\Upsilon_{Y}|t-s|^{\lambda^{\prime}}.$

Furthermore, by Theorem 2.5 (for $\alpha\in\left[\frac{3}{2},1\right)$) and
Theorem 2.6 (for $\alpha\in\left(\frac{1}{2},\frac{3}{2}\right)$), it is clear
that there exists a random variable $\Theta>0$ that does not depend on the
partition and has moments of all orders such that $\Theta_{k}<\Theta$, whence

$\sum_{k=1}^{n}\left(\Theta_{k}^{\frac{3\alpha-2}{1-\alpha}}(Y_{t_{k}}-Y_{t_{k-1}})^{3}\right)\leq\Theta^{\frac{3\alpha-2}{1-\alpha}}\Upsilon_{Y}^{3}\sum_{k=1}^{n}(t_{k}-t_{k-1})^{3\lambda^{\prime}}\xrightarrow{L^{2}}0,\quad|\Delta
t|\to 0.$

Using Step 1, it is also straightforward to verify that

$\displaystyle\frac{1}{1-\alpha}\sum_{k=0}^{n-1}\left(Y^{\frac{\alpha}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k}-1})\right)\xrightarrow{L^{2}}$
$\displaystyle\frac{1}{1-\alpha}\int_{0}^{t}\left(\kappa-\theta
X_{s}\right)ds+\frac{\nu_{1}}{1-\alpha}\int_{0}^{t}X_{s}^{\alpha}dB_{s}$
$\displaystyle\quad+\frac{\nu_{2}}{1-\lambda}\int_{0}^{t}X_{s}^{\alpha}dB^{H}_{s}-\frac{\alpha\nu_{1}^{2}}{2(1-\alpha)^{2}}\int_{0}^{t}Y^{\frac{2\alpha-1}{1-\alpha}}_{s}ds,\quad|\Delta
t|\to 0,$

and

$\displaystyle\frac{1}{2}\frac{\alpha}{(1-\alpha)^{2}}\sum_{k=0}^{n-1}\left(Y^{\frac{2\alpha-1}{1-\alpha}}_{t_{k-1}}(Y_{t_{k}}-Y_{t_{k}-1})^{2}\right)\xrightarrow{L^{2}}\frac{\alpha\nu_{1}^{2}}{2(1-\alpha)^{2}}\int_{0}^{t}Y^{\frac{2\alpha-1}{1-\alpha}}_{s}ds,\quad|\Delta
t|\to 0,$

which concludes the proof. ∎

##  5 Semi-heuristic Euler discretization scheme and simulations

In this section, we present simulated paths of the sandwiched process based on
a semi-heuristic approximation approach. One must note that it does not have
the virtue of giving sandwiched discretized process and has worse convergence
type in comparison to some alternative schemes (see, for example, [24, 38] for
the case of fractional Brownian motion), but, on the other hand, allows much
weaker assumptions on both the drift and the noise and is much simpler from
the implementation point of view.

###  5.1 Numerical scheme and convergence results

We first consider the setting of assumptions (A1)–(A4). Additionally, we
require local Hölder continuity of the drift $b$ with respect to $t$ in the
following sense:

  * (A5)

for any $\varepsilon>0$ there is $c_{\varepsilon}>0$ such that for any
$(t,y)$, $(s,y)\in\mathcal{D}_{\varepsilon}$:

$|b(t,y)-b(s,y)|\leq c_{\varepsilon}|t-s|^{\lambda}.$

Obviously, without loss of generality one can assume that the constant
$c_{\varepsilon}$ is the same for assumptions (A2) and (A5).

We stress that the drift $b$ is not globally Lipschitz and, furthermore, for
any $t\in[0,T]$, the value $b(t,y)$ is not defined for $y<\varphi(t)$. Hence
classical Euler approximations applied directly to the equation (0.3) fail
since such scheme does not guarantee that the discretized version of the
process stays above $\varphi$.

A straightforward way to overcome this issue is to discretize not the process
$Y$ itself, but its approximation $\tilde{Y}^{(n)}$ obtained by “leveling” the
singularity in the drift. Namely, let
$n_{0}>\max_{t\in[0,T]}|b(t,\varphi(t)+y_{*})|$, where $y_{*}$ is from
Assumption (A3). For an arbitrary $n\geq n_{0}$ define the set

$\begin{gathered}\mathcal{G}_{n}:=\\{(t,y)\in\mathcal{D}_{0}\setminus\mathcal{D}_{y_{*}}~{}|~{}b(t,y)<n\\}\end{gathered}$

and consider the functions $\tilde{b}_{n}$:
$[0,T]\times\mathbb{R}\to\mathbb{R}$ of the form

$\tilde{b}_{n}(t,y):=\begin{cases}b(t,y),&\quad(t,y)\in\mathcal{G}_{n}\cup\mathcal{D}_{y_{*}},\\\
n,&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus\left(\mathcal{G}_{n}\cup\mathcal{D}_{y_{*}}\right).\end{cases}$
(5.1)

Denote $\varepsilon_{n}:=\left(\frac{c}{n}\right)^{\frac{1}{\gamma}}$, and
observe that

$\begin{gathered}|\tilde{b}_{n}(t,y_{1})-\tilde{b}_{n}(t,y_{2})|\leq
c_{n}|y_{1}-y_{2}|,\quad t\in[0,T],\quad y_{1},y_{2}\in\mathbb{R},\\\
|\tilde{b}_{n}(t_{1},y)-\tilde{b}_{n}(t_{2},y)|\leq
c_{n}|t_{1}-t_{2}|^{\lambda},\quad t_{1},t_{2}\in[0,T],\quad
y\in\mathbb{R},\end{gathered}$ (5.2)

where $c_{n}:=c_{\varepsilon_{n}}$ denotes the constant from assumptions (A2)
and (A5) which corresponds to
$\varepsilon_{n}=\left(\frac{c}{n}\right)^{\frac{1}{\gamma}}$. In particular,
this implies that the SDE

$d\tilde{Y}^{(n)}_{t}=\tilde{b}_{n}(t,\tilde{Y}^{(n)}_{t})dt+dZ_{t},\quad\tilde{Y}^{(n)}_{0}=Y_{0}>\varphi(0),$
(5.3)

has a unique pathwise solution which can be approximated by the Euler scheme.

######  Remark 5.1.

In this section, by $C$ we will denote any positive constant that does not
depend on order of approximation $n$ or the the partition and the exact value
of which is not important. Note that $C$ may change from line to line (or even
within one line).

Regarding the process $\tilde{Y}^{(n)}$, we have the following result.

######  Proposition 5.2.

Let assumptions (A1)–(A4) hold. Then, for any $r>0$, there exists a constant
$C>0$ that does not depend on $n$ such that

$\max_{t\in[0,T]}|\tilde{Y}^{(n)}_{t}|^{r}\leq
C\left(1+\left(\max_{t\in[0,T]}|Z_{t}|\right)^{r}\right).$

###### Proof.

First, consider $n=n_{0}$. It is easy to see that there exists $C>0$ that does
not depend $n$ such that

$|b_{n_{0}}(t,y)|\leq C(1+|y|),$

therefore there exists $C>0$ such that

$\displaystyle|\tilde{Y}^{(n_{0})}_{t}|$ $\displaystyle\leq
Y_{0}+\int_{0}^{t}|b_{n_{0}}(s,\tilde{Y}^{(n_{0})}_{s})|ds+Z_{t}$
$\displaystyle\leq
C+\max_{t\in[0,T]}Z_{t}+\int_{0}^{t}|\tilde{Y}^{(n_{0})}_{s}|ds,$

hence, by Gronwall’s inequality,

$\max_{t\in[0,T]}|\tilde{Y}^{(n_{0})}_{t}|\leq C(1+\max_{t\in[0,T]}|Z_{t}|)$

for some constant $C>0$.

Next, the process $\tilde{Y}^{(n)}$ satisfies the SDE of the form

$d\tilde{Y}^{(n)}_{t}=b_{n}(t,\tilde{Y}^{(n)}_{t})dt+d\tilde{Z}_{t},$

where $b_{n}:=\tilde{b}_{n}-\frac{1}{n}$ and
$\tilde{Z}_{t}:=Z_{t}+\frac{t}{n}$ and thus, similarly to (A.7) from
Proposition A.2, one can verify that

$\displaystyle|\tilde{Y}^{(n)}_{t}|$ $\displaystyle\leq
C\left(1+\max_{t\in[0,T]}|\tilde{Y}^{(n_{0})}_{t}|+\max_{t\in[0,T]}|\tilde{Z}_{t}|\right)$
$\displaystyle\leq C(1+\max_{t\in[0,T]}|Z_{t}|),$

where $C>0$ is some constant that does not depend on $n$. ∎

Since

$|Z_{t}|=|Z_{t}-Z_{0}|\leq\Lambda t^{\lambda}\leq T^{\lambda}\Lambda,\quad
t\in[0,T],$

we immediately get the following corollary.

######  Corollary 5.1.

Under assumptions (A1)–(A4), there exists a constant $C>0$ that does not
depend on $n$ such that

$\max_{t\in[0,T]}\left|\tilde{Y}^{(n)}_{t}\right|^{r}\leq C(1+\Lambda^{r}).$

Before proceeding to the main theorem of the section, let us provide another
simple auxiliary proposition.

######  Proposition 5.3.

Let assumptions (A1)–(A4) hold. Assume also that the noise $Z$ satisfying
Assumptions (Z1)–(Z2) is such that

$\mathbb{E}\left[|Z(t)-Z(s)|^{p}\right]\leq C_{\lambda,p}|t-s|^{\lambda
p},\quad s,t\in[0,T],$

for some $p\geq 1$ such that
$\lambda_{p}:=\lambda-\frac{2}{p}>\frac{1}{1+\gamma}$ with $\gamma$ from
assumption (A3) and a positive constant $C_{\lambda,p}>0$. Then

$\mathbb{P}\left(\min_{t\in[0,T]}(Y(t)-\varphi(t))\leq\varepsilon\right)=O(\varepsilon^{\gamma\lambda_{p}+\lambda_{p}-1}),\quad\varepsilon\to
0.$

###### Proof.

By Lemma 1.1,

$|Z(t)-Z(s)|\leq
A_{\lambda,p}|t-s|^{\lambda-\frac{2}{p}}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z_{x}-Z_{y}|^{p}}{|x-y|^{\lambda
p}}dxdy\right)^{\frac{1}{p}},$

where

$A_{\lambda,p}=2^{3+\frac{2}{p}}\left(\frac{\lambda p}{\lambda p-2}\right).$

Note that the random variable

$\Lambda_{p}:=A_{\lambda,p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z_{x}-Z_{y}|^{p}}{|x-y|^{\lambda
p}}dxdy\right)^{\frac{1}{p}}$

is finite a.s. since

$\displaystyle\mathbb{E}\Lambda^{p}_{p}$
$\displaystyle=A^{p}_{\lambda,p}\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}|Z_{x}-Z_{y}|^{p}}{|x-y|^{\lambda
p}}dxdy$ $\displaystyle\leq T^{2}A^{p}_{\lambda,p}C_{\lambda,p}$
$\displaystyle<\infty.$

Now, by applying Theorem 2.6 and Remark 2.8 with respect to the Hölder order
$\lambda_{p}=\lambda-\frac{2}{p}$, one can deduce that for all $t\in[0,T]$

$Y(t)-\varphi(t)\geq\frac{1}{M_{3,p}(1,T)\tilde{\Lambda}_{p}^{\frac{1}{\gamma\lambda_{p}+\lambda_{p}-1}}},$

where

$M_{3,p}(1,T):=2^{\frac{\gamma\lambda_{p}}{\gamma\lambda_{p}+\lambda_{p}-1}}\beta^{\frac{1-\lambda_{p}}{\gamma\lambda_{p}+\lambda_{p}-1}}>0,$
$\beta_{p}:=\frac{\lambda_{p}^{\frac{\lambda_{p}}{1-\lambda_{p}}}-\lambda_{p}^{\frac{1}{1-\lambda_{p}}}}{c^{\frac{\lambda_{p}}{1-\lambda_{p}}}}>0$

and

$\tilde{\Lambda}_{p}:=\max\left\\{\Lambda_{p},K_{p},\left(2\beta_{p}\right)^{\lambda_{p}-1}\left(\frac{(Y_{0}-\varphi(0))\wedge
y_{*}}{2}\right)^{1-\lambda_{p}-\gamma\lambda_{p}}\right\\}$

with $y_{*}$, $c$ and $\gamma$ being from assumption (A3) and $K_{p}$ being
such that

$|\varphi(t)-\varphi(s)|\leq K_{p}|t-s|^{\lambda_{p}},\quad s,t\in[0,T].$

Therefore

$\displaystyle\mathbb{P}\left(\min_{t\in[0,T]}(Y(t)-\varphi(t))\leq\varepsilon\right)$
$\displaystyle\leq\mathbb{P}\left(\frac{1}{M_{3,p}(1,T)\tilde{\Lambda}_{p}^{\frac{1}{\gamma\lambda_{p}+\lambda_{p}-1}}}\leq\varepsilon\right)$
$\displaystyle=\mathbb{P}\left(\tilde{\Lambda}_{p}\geq\left(\frac{1}{M_{3,p}(1,T)\varepsilon}\right)^{\gamma\lambda_{p}+\lambda_{p}-1}\right)$
$\displaystyle\leq(M_{3,p}(1,T))^{\gamma\lambda_{p}+\lambda_{p}-1}\mathbb{E}[\tilde{\Lambda}_{p}]\varepsilon^{\gamma\lambda_{p}+\lambda_{p}-1}$
$\displaystyle=O(\varepsilon^{\gamma\lambda_{p}+\lambda_{p}-1}),\quad\varepsilon\to
0.$

∎

Finally, let $\Delta=\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ be a uniform partition
of $[0,T]$, $t_{k}=\frac{Tk}{N}$, $k=0,1,...,N$, $|\Delta|:=\frac{T}{N}$. For
the given partition, we introduce

$\begin{gathered}\tau_{-}(t):=\max\\{t_{k},~{}t_{k}\leq t\\},\\\
\kappa_{-}(t):=\max\\{k,~{}t_{k}\leq t\\},\\\
\tau_{+}(t):=\min\\{t_{k},~{}t_{k}\geq t\\},\\\
\kappa_{+}(t):=\min\\{k,~{}t_{k}\geq t\\}.\end{gathered}$ (5.4)

For any $n\geq n_{0}$, denote

$\hat{Y}^{N,n}_{t}:=Y_{0}+\int_{0}^{t}\tilde{b}_{n}\left(\tau_{-}(s),\hat{Y}^{N,n}_{\tau_{-}(s)}\right)ds+Z_{\tau_{-}(t)}.$
(5.5)

For each $n>n_{0}$, define also the function $y_{n}$:
$[0,T]\to\mathcal{D}_{0}$ by

$y_{n}(t):=\min\\{y>\varphi(t):~{}b(t,y)\leq n\\}$

and consider

$\delta_{n}:=\inf_{t\in[0,T]}(y_{n}(t)-\varphi(t)).$ (5.6)

######  Remark 5.4.

It is easy to see that
$\varepsilon_{n}:=\left(\frac{c}{n}\right)^{\frac{1}{\gamma}}\leq\delta_{n}$.
Moreover, $\delta_{n}\to 0$ as $n\to\infty$. Indeed, by the definition of
$y_{n}$, for any fixed $t\in[0,T]$ and $n>n_{0}$

$y_{n}(t)\geq y_{n+1}(t)$

and hence $\delta_{n}\geq\delta_{n+1}$. Now, consider an arbitrary
$\varepsilon\in(0,y_{*})$ and take
$n_{\varepsilon}:=[\max_{t\in[0,T]}b(t,\varphi(t)+\varepsilon)]$ with
$[\cdot]$ denoting the integer part. Then

$b(t,\varphi(t)+\varepsilon)<n_{\varepsilon}+1$

for all $t\in[0,T]$. On the other hand, by assumption (A3),

$b(t,\varphi(t)+\varepsilon^{\prime})\geq n_{\varepsilon}+1$

for all
$\varepsilon^{\prime}<\left(\frac{c}{n_{\varepsilon}+1}\right)^{\frac{1}{\gamma}}$
which, by assumption (A1), implies that for each $t\in[0,T]$

$y_{n_{\varepsilon}+1}(t)-\varphi(t)<\varepsilon,$

i.e. $\delta_{n_{\varepsilon}+1}<\varepsilon$. This, together with
$\delta_{n}$ being decreasing, yields that $\delta_{n}\to 0$ as $n\to\infty$.

######  Theorem 5.5.

Let assumptions (A1)–(A5) hold and the noise $Z$ satisfying Assumptions
(Z1)–(Z2) is such that

$\mathbb{E}\left[|Z(t)-Z(s)|^{p}\right]\leq C_{\lambda,p}|t-s|^{\lambda
p},\quad s,t\in[0,T],$

for some $p\geq 2$ such that
$\lambda_{p}:=\lambda-\frac{2}{p}>\frac{1}{1+\gamma}$ with $\gamma$ from
assumption (A3) and a positive constant $C_{\lambda,p}>0$. Then

$\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\hat{Y}^{N,n}_{t}\right|\right]\leq
C\left(\delta_{n}^{\frac{\gamma\lambda_{p}+\lambda_{p}-1}{2}}+\frac{(1+c_{n})e^{c_{n}}}{N^{\lambda_{p}}}\right),$

where $C$ is some positive constant that does not depend on $n$ or the mesh of
the partition $|\Delta|=\frac{T}{N}$, $\delta_{n}$ is defined by (5.6),
$\delta_{n}\to 0$, $n\to\infty$, and $c_{n}$ is from (5.2).

###### Proof.

Just like in the proof of Proposition 5.3, observe that

$|Z_{t}-Z_{s}|\leq\Lambda_{p}|t-s|^{\lambda_{p}},$

where

$\Lambda_{p}:=A_{\lambda,p}\left(\int_{0}^{T}\int_{0}^{T}\frac{|Z_{x}-Z_{y}|^{p}}{|x-y|^{\lambda
p}}dxdy\right)^{\frac{1}{p}},$

and note that the condition $p\geq 2$ implies that

$\mathbb{E}\Lambda_{p}^{2}\leq\left(\mathbb{E}\Lambda^{p}_{p}\right)^{\frac{2}{p}}<\infty.$

It is clear that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\hat{Y}^{N,n}_{t}\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\right]+\mathbb{E}\left[\sup_{t\in[0,T]}\left|\tilde{Y}^{(n)}_{t}-\hat{Y}^{N,n}_{t}\right|\right].$

Let is estimate both terms in the right-hand side of the inequality above
separately. Observe that

$b(t,y)=\tilde{b}_{n}(t,y),\quad(t,y)\in\mathcal{D}_{\delta_{n}}$

with $\delta_{n}$ defined by (5.6). Consider the set

$\mathcal{A}_{n}:=\\{\omega\in\Omega~{}|~{}\min_{t\in[0,T]}(Y_{t}(\omega)-\varphi(t))>\delta_{n}\\}$

and note that

$b(t,Y_{t})\mathbbm{1}_{\mathcal{A}_{n}}=b_{n}(t,Y_{t})\mathbbm{1}_{\mathcal{A}_{n}},$

i.e., for all $\omega\in\mathcal{A}_{n}$ the path $Y_{t}(\omega)$ satisfies
the equation (5.3) and thus coincides with $\tilde{Y}^{(n)}_{t}(\omega)$.
Whence

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\right]$
$\displaystyle=\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\mathbbm{1}_{\mathcal{A}_{n}}\right]+\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\mathbbm{1}_{\Omega\setminus\mathcal{A}_{n}}\right]$
$\displaystyle=\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\mathbbm{1}_{\Omega\setminus\mathcal{A}_{n}}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\right)^{2}\right]\right)^{\frac{1}{2}}\sqrt{\mathbb{P}\left(\min_{t\in[0,T]}(Y_{t}-\varphi(t))>\delta_{n}\right)}.$

By Theorem 2.5 and Corollary 5.1 applied w.r.t.
$\lambda_{p}=\lambda-\frac{2}{p}$,

$\displaystyle\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\right)^{2}\right]$
$\displaystyle\leq
C\left(\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|Y_{t}\right|\right)^{2}\right]+\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|\tilde{Y}^{(n)}_{t}\right|\right)^{2}\right]\right)$
$\displaystyle\leq C\left(1+\mathbb{E}\Lambda_{p}^{2}\right)<\infty,$

and, by Proposition 5.3, there exists a constant $C>0$ such that

$\sqrt{\mathbb{P}\left(\min_{t\in[0,T]}(Y_{t}(\omega)-\varphi(t))>\delta_{n}\right)}\leq
C\delta_{n}^{\frac{\gamma\lambda_{p}+\lambda_{p}-1}{2}}.$

Therefore, there exists a constant $C>0$ that does not depend on $n$ or $N$
such that

$\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\tilde{Y}^{(n)}_{t}\right|\right]\leq
C\delta_{n}^{\frac{\gamma\lambda_{p}+\lambda_{p}-1}{2}}.$ (5.7)

Next, taking into account (5.2), for any $t\in[0,T]$ we can write

$\displaystyle\left|\tilde{Y}^{(n)}_{t}-\hat{Y}^{N,n}_{t}\right|$
$\displaystyle\leq\int_{0}^{t}\left|\tilde{b}_{n}(s,\tilde{Y}^{(n)}_{s})-\tilde{b}_{n}\left(\tau_{-}(s),\tilde{Y}^{(n)}_{s}\right)\right|ds$
$\displaystyle\qquad+\int_{0}^{t}\left|\tilde{b}_{n}(\tau_{-}(s),\tilde{Y}^{(n)}_{s})-\tilde{b}_{n}\left(\tau_{-}(s),\hat{Y}^{N,n}_{\tau_{-}(s)}\right)\right|ds$
$\displaystyle\qquad+\Lambda_{p}|\Delta|^{\lambda_{p}}$ $\displaystyle\leq
c_{n}|\Delta|^{\lambda_{p}}+c_{n}\int_{0}^{t}\left|\tilde{Y}^{(n)}_{s}-\hat{Y}^{N,n}_{s}\right|ds+\Lambda_{p}|\Delta|^{\lambda_{p}},$

whence, since $\mathbb{E}\Lambda_{p}<\infty$,

$\mathbb{E}\left[\sup_{s\in[0,t]}\left|\tilde{Y}^{(n)}_{t}-\hat{Y}^{N,n}_{t}\right|\right]\leq
c_{n}|\Delta|^{\lambda_{p}}+c_{n}\int_{0}^{t}\mathbb{E}\left[\sup_{u\in[0,s]}\left|\tilde{Y}^{(n)}_{u}-\hat{Y}^{N,n}_{u}\right|\right]ds+C|\Delta|^{\lambda_{p}},$

and, by Gronwall’s inequality, there exists constant $C>0$ such that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|\tilde{Y}^{(n)}_{t}-\hat{Y}^{N,n}_{t}\right|\right]\leq\frac{C(1+c_{n})e^{c_{n}}}{N^{\lambda_{p}}}.$

This, together with (5.7), finalizes the proof. ∎

######  Remark 5.6.

  1. 1.

As it is stated in [5], the condition

$\mathbb{E}\left[|Z(t)-Z(s)|^{p}\right]\leq C_{\lambda,p}|t-s|^{\lambda
p},\quad s,t\in[0,T],$

from Theorem 5.5 is satisfied by any $\lambda$-Hölder continuous Gaussian
process $Z$, i.e. processes from Examples 1.2–1.5 fit into this framework.

  2. 2.

The process from Example 1.6 also satisfies conditions of Theorem 5.5 as shown
by (1.6).

The sandwiched case presented in section 3 can be treated in the same manner.
Instead of assumption (A5), one should use the following one:

  * (B5)

for any $\varepsilon_{1},\varepsilon_{2}>0$,
$\varepsilon_{1}+\varepsilon_{2}\leq\lVert\varphi-\psi\rVert_{\infty}$, there
is a constant $c_{\varepsilon_{1},\varepsilon_{2}}>0$ such that for any
$(t,y)$, $(s,y)\in\mathcal{D}_{\varepsilon_{1},\varepsilon_{2}}$:

$|b(t,y)-b(s,y)|\leq c_{\varepsilon_{1},\varepsilon_{2}}|t-s|^{\lambda},$

where $\mathcal{D}_{\varepsilon_{1},\varepsilon_{2}}$ is defined by (1.7).
Namely, let

$n_{0}>\max\left\\{\max_{t\in[0,T]}|b(t,\varphi(t)+y_{*})|,\max_{t\in[0,T]}|b(t,\psi(t)-y_{*})|\right\\},$

where $y_{*}$ is from Assumption (B3). For an arbitrary $n\geq n_{0}$ define

$\begin{gathered}\mathcal{G}^{\varphi}_{n}:=\\{(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{y_{*},0}~{}|~{}b(t,y)<n\\}\\\
\mathcal{G}^{\psi}_{n}:=\\{(t,y)\in\mathcal{D}_{0,0}\setminus\mathcal{D}_{0,y_{*}}~{}|~{}b(t,y)>-n\\}\end{gathered}$

and consider the functions $\tilde{b}_{n}$:
$[0,T]\times\mathbb{R}\to\mathbb{R}$ of the form

$\tilde{b}_{n}(t,y):=\begin{cases}b(t,y),&\quad(t,y)\in\mathcal{G}^{\varphi}_{n}\cup\mathcal{G}^{\psi}_{n}\cup\mathcal{D}_{y_{*},y_{*}},\\\
n,&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus\left(\mathcal{G}^{\varphi}_{n}\cup\mathcal{D}_{y_{*},-\infty}\right),\\\
-n,&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus\left(\mathcal{G}^{\psi}_{n}\cup\mathcal{D}_{,-\infty,y_{*}}\right),\end{cases}$
(5.8)

where

$\mathcal{D}_{y_{*},-\infty}:=\\{(t,y)~{}|~{}t\in[0,T],y\in(\varphi(t)+y_{*},\infty)\\}$

and

$\mathcal{D}_{-\infty,y_{*}}:=\\{(t,y)~{}|~{}t\in[0,T],y\in(-\infty,\psi(t)-y_{*})\\}.$

Denote $\varepsilon_{n}:=\left(\frac{c}{n}\right)^{\frac{1}{\gamma}}$, and
observe that

$\begin{gathered}|\tilde{b}_{n}(t,y_{1})-\tilde{b}_{n}(t,y_{2})|\leq
c_{n}|y_{1}-y_{2}|,\quad t\in[0,T],\quad y_{1},y_{2}\in\mathbb{R},\\\
|\tilde{b}_{n}(t_{1},y)-\tilde{b}_{n}(t_{2},y)|\leq
c_{n}|t_{1}-t_{2}|^{\lambda},\quad t_{1},t_{2}\in[0,T],\quad
y\in\mathbb{R},\end{gathered}$ (5.9)

where $c_{n}:=c_{\varepsilon_{n},\varepsilon_{n}}$ denotes the constant from
assumptions (B2) and (B5) which corresponds to
$\varepsilon_{n}=\left(\frac{c}{n}\right)^{\frac{1}{\gamma}}$. In particular,
this implies that the SDE

$d\tilde{Y}^{(n)}_{t}=\tilde{b}_{n}(t,\tilde{Y}^{(n)}_{t})dt+dZ_{t},\quad\tilde{Y}^{(n)}_{0}=Y_{0}>\varphi(0),$
(5.10)

has a unique pathwise solution and, just like in the one-sided case, can be
simulated via the standard Euler scheme:

$\hat{Y}^{N,n}_{t}:=Y_{0}+\int_{0}^{t}\tilde{b}_{n}\left(\tau_{-}(s),\hat{Y}^{N,n}_{\tau_{-}(s)}\right)ds+Z_{\tau_{-}(t)}.$
(5.11)

Now, for each $n\geq n_{0}$ denote

$y^{\varphi}_{n}(t):=\min\\{y\in(\varphi(t),\psi(t)):~{}b(t,y)\leq n\\},\quad
y^{\psi}_{n}(t):=\max\\{y\in(\varphi(t),\psi(t)):~{}b(t,y)\geq-n\\}$

and define

$\delta_{n}:=\max\left\\{\inf_{t\in[0,T]}(y^{\varphi}_{n}(t)-\varphi(t)),\inf_{t\in[0,T]}(\psi(t)-y^{\psi}_{n}(t))\right\\}.$
(5.12)

Similarly to the one-seded case, it is straightforward to prove that
$\delta_{n}\to 0$, $n\to\infty$.

Now we are ready to formulate the two-sided counterpart of Theorem 5.5.

######  Theorem 5.7.

Let assumptions (B1)–(B5) hold and the noise $Z$ satisfying Assumptions
(Z1)–(Z2) is such that

$\mathbb{E}\left[|Z(t)-Z(s)|^{p}\right]\leq C_{\lambda,p}|t-s|^{\lambda
p},\quad s,t\in[0,T],$

for some $p\geq 2$ such that
$\lambda_{p}:=\lambda-\frac{2}{p}>\frac{1}{1+\gamma}$ with $\gamma$ from
assumption (B3) and a positive constant $C_{\lambda,p}>0$. Then

$\mathbb{E}\left[\sup_{t\in[0,T]}\left|Y_{t}-\hat{Y}^{N,n}_{t}\right|\right]\leq
C\left(\delta_{n}^{\frac{\gamma\lambda_{p}+\lambda_{p}-1}{2}}+\frac{(1+c_{n})e^{c_{n}}}{N^{\lambda_{p}}}\right),$

where $C$ is some positive constant that does not depend on $n$ or the mesh of
the partition $|\Delta|=\frac{T}{N}$, $\delta_{n}$ is defined by (5.12),
$\delta_{n}\to 0$, $n\to\infty$, and $c_{n}$ is from (5.9).

######  Remark 5.8.

Theorems 5.5 and 5.7 guarantee convergence for all $\lambda\in(0,1)$, but in
practice the scheme performs much better for $\lambda$ close to 1. The reason
is as follows: in order to make
$\delta_{n}^{\frac{\gamma\lambda_{p}+\lambda_{p}-1}{2}}$ small, one has to
consider large values of $n$; this results in larger values of
$(1+c_{n})e^{c_{n}}$ that, in turn, have to be “compensated” by the
denominator $N^{\lambda_{p}}$. The bigger is $\lambda_{p}$, the smaller values
of $n$ (and hence of $N$) can be.

###  5.2 Simulations

To conclude the work, we illustrate the results presented in this paper using
the semi-heuristic Euler approximation scheme considered previously. Note that
the scheme does not guarantee that the discretized process remains between
$\varphi$ and $\psi$, but in practice the property of being sandwiched is not
violated to a big extent (see below). All the simulations are performed in the
R programming language on the system with Intel Core i9-9900K CPU and 64 Gb
RAM. In order to simulate paths of fractional Brownian motion, R package
somebm is used.

####  5.2.1 Simulation 1: square root of fractional Cox–Ingersoll–Ross process

As the first example, consider a particular example of the process described
in subsection 4.1, namely the square root of the fractional Cox–Ingersoll-Ross
process:

$Y_{t}=Y_{0}+\frac{1}{2}\int_{0}^{t}\left(\frac{\kappa}{Y_{s}}-\theta
Y_{s}\right)ds+\frac{\sigma}{2}B^{H}_{t},\quad t\in[0,T]$ (5.13)

where $Y_{0}$, $\kappa$, $\theta$ and $\sigma$ are positive constants and
$B^{H}$ is a fractional Brownian motion with Hurst index $H>\frac{1}{2}$. Note
that such process is a convenient subject for testing our approximation scheme
since [24, Theorem 4.1] gives an alternative backward Euler approximation
method for it and one can use that algorithm as a reference for performance
evaluation.

In our simulations, we take $N=100000$, $T=1$, $Y_{0}=1$, $\kappa=3$,
$\theta=1$, $\sigma=1$, $H=0.7$ (these values of parameters satisfy conditions
of [24, Theorem 4.1]). For the semi-heuristic Euler scheme, we take $n=20$. We
generated 1000 paths of fractional Brownian motion and used them to simulate
the process (5.13) with the semi-heuristic Euler approximation method (10
trajectories are given on Fig. 1) and with the algorithm from [24]. On
average, simulation time of one path using the semi-heuristic Euler scheme was
slightly slower in comparison to the backward Euler scheme (0.08577108 seconds
versus 0.03191614 seconds).

Figure 1: Ten sample paths of (5.13) generated using the semi-heuristic Euler
approximation scheme; $N=100000$, $T=1$, $Y_{0}=1$, $\kappa=3$, $\theta=1$,
$\sigma=1$, $H=0.7$, $n=20$.

Afterwards, we calculated

$\sup_{k=0,...,N}|\hat{Y}^{N,n}_{t_{k}}-\hat{Y}_{t_{k}}|$ (5.14)

for the corresponding generated paths ($\hat{Y}$ above denotes the backward
Euler scheme from [24]). The boxplot of values (5.14) based on 1000
simulations is given on Fig. 2(a). As we can see, both schemes give almost
identical results with the value (5.14) taking values roughly between
$10^{-6}$ and $10^{-5}$. In fact, paths generated with these two methods are
indistinguishable on the plots (see Fig. 2(b)) despite worse convergence rate
of the semi-heuristic scheme provided by Theorem 5.5. Moreover, all 1000 paths
of (5.13) simulated with the semi-heuristic method stay above 0.

(a)

(b)

Figure 2:  Comparison of two simulation methods. (a) Boxplot of (5.14) based
on 1000 simulations. (b) Path of (5.13) generated by the semi-heuristic
(black) and backward (red) Euler methods. Note that both trajectories
completely coincide and therefore the black path is not visible on the
picture.

####  5.2.2 Simulation 2: two-sided sandwiched process with equidistant bounds

As the second example, we take

$Y_{t}=2.5+\int_{0}^{t}\left(\frac{1}{(Y_{s}-\cos(5s))^{4}}-\frac{1}{(3+\cos(5s)-Y_{s})^{4}}\right)ds+3B^{H}_{t},\quad
t\in[0,1],$ (5.15)

with

$\psi(t)-\varphi(t)=3+\cos(5t)-\cos(5t)=3,\quad t\in[0,1].$

In all simulations, $N=100000$, $n=20$. Simulation of one trajectory takes
approximately 0.1601701 seconds. All 1000 generated paths (10 of simulated
paths are presented on Fig. 3) stay between the bounds without crossing them.

Figure 3: Ten sample paths of (5.15) generated using the semi-heuristic Euler
approximation scheme, $N=100000$, $n=20$.

####  5.2.3 Simulation 3: two-sided sandwiched process with shrinking bounds

As the final example, we consider

$Y_{t}=\int_{0}^{t}\left(\frac{1}{(Y_{s}+e^{-s})^{4}}-\frac{1}{(e^{-s}-Y_{s})^{4}}\right)ds+B^{H}_{t},\quad
t\in[0,1],$ (5.16)

with

$\psi(t)-\varphi(t)=2e^{-t}\to 0,\quad t\to\infty.$

Once again, $N=100000$, $n=20$. Simulation of one trajectory takes
approximately 0.1575801 seconds and all 1000 generated paths (10 of simulated
paths are presented on Fig. 4) stay between the bounds without crossing them.

Figure 4: Ten sample paths of (5.16) generated using the semi-heuristic Euler
approximation scheme, $N=100000$, $n=20$.

## Acknowledgements

The present research is carried out within the frame and support of the
ToppForsk project nr. 274410 of the Research Council of Norway with title
STORM: Stochastics for Time-Space Risk Models. The second author is supported
by the Ukrainian research project ”Exact formulae, estimates, asymptotic
properties and statistical analysis of complex evolutionary systems with many
degrees of freedom” (state registration number 0119U100317).

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##  Appendix A Appendix: Existence of the local solution

In this Appendix, we give a proof of Theorem 2.2 on the existence of the
solution to (0.3) under assumptions (A1)–(A3) until the first moment of
hitting $\varphi$ by the latter. Note that it would be possible to prove this
result using a modification of the standard Picard iteration argument, but we
choose a different strategy: we approximate the non-Lipschitz drift of (0.3)
by a sequence of the Lipschitz ones, obtain a monotonically increasing
sequence of processes and prove that their limit is the only solution. Choice
of such a method is explained by two points. First, without assumption (A4),
the solution may hit $\varphi$ and the limiting procedure described in this
Appendix allows to see (up to some extent) what happens beyond this moment.
Second, the pre-limit processes are very easy to simulate, so they can be used
for numerical schemes.

Before going to the proof of Theorem 2.2, we will require several auxiliary
results. Let $n_{0}>\max_{t\in[0,T]}|b(t,\varphi(t)+y_{*})|$, where $y_{*}$ is
from Assumption (A3). For an arbitrary $n\geq n_{0}$ define the set

$\begin{gathered}\mathcal{G}_{n}:=\\{(t,y)\in\mathcal{D}_{0}\setminus\mathcal{D}_{y_{*}}~{}|~{}b(t,y)<n\\}\end{gathered}$

and consider the functions $\tilde{b}_{n}$:
$[0,T]\times\mathbb{R}\to\mathbb{R}$ of the form

$\tilde{b}_{n}(t,y):=\begin{cases}b(t,y),&\quad(t,y)\in\mathcal{G}_{n}\cup\mathcal{D}_{y_{*}},\\\
n,&\quad(t,y)\in[0,T]\times\mathbb{R}\setminus\left(\mathcal{G}_{n}\cup\mathcal{D}_{y_{*}}\right),\end{cases}$
(A.1)

$b_{n}(t,y):=\tilde{b}_{n}(t,y)-\frac{1}{n}$.

Note that each $b_{n}$ is Lipschitz continuous, i.e. for all
$(t,y_{1}),(t,y_{2})\in[0,T]\times\mathbb{R}$ there exists the constant $C$
that depends on $n$ but does not depend on $t$ such that

$|b_{n}(t,y_{1})-b_{n}(t,y_{2})|\leq C|y_{1}-y_{2}|.$

Using this fact, it is straightforward to prove by the standard fixed point
argument that the stochastic differential equation of the form

$dY^{(n)}_{t}=b_{n}(t,Y^{(n)}_{t})dt+dZ_{t},\quad Y^{(n)}_{0}=Y_{0}>0,$ (A.2)

has a pathwisely unique solution.

In order to progress, we will require a simple comparison-type result.

######  Lemma A.1.

Assume that continuous random processes $\\{X_{1}(t),~{}t\geq 0\\}$ and
$\\{X_{2}(t),t\geq 0\\}$ satisfy (a.s.) the equations of the form

$X_{i}(t)=X_{0}+\int_{0}^{t}f_{i}(s,X_{i}(s))ds+Z_{t},\quad t\geq 0,\quad
i=1,2,$

where $X_{0}$ is a constant and $f_{1}$, $f_{2}$:
$[0,\infty)\times\mathbb{R}\to\mathbb{R}$ are continuous functions such that
for any $(t,x)\in[0,\infty)\times\mathbb{R}$:

$f_{1}(t,x)<f_{2}(t,x).$

Then $X_{1}(t)<X_{2}(t)$ a.s. for any $t>0$.

###### Proof.

The proof is straightforward. Denote

$\Delta(t):=X_{2}(t)-X_{1}(t)=\int_{0}^{t}\left(f_{2}(s,X_{2}(s))-f_{1}(s,X_{1}(s))\right)ds,\quad
t\geq 0,$

and observe that $\Delta(0)=0$ and that the function $\Delta$ is
differentiable with

$\Delta^{\prime}_{+}(0)=f_{2}(0,X_{0})-f_{1}(0,X_{0})>0.$

It is clear that $\Delta(t)=\Delta^{\prime}_{+}(0)t+o(t)$, $t\to 0+$, whence
there exists the maximal interval $(0,t^{*})\subset(0,\infty)$ such that
$\Delta(t)>0$ for all $t\in(0,t^{*})$. It is also clear that

$t^{*}=\sup\\{t>0~{}|~{}\forall s\in(0,t):\Delta(s)>0\\}.$

Assume that $t^{*}<\infty$. By the definition of $t^{*}$ and continuity of
$\Delta$, $\Delta(t^{*})=0$. Hence $X_{1}(t^{*})=X_{2}(t^{*})=X^{*}$ and

$\Delta^{\prime}(t^{*})=f_{2}(t^{*},X^{*})-f_{1}(t^{*},X^{*})>0.$

As $\Delta(t)=\Delta^{\prime}(t^{*})(t-t^{*})+o(t-t^{*})$, $t\to t^{*}$, there
exists such $\varepsilon>0$ that $\Delta(t)<0$ for all
$t\in(t^{*}-\varepsilon,t^{*})$ which contradicts the definition of $t^{*}$.
Therefore $t^{*}=\infty$ and for all $t>0$:

$X_{1}(t)<X_{2}(t).$

∎

It is easy to observe that $b_{n}(t,y)<b_{n+1}(t,y)$ for any $n\geq 1$ and
$(t,y)\in[0,T]\times\mathbb{R}$, whence $Y^{(n)}_{t}<Y^{(n+1)}_{t}$ for all
$t\in(0,T]$ and therefore one can define a limit
$Y^{(\infty)}_{t}:=\lim_{n\to\infty}Y^{(n)}_{t}\in(-\infty,\infty]$,
$t\in[0,T]$.

######  Proposition A.2.

Let assumptions (A1)–(A3) hold. Then, there is a random variable
$\Psi>\max_{t\in[0,T]}|\varphi(t)|$ such that for any $t\in[0,T]$:

$|Y^{(\infty)}_{t}|\leq\Psi<\infty.$

###### Proof.

Denote $\eta:=\frac{Y_{0}-\varphi(0)}{2}$ and consider

$\displaystyle\tau^{n}_{1}:$
$\displaystyle=\sup\left\\{s\in[0,T]~{}\bigg{|}~{}\forall
u\in[0,s]:~{}Y^{(n)}_{u}\geq\varphi(u)+\eta\right\\}$
$\displaystyle=\inf\left\\{s\in[0,T]~{}\bigg{|}~{}Y^{(n)}_{s}<\varphi(s)+\eta\right\\}\wedge
T.$

We shall first prove that for all $n\geq n_{0}$:

$\displaystyle|Y^{(n)}_{t}|$
$\displaystyle\leq|Y_{0}|+2\max_{s\in[0,T]}|Y^{(n_{0})}_{s}|+5\max_{s\in[0,T]}|\varphi(s)|+\eta$
$\displaystyle\quad+Ct+C\int_{0}^{t}|Y^{(n)}_{s}|ds+2\max_{s\in[0,T]}|Z_{s}|,$

with $C>0$ being a constant that does not depend on $n$. Then the required
result follows by Gronwall’s inequality.

For the reader’s convenience, we will divide the proof into several steps to
separate cases $t\in[0,\tau^{n}_{1}]$ and $t\in(\tau^{n}_{1},T]$.
Step 1. Fix an arbitrary $n\geq n_{0}$ and assume that $t\in[0,\tau^{n}_{1}]$,
i.e. $Y^{(n)}_{s}\geq\varphi(s)+\eta$ for each $s\leq t$. Observe that for all
$(s,y)\in\mathcal{D}_{\eta}$

$|b_{n}(s,y)|\leq C(1+|y|),$ (A.3)

where $C>0$ is some constant that depends neither on $n$ nor on $s$. Indeed,
it is easy to verify using definition of $b_{n}$ and assumption (A3) that for
all $(s,y)\in\mathcal{D}_{\eta}$

$|b_{n}(s,y)|\leq\left|b_{n}(s,y)+\frac{1}{n}\right|+1\leq|b(s,y)|+1.$

Furthermore, by assumption (A2), for all $(s,y)\in\mathcal{D}_{\eta}$

$\displaystyle|b(s,y)|$
$\displaystyle\leq\left|b(s,y)-b\left(s,\varphi(s)+\eta\right)\right|+\left|b\left(s,\varphi(s)+\eta\right)\right|$
$\displaystyle\leq
c_{\eta}(y-\varphi(s)-\eta)+\max_{s\in[0,T]}\left|b\left(s,\varphi(s)+\eta\right)\right|$
$\displaystyle\leq\left(\max_{s\in[0,T]}|b\left(s,\varphi(s)+\eta\right)|+c_{\eta}\max_{s\in[0,T]}|\varphi(s)|+c_{\eta}(\eta+1)\right)(1+|y|).$

Using (A.3), for an arbitrary $n\geq n_{0}$:

$\displaystyle\left|Y^{(n)}_{t}\right|$
$\displaystyle=\left|Y_{0}+\int_{0}^{t}b_{n}(s,Y^{(n)}_{s})ds+Z_{t}\right|$
(A.4) $\displaystyle\leq|Y_{0}|+\int_{0}^{t}|b_{n}(s,Y^{(n)}_{s})|ds+|Z_{t}|$
$\displaystyle\leq|Y_{0}|+Ct+C\int_{0}^{t}|Y^{(n)}_{s}|ds+\max_{s\in[0,T]}|Z_{s}|.$

Step 2. Assume $t>\tau^{n}_{1}$. Consider

$\tau_{2}^{n}(t):=\sup\left\\{s\in(\tau^{n}_{1},t]~{}\bigg{|}~{}|Y^{(n)}_{s}-\varphi(s)|<\eta\right\\}.$

Note that
$\left|Y^{(n)}_{{\tau^{n}_{2}(t)}}\right|\leq|\varphi\left(\tau_{2}^{n}(t)\right)|+\eta\leq\max_{s\in[0,T]}|\varphi(s)|+\eta$
and, therefore,

$\displaystyle|Y^{(n)}_{t}|$
$\displaystyle=\left|\left(Y^{(n)}_{t}-Y^{(n)}_{\tau^{n}_{2}(t)}\right)+Y^{(n)}_{\tau^{n}_{2}(t)}\right|$
$\displaystyle\leq\left|Y^{(n)}_{t}-Y^{(n)}_{\tau^{n}_{2}(t)}\right|+\max_{s\in[0,T]}|\varphi(s)|+\eta.$

If $\tau^{n}_{2}(t)=t$, then $|Y^{(n)}_{t}-Y^{(n)}_{\tau^{n}_{2}(t)}|=0$, so
$|Y^{(n)}_{t}|<\max_{s\in[0,T]}|\varphi(s)|+\eta$. Otherwise, if
$\tau^{n}_{2}(t)<t$, then, for any $s\in[\tau^{n}_{2}(t),t]$:
$|Y^{(n)}_{s}-\varphi(s)|\geq\eta$ which means that either
$Y^{(n)}_{s}\leq\varphi(s)-\eta$ or $Y^{(n)}_{s}\geq\varphi(s)+\eta$ for all
$s\in[\tau^{n}_{2}(t),t]$. In the first case, taking into account the
monotonicity of $Y^{(n)}_{s}$ with respect to $n$, we have

$Y^{(n_{0})}_{s}-\varphi(s)\leq Y^{(n)}_{s}-\varphi(s)\leq-\eta,$

i.e.

$\eta\leq|Y^{(n)}_{s}-\varphi(s)|\leq|Y^{(n_{0})}_{s}-\varphi(s)|,$

so

$\displaystyle|Y^{(n)}_{t}-Y^{(n)}_{\tau^{n}_{2}(t)}|$
$\displaystyle\leq|Y^{(n)}_{t}-\varphi(t)|+|Y^{(n)}_{\tau^{n}_{2}(t)}-\varphi({\tau^{n}_{2}(t)})|+|\varphi(t)-\varphi(\tau^{n}_{2}(t))|$
(A.5)
$\displaystyle\leq|Y^{(n_{0})}_{t}-\varphi(t)|+|Y^{(n_{0})}_{\tau^{n}_{2}(t)}-\varphi({\tau^{n}_{2}(t)})|+|\varphi(t)-\varphi(\tau^{n}_{2}(t))|$
$\displaystyle\leq
2\max_{s\in[0,T]}|Y^{(n_{0})}_{t}|+4\max_{s\in[0,T]}|\varphi(s)|.$

In the second case, since $(s,Y^{(n)}_{s})\in\mathcal{D}_{\eta}$, we can use
(A.3) to obtain that

$\displaystyle|Y^{(n)}_{t}-Y^{(n)}_{\tau^{n}_{2}(t)}|$
$\displaystyle=\bigg{|}\int_{\tau^{n}_{2}(t)}^{t}b(s,Y^{(n)}_{s})ds+(Z_{t}-Z_{\tau^{n}_{2}(t)})\bigg{|}$
$\displaystyle\leq
C(t-\tau^{n}_{2}(t))+C\int_{\tau^{n}_{2}(t)}^{t}|Y^{(n)}_{s}|ds+2\max_{s\in[0,T]}|Z_{s}|$
$\displaystyle\leq Ct+C\int_{0}^{t}|Y^{(n)}_{s}|ds+2\max_{s\in[0,T]}|Z_{s}|.$

In any situation, for all $t>\tau^{n}_{1}$:

$\displaystyle|Y^{(n)}_{t}|$ $\displaystyle\leq
2\max_{s\in[0,T]}|Y^{(n_{0})}_{s}|+5\max_{s\in[0,T]}|\varphi(s)|+\eta$ (A.6)
$\displaystyle\quad+Ct+C\int_{0}^{t}|Y^{(n)}_{s}|ds+2\max_{s\in[0,T]}|Z_{s}|.$

Step 3. Taking into account (A.4) and (A.6), it is easy to see that for all
$t\geq 0$:

$\displaystyle|Y^{(n)}_{t}|$
$\displaystyle\leq|Y_{0}|+2\max_{s\in[0,T]}|Y^{(n_{0})}_{s}|+5\max_{s\in[0,T]}|\varphi(s)|+\eta$
$\displaystyle\quad+Ct+C\int_{0}^{t}|Y^{(n)}_{s}|ds+2\max_{s\in[0,T]}|Z_{s}|,$

so, by Gronwall’s inequality, for all $n\geq 1$:

$\displaystyle|Y^{(n)}_{t}|$ $\displaystyle\leq\Psi<\infty,$ (A.7)

where

$\Psi:=\left(|Y_{0}|+2\max_{s\in[0,T]}|Y^{(n_{0})}_{s}|+5\max_{s\in[0,T]}|\varphi(s)|+\eta+CT+2\max_{s\in[0,T]}|Z_{s}|\right)e^{CT}.$

Since the right-hand side of (A.7) does not depend on $n$, the claim of the
proposition holds for $Y^{(\infty)}$. ∎

######  Proposition A.3.

For all $t\in[0,T]$: $Y^{(\infty)}_{t}\geq\varphi(t)$.

###### Proof.

Step 1. Fix an arbitrary $t\in[0,T]$ and denote

$b_{n}^{+}(s,y):=b_{n}(s,y)\vee 0,\qquad b_{n}^{-}(s,y):=-(b_{n}(s,y)\wedge
0),$

$b_{n}(s,y)=b_{n}^{+}(s,y)-b_{n}^{-}(s,y)$.
Observe that, by assumption (A3), $b_{n}^{-}(s,y)=0$ for all
$(s,y)\in\mathcal{D}_{0}\setminus\mathcal{D}_{y_{*}}$, and, by assumption
(A2), $b_{n}^{-}$ is globally Lipschitz continuous. From Proposition A.2 we
obtain that, for some constant $L>0$ that does not depend on $n$ and for all
$s\in[0,t]$:

$\displaystyle|b_{n}^{-}(s,Y^{(n)}_{s})|$ $\displaystyle\leq
L(1+|Y^{(n)}_{s}|)\leq L(1+\Psi)=:\widetilde{\Psi},$

where $\widetilde{\Psi}$ is a finite random variable. Hence, since
$b^{-}_{n}(s,Y^{(n)}_{s})\to b^{-}(s,Y^{(\infty)}_{s})$ pointwise as
$n\to\infty$, by the dominated convergence theorem,

$\int_{0}^{t}b^{-}_{n}(s,Y^{(n)}_{s})ds\to\int_{0}^{t}b^{-}(s,Y^{(\infty)}_{s})ds,\quad
n\to\infty.$

Taking into account the convergence above and Proposition A.2, the left hand
side of

$Y^{(n)}_{t}-Y_{0}-Z_{t}+\int_{0}^{t}b^{-}_{n}(s,Y^{(n)}_{s})ds=\int_{0}^{t}b^{+}_{n}(s,Y^{(n)}_{s})ds$

converges to a finite value as $n\to\infty$ for each $t\in[0,T]$. Therefore
there exists the limit

$\lim_{n\to\infty}\int_{0}^{t}b^{+}_{n}(s,Y^{(n)}_{s})ds<\infty.$ (A.8)

Step 2. Let us now prove that

$\mu\\{s\in[0,T]~{}|~{}Y^{(n)}_{s}\leq\varphi(s)\\}\to 0,\quad n\to\infty,$

with $\mu$ being the Lebesgue measure on $[0,T]$. Assume that it is not true.
i.e. there exist $\epsilon>0$ and a subsequence $\\{n_{k}:~{}k\geq 1\\}$ such
that for all $k\geq 1$:

$\mu\\{s\in[0,T]~{}|~{}Y^{(n_{k})}_{s}\leq\varphi(s)\\}\geq\epsilon>0.$

In this case,

$\displaystyle\int_{0}^{T}b^{+}_{n_{k}}(s,Y^{(n_{k})}_{s})ds$
$\displaystyle=\int_{\\{s\in[0,T]~{}|~{}Y^{(n_{k})}_{s}>\varphi(s)\\}}b^{+}_{n_{k}}(s,Y^{(n_{k})}_{s})ds$
$\displaystyle\quad+\int_{\\{s\in[0,T]~{}|~{}Y^{(n_{k})}_{s}\leq\varphi(s)\\}}b^{+}_{n_{k}}(s,Y^{(n_{k})}_{s})ds$
$\displaystyle\geq\int_{\\{s\in[0,T]~{}|~{}Y^{(n_{k})}_{s}\leq\varphi(s)\\}}b^{+}_{n_{k}}(s,Y^{(n_{k})}_{s})ds$
$\displaystyle=\int_{\\{s\in[0,T]~{}|~{}Y^{(n_{k})}_{s}\leq\varphi(s)\\}}\left(n_{k}-\frac{1}{n_{k}}\right)ds$
$\displaystyle\geq n_{k}\epsilon-\frac{\epsilon}{n_{k}}\to\infty,\quad
k\to\infty,$

that contradicts (A.8).
This implies that $\mu\\{s\in[0,T]~{}|~{}Y^{(\infty)}_{s}\leq\varphi(s)\\}=0$,
i.e. $Y^{(\infty)}$ exceeds $\varphi$ a.e. on $[0,T]$.
Step 3. Assume that there is such $\tau\in(0,T]$ that
$Y^{(\infty)}_{\tau}<\varphi(\tau)$. Then, for all $n\geq 1$:

$Y^{(n)}_{\tau}<Y^{(\infty)}_{\tau}\leq\varphi(\tau).$

Fix an arbitrary $n\geq n_{0}$ and denote

$\begin{gathered}\tau_{-}^{n}:=\sup\\{t\in[0,\tau)~{}|~{}Y^{(n)}_{t}>\varphi(t)\\}.\end{gathered}$

Note that, due to continuity of $Y^{(n)}$ and Step 2, $0<\tau_{-}^{n}<\tau\leq
T$. Furthermore, $Y^{(n)}_{\tau_{-}^{n}}-\varphi(\tau_{-}^{n})=0$ and for all
$t\in(\tau_{-}^{n},\tau]$: $Y^{(n)}_{t}\leq\varphi(t)$. Next, for an arbitrary
$t\in(\tau_{-}^{n},\tau)$:

$\displaystyle\varphi(t)\geq$ $\displaystyle
Y^{(n)}_{t}=Y^{(n)}_{t}-Y^{(n)}_{\tau_{-}^{n}}+\varphi(\tau_{-}^{n})$
$\displaystyle=$
$\displaystyle\varphi(\tau_{-}^{n})+\int_{\tau_{-}^{n}}^{t}b_{n}(s,Y^{(n)}_{s})ds+(Z_{t}-Z_{\tau_{-}^{n}})$
$\displaystyle=$
$\displaystyle\varphi(\tau_{-}^{n})+\left(n-\frac{1}{n}\right)(t-\tau_{-}^{n})+(Z_{t}-Z_{\tau_{-}^{n}})$
$\displaystyle\geq$
$\displaystyle\varphi(\tau_{-}^{n})+\left(n-\frac{1}{n}\right)(t-\tau_{-}^{n})-\Lambda(t-\tau_{-}^{n})^{\lambda},$

therefore, for any $n\geq n_{0}$:

$0>Y^{(\infty)}_{\tau}-\varphi(\tau)>Y^{(n)}_{\tau}-\varphi(\tau)\geq\varphi(\tau_{-}^{n})-\varphi(\tau)+\min_{t\in[\tau_{-}^{n},\tau]}F^{(n)}(t),$
(A.9)

with
$F^{(n)}(t):=\left(n-\frac{1}{n}\right)(t-\tau_{-}^{n})-\Lambda(t-\tau_{-}^{n})^{\lambda}$.
However, $\min_{t\in[\tau_{-}^{n},\tau]}F^{(n)}(t)\to 0$, $n\to\infty$.
Indeed,

$\min_{t\in[0,\infty)}F^{(n)}(t)\leq\min_{t\in[\tau_{-}^{n},\tau]}F^{(n)}(t)\leq
0$

and it is straightforward to verify that $F^{(n)}(t)$ takes its minimal value
on $[0,\infty)$ at

$t_{*}:=\tau_{-}^{n}+\left(\frac{\lambda\Lambda}{n-\frac{1}{n}}\right)^{\frac{1}{1-\lambda}}$

with

$F^{(n)}(t_{*})=\frac{\Lambda^{\frac{1}{1-\lambda}}(\lambda^{\frac{1}{1-\lambda}}-\lambda^{\frac{\lambda}{1-\lambda}})}{\left(n-\frac{1}{n}\right)^{\frac{\lambda}{1-\lambda}}}\to
0,\quad n\to\infty.$

Furthermore, it is easy to see from Step 2 that $\tau^{n}_{-}\to\tau$,
$n\to\infty$, so $\varphi(\tau_{-}^{n})-\varphi(\tau)\to 0$, $n\to\infty$, and
therefore (A.9) cannot hold for all $n$. The obtained contradiction finalizes
the proof. ∎

For arbitrary positive
$\varepsilon<\min_{t\in[0,T]}\left(\Psi-\varphi(t)\right)$ and $0\leq
t_{1}<t_{2}\leq T$ denote

$\tilde{\mathcal{D}}^{[t_{1},t_{2}]}_{\varepsilon}:=\\{(t,y)~{}|~{}t\in[t_{1},t_{2}],y\in[\varphi(t)+\varepsilon,\Psi]\\},$

where $\Psi$ is from Proposition A.2, and observe that
$\tilde{\mathcal{D}}^{[t_{1},t_{2}]}_{\varepsilon}$ is a compact set and $b$
is continuous on it. Consider also

$\tau_{0}:=\sup\\{t\in[0,T]~{}|~{}\forall
s\in[0,t):~{}Y^{(\infty)}_{s}>\varphi(s)\\}.$

It is clear that $\tau_{0}>0$ because $Y^{(\infty)}$ is bounded from below by
continuous processes $Y^{(n)}$ which start from the level $Y_{0}>\varphi(0)$.

######  Proposition A.4.

  1. 1.

$Y^{(\infty)}$ is continuous at any $t$ such that
$Y^{(\infty)}_{t}>\varphi(t)$.

  2. 2.

For any $t<\tau_{0}$:

$Y^{(\infty)}_{t}=Y_{0}+\int_{0}^{t}b(s,Y^{(\infty)}_{s})ds+Z_{t}.$

  3. 3.

$Y^{(\infty)}_{\tau_{0}}=\varphi(\tau_{0})$ and, furthermore, $Y^{(\infty)}$
is left continuous at $\tau_{0}$:

$\lim_{t\to\tau_{0}-}Y^{(\infty)}_{t}=\varphi(\tau_{0}).$

###### Proof.

1. Let $t\in[0,T]$ be such that $Y^{(\infty)}_{t}>\varphi(t)$. Then there exists $n_{1}\geq n_{0}$ such that for all $n\geq n_{1}$: $Y^{(n)}_{t}>\varphi(t)$. Furthermore, because of monotonicity with respect to $n$ and continuity of both $Y^{(n)}_{\cdot}$ and $\varphi$, there is such $\varepsilon_{1}=\varepsilon_{1}(n_{1})$ that for any $s\in[t-\varepsilon_{1},t+\varepsilon_{1}]$: $Y^{(n)}_{s}>\varphi(s)$, $n\geq n_{1}$. Furthermore, since for all $s\in[t-\varepsilon_{1},t+\varepsilon_{1}]$ and $n\geq n_{0}$: $Y^{(n)}_{s}<\Psi$, for all $n\geq n_{1}$:

$\left(s,Y^{(n)}_{s}\right)\in\tilde{\mathcal{D}}^{[t-\varepsilon_{1},t+\varepsilon_{1}]}_{\varepsilon_{0}},$

with
$\varepsilon_{0}:=\min_{r\in[t-\varepsilon_{1},t+\varepsilon_{1}]}\left(Y^{(n_{1})}_{r}-\varphi(r)\right)>0$.
Therefore, if $n_{2}\geq n_{1}$ is such that

$n_{2}>\max_{(s,y)\in\tilde{\mathcal{D}}^{[t-\varepsilon_{1},t+\varepsilon_{1}]}_{\varepsilon_{0}}}b(s,y),$

for any $n\geq n_{2}$ and $s\in[t-\varepsilon_{1},t+\varepsilon_{1}]$:
$b_{n}(s,Y^{(n)}_{s})=b(s,Y^{(n)}_{s})-\frac{1}{n}$, whence

$\displaystyle Y^{(n)}_{s}$
$\displaystyle=Y^{(n)}_{t-\varepsilon_{1}}+\int_{t-\varepsilon_{1}}^{s}b_{n}(r,Y^{(n)}_{r})dr+Z_{s}-Z_{t-\varepsilon_{1}}$
(A.10)
$\displaystyle=Y^{(n)}_{t-\varepsilon_{1}}+\int_{t-\varepsilon_{1}}^{s}b(r,Y^{(n)}_{r})dr-\frac{s-t+\varepsilon_{1}}{n}+Z_{s}-Z_{t-\varepsilon_{1}}.$

From the choice of $n_{2}$, for any $n\geq n_{2}$ and
$u\in[t-\varepsilon_{1},s]$:
$(u,Y^{(n)}_{u})\in\mathcal{D}_{\varepsilon_{0}}$, therefore, by assumption
(A2) and Proposition A.2, there exists a constant $L>0$ that does not depend
on $n$ such that for any
$u\in[t-\varepsilon_{1},s]\subset[t-\varepsilon_{1},t+\varepsilon_{1}]$

$|b(u,Y^{(n)}_{u})|\leq L(1+|Y^{(n)}_{u}|)\leq L(1+\Psi)<\infty,$

therefore, by dominated convergence,

$\lim_{n\to\infty}\int_{t-\varepsilon_{1}}^{s}b(r,Y^{(n)}_{r})dr=\int_{t-\varepsilon_{1}}^{s}b(r,Y^{(\infty)}_{r})dr$

which, together with (A.10), implies

$Y^{(\infty)}_{s}=Y^{(\infty)}_{t-\varepsilon_{1}}+\int_{t-\varepsilon_{1}}^{s}b(r,Y^{(\infty)}_{r})dr+Z_{s}-Z_{t-\varepsilon_{1}},\quad
s\in[t-\varepsilon_{1},t+\varepsilon_{1}].$

Hence $Y^{(\infty)}$ is continuous on $[t-\varepsilon_{1},t+\varepsilon_{1}]$
and, in particular, at point $t$.
2. Since $Y^{(\infty)}$ is greater than $\varphi$ on an arbitrary interval $[0,t]\subset[0,\tau_{0})$, it is continuous on this interval. Therefore, by Dini’s theorem, $Y^{(n)}$ converges uniformly to $Y^{(\infty)}$, $n\to\infty$, on $[0,t]$. Let $n_{3}$ be such that for all $n\geq n_{3}$: $Y^{(n)}_{s}-\varphi(s)>\frac{\min_{r\in[0,t]}(Y^{(\infty)}_{r}-\varphi(r))}{2}=:\varepsilon_{\infty}$, $s\in[0,t]$. For any $s\in[0,t]$ and $n\geq n_{3}$ it holds that

$(s,Y^{(n)}_{s})\in\tilde{\mathcal{D}}^{[0,t]}_{\varepsilon_{\infty}},$

so, if $n_{4}\geq n_{3}$ is such that
$n_{4}>\max_{(s,y)\in\tilde{\mathcal{D}}^{[0,t]}_{\varepsilon_{\infty}}}b(s,y)$,
for any $s\in[0,t]$ and $n\geq n_{4}$:
$b_{n}(s,Y^{(n)}_{s})=b(s,Y^{(n)}_{s})-\frac{1}{n}$. Taking into account that

$(s,Y^{(n)}_{s}),(s,Y^{(\infty)}_{s})\in\mathcal{D}_{\varepsilon}$

for any $s\in[0,t]$ with $\varepsilon\in(0,\varepsilon_{\infty})$, we have
that, by assumption (A2), there exists a constant $c_{\varepsilon}$ that does
not depend on $n$ such that

$|b_{n}(s,Y^{(n)}_{s})-b(s,Y^{(\infty)}_{s})|\leq
c_{\varepsilon}|Y^{(n)}_{s}-Y^{(\infty)}_{s}|+\frac{1}{n},\quad s\in[0,t],$

whence $b_{n}(s,Y^{(n)}_{s})\rightrightarrows b(s,Y^{(\infty)}_{s})$ on
$[0,t]$, $n\to\infty$. Now the claim can be verified by transition to the
limit under the integral.
3. First, note that $Y^{(\infty)}_{\tau_{0}}=\varphi(\tau_{0})$. Indeed, by Proposition A.3, $Y^{(\infty)}_{t}\geq\varphi(t)$ for all $t\in[0,T]$ and, if $Y^{(\infty)}_{\tau_{0}}>\varphi(\tau_{0})$, then $Y^{(\infty)}$ is continuous at $\tau_{0}$ and therefore exceeds $\varphi$ on some interval $[\tau_{0},\tau_{0}+\delta)$, that contradicts the definition of $\tau_{0}$. Now it is sufficient to verify that

$\limsup_{t\to\tau_{0}-}Y^{(\infty)}_{t}=\varphi(\tau_{0}).$

Assume it is not true and there is such $x\in(0,\infty)$ that

$\limsup_{t\to\tau_{0}-}Y^{(\infty)}_{t}=\varphi(\tau_{0})+x.$

Note also that $x<\infty$ since, by Proposition A.2, $Y^{(\infty)}$ is bounded
from above by the (random) constant $\Psi$.
Let $\delta_{x}$ be such that for any $t\in[\tau_{0}-\delta_{x},\tau_{0}]$:
$|\varphi(t)-\varphi(\tau_{0})|<\frac{x}{4}$. Denote

$\varepsilon_{x}:=\min_{t\in[\tau_{0}-\delta,\tau_{0}]}\left(\varphi(\tau_{0})+\frac{x}{4}-\varphi(t))\right)$

and observe that $\varepsilon_{x}>0$ and
$\varphi(t)+\varepsilon_{x}\leq\varphi(\tau_{0})+\frac{x}{4}$ whenever
$t\in[\tau_{0}-\delta,\tau_{0}]$.
If $x>0$, for any $\delta\in(0,\delta_{x})$ there is such
$t_{\delta}\in(\tau_{0}-\delta,\tau_{0})$ that
$Y^{(\infty)}_{t_{\delta}}\geq\varphi(\tau_{0})+\frac{3x}{4}$. Let such
$\delta\in(0,\delta_{x})$ and $t_{\delta}$ be fixed. Since
$Y^{(n)}_{t_{\delta}}\uparrow Y^{(\infty)}_{t_{\delta}}$, $n\geq 1$, there is
such $n_{\delta}$ that for all $n\geq n_{\delta}$:

$Y^{(n)}_{t_{\delta}}\geq\varphi(\tau_{0})+\frac{x}{2}.$

It is clear that $Y^{(n)}_{\tau_{0}}<\varphi(\tau_{0})$ therefore, for $n\geq
n_{\delta}$ one can consider the moment

$\theta^{n}:=\inf\left\\{t\in(t_{\delta},\tau_{0})~{}|~{}Y^{(n)}_{t}=\varphi(\tau_{0})+\frac{x}{4}\right\\}.$

From the continuity of $Y^{(n)}$,
$Y^{(n)}_{\theta^{n}}=\varphi(\tau_{0})+\frac{x}{4}$, so
$Y^{(n)}_{\theta^{n}}-Y^{(n)}_{t_{\delta}}<-\frac{x}{4}$. On the other hand,
from definition of $\theta^{n}$ and Proposition A.2, for all
$t\in[t_{\delta},\theta^{n}]$:

$(t,Y^{(n)}_{t})\in[t_{\delta},\theta^{n}_{x}]\times\left[\varphi(\tau_{0})+\frac{x}{4},\Psi\right]\subset\tilde{\mathcal{D}}^{[\tau_{0}-\delta_{x},\tau_{0}]}_{\varepsilon_{x}}.$

Let $\tilde{n}_{\delta}>n_{\delta}$ be such that

$\tilde{n}_{\delta}>\max_{(s,y)\in\tilde{\mathcal{D}}^{[\tau_{0}-\delta_{x},\tau_{0}]}_{\varepsilon_{x}}}b(s,y).$

For any $t\in[t_{\delta},\theta_{x}]$ and $n\geq\tilde{n}_{\delta}$:

$b_{n}(t,Y^{(n)}_{t})=b(t,Y^{(n)}_{t})-\frac{1}{n}$

and, therefore, we obtain that

$\displaystyle-\frac{x}{4}$
$\displaystyle>Y^{(n)}_{\theta^{n}}-Y^{(n)}_{t_{\delta}}=\int_{t_{\delta}}^{\theta^{n}}b_{n}(s,Y^{(n)}_{s})ds+(Z_{\theta^{n}}-Z_{t_{\delta}})$
$\displaystyle=\int_{t_{\delta}}^{\theta^{n}}b(s,Y^{(n)}_{s})ds-\frac{1}{n}(\theta^{n}-t_{\delta})+(Z_{\theta^{n}}-Z_{t_{\delta}})$
$\displaystyle=\int_{t_{\delta}}^{\theta^{n}}b^{+}(s,Y^{(n)}_{s})ds-\int_{t_{\delta}}^{\theta^{n}}b^{-}(s,Y^{(n)}_{s})ds-\frac{1}{n}(\theta^{n}-t_{\delta})+(Z_{\theta^{n}}-Z_{t_{\delta}})$
$\displaystyle\geq-\int_{t_{\delta}}^{\theta^{n}}b^{-}(s,Y^{(n)}_{s})ds-\frac{1}{n}(\theta^{n}-t_{\delta})-\Lambda(\theta^{n}-t_{\delta})^{\alpha}$
$\displaystyle\geq-\left(\max_{(s,y)\in\tilde{\mathcal{D}}^{[\tau_{0}-\delta_{x},\tau_{0}]}_{\varepsilon_{x}}}b^{-}(s,y)+\frac{1}{n}\right)(\theta^{n}_{x}-t_{\delta})-\Lambda(\theta^{n}_{x}-t_{\delta})^{\lambda}$
$\displaystyle\geq-\left(\max_{y\in\tilde{\mathcal{D}}^{[\tau_{0}-\delta_{x},\tau_{0}]}_{\varepsilon_{x}}}b^{-}(s,y)+\frac{1}{n}\right)\delta-\Lambda\delta^{\lambda},$

i.e. for any $\delta\in(0,\delta_{x})$:

$\delta\max_{y\in\tilde{\mathcal{D}}^{[\tau_{0}-\delta_{x},\tau_{0}]}_{\varepsilon_{x}}}b^{-}(s,y)+\Lambda\delta^{\lambda}\geq\frac{x}{4},$

which is not possible. The obtained contradiction implies that $x=0$, i.e.

$\limsup_{t\to\tau_{0}-}Y^{(\infty)}_{t}=\varphi(\tau_{0}).$

∎

Now, let us move to the proof of Theorem 2.2.

###### Proof of Theorem 2.2.

By Proposition A.4, $Y=Y^{(\infty)}$ indeed satisfies the equation of the
required form. Let $\tilde{Y}$ satisfy the equation (0.3) on
$[0,t]\subset[0,\tilde{\tau}_{0}\wedge\tau_{0})$. Then it is continuous on
$[0,t]$ and therefore $\min_{s\in[0,t]}(\tilde{Y}_{s}-\varphi(s))>0$. Let
$\varepsilon:=\min_{s\in[0,t]}(\tilde{Y}_{s}-\varphi(s))\wedge\min_{s\in[0,t]}(Y^{(\infty)}_{s}-\varphi(s))$
and choose $\tilde{n}$ such that for all $n\geq\tilde{n}$:
$n>\max_{(s,y)\in\tilde{\mathcal{D}}^{[0,t]}_{\varepsilon}}b(s,y)$. Then
$b(s,\tilde{Y}_{s})=\tilde{b}_{n}(s,\tilde{Y}_{s})$, $s\in[0,t]$,
$n\geq\tilde{n}$, where $\tilde{b}_{n}$ is defined by (A.1), whence

$\tilde{Y}_{s}=Y_{0}+\int_{0}^{s}\tilde{b}_{n}(s,\tilde{Y}_{u})du+Z_{s},\quad
s\in[0,t].$

However, $Y^{(\infty)}$ also satisfies the equation above and, since the
latter has a unique solution, $\tilde{Y}_{s}=Y^{(\infty)}_{s}$ for all
$s\in[0,t]$. Now it is easy to deduce that $\tau_{0}=\tilde{\tau}_{0}$ and
$\tilde{Y}_{t}=Y^{(\infty)}_{t}=Y_{t}$ for all $t\in[0,\tau_{0}]$. ∎