# Wong–Zakai approximations with convergence rate for stochastic differential
equations
with regime switching

Giang T. Nguyen  Oscar Peralta

###### Abstract

We construct Wong–Zakai approximations of time–inhomogeneous stochastic
differential equations with regime switching (RSSDEs), and provide a
convergence rate. In the proposed approximations, the standard Brownian motion
driving the time-inhomogeneous RSSDEs is replaced by a family of
finite–variation processes $\\{\mathcal{F}^{\lambda}\\}_{\lambda>0}$. We show
that if $\mathcal{F}^{\lambda}$ strongly converges to $\mathcal{B}$ at rate
$\delta(\lambda)$, then the Wong–Zakai approximation strongly converges to the
original solution of the time–inhomogeneous RSSDE at rate
$\delta(\lambda)\lambda^{\varepsilon}$, for any $\varepsilon>0$. This is the
first paper on Wong–Zakai approximations for time–inhomogeneous RSSDEs, and
significantly extends the counterparts for time–homogeneous SDEs without
regime switching in Römisch and Wakolbinger [24].

##  1 Introduction

Let $\mathcal{B}=\\{B_{t}\\}_{t\geq 0}$ be a standard Brownian motion and let
$\mathcal{J}=\\{J_{t}\\}_{t\geq 0}$ be a càdlàg jump process with finite state
space $\mathcal{E}$. Suppose that $\mathcal{J}$ is independent of
$\mathcal{B}$ and has finite jump activity on compact intervals. For
$i\in\mathcal{E}$, let
$\mu_{i}:\mathds{R}_{+}\times\mathds{R}\mapsto\mathds{R}$ and
$\sigma_{i}:\mathds{R}_{+}\times\mathds{R}\mapsto\mathds{R}$, where
$\mathds{R}_{+}=[0,\infty)$. A _regime–switching time-inhomogeneous stochastic
differential equation_ (RSSDE) takes the form

$\displaystyle\mathrm{d}X_{t}$
$\displaystyle=\mu_{J_{t}}(t,X_{t})\mathrm{d}t+\sigma_{J_{t}}(t,X_{t})\mathrm{d}B_{t},\quad{X_{0}=x_{0}\in\mathds{R}}.$
(1.1)

RSSDEs are used to model systems in which randomness arise from two sources:
continuous (diffusive) and discrete (environmental). These processes are
widely used in areas such as economics, finance, ecology, and automatic
control [30]. Although (1.1) is well–defined in the context of Itô stochastic
differential equations (SDE), most of its properties such as existence and
uniqueness of solutions and stability do not follow from standard SDE theory
and thus need to be properly verified; see [18] for the particular case when
$\mathcal{J}$ is Markovian.

Suppose that there exists a unique continuous solution
$\mathcal{X}=\\{X_{t}\\}_{t\geq 0}$ of (1.1). Our goals in paper are twofold.
First, we construct a family of the so–called Wong–Zakai approximations
$\\{\mathcal{X}^{\lambda}\\}_{\lambda>0}=\\{\\{X^{\lambda}_{t}\\}_{t\geq
0}\\}_{\lambda>0}$. On a fixed time interval $[0,T]$, $T<\infty$, we replace
the standard Brownian motion $\mathcal{B}$ in (1.1) by a family of finite-
variation processes
$\\{\mathcal{F}^{\lambda}\\}_{\lambda>0}=\\{\\{F^{\lambda}_{t}\\}_{t\geq
0}\\}_{\lambda>0}$ that strongly converges to $\mathcal{B}$; that is,

$\lim_{\lambda\rightarrow\infty}\sup_{t\in[0,T]}\left|B_{t}-F^{\lambda}_{t}\right|=0\quad\mbox{almost
surely.}$

For each $\lambda>0$, consider the Wong–Zakai approximation
$\mathcal{X}^{\lambda}$ that is the solution of the following regime-switching
SDE

$\displaystyle\mathrm{d}X_{t}^{\lambda}$
$\displaystyle=\left(\mu_{J_{t}}(t,X_{t}^{\lambda})-\frac{1}{2}\sigma_{J_{t}}(t,X^{\lambda}_{t})\partial_{2}\left[\sigma_{J_{t}}(t,X^{\lambda}_{t})\right]\right)\mathrm{d}t+\sigma_{J_{t}}(t,X_{t}^{\lambda})\mathrm{d}F_{t}^{\lambda},$
(1.2)

where $\partial_{k}$ denotes the partial differential operator with respect to
the $k$th entry. We show in Proposition 4.1 that this solution is unique.

Second, under certain regularity conditions on $\mu_{i}$ and $\sigma_{i}$, and
boundedness on $\mathcal{J}$, we show that the Wong–Zakai approximations
$\\{\mathcal{X}^{\lambda}\\}_{\lambda>0}$ converge to the solution
$\mathcal{X}$ of (1.1). More precisely, we obtain the rate of strong
convergence of $\mathcal{X}^{\lambda}$ to $\mathcal{X}$, and show that it is
_almost the same_ as that of $\mathcal{F}^{\lambda}$ to $\mathcal{B}$. The
precise rate of strong convergence and the conditions under which this holds
are stated in Theorem 1.1, our main result.

######  Theorem 1.1.

Under Assumptions 1–4, suppose that there exists a family of finite–variation
processes $\\{\mathcal{F}^{\lambda}\\}_{\lambda>0}$ such that for some
$\delta:\mathds{R}_{+}\mapsto\mathds{R}_{+}$ with
$\lim_{\lambda\rightarrow\infty}\delta(\lambda)=0$, we have that for all $q>0$

$\displaystyle\mathds{P}\left(\sup_{t\in[0,T]}|F^{\lambda}_{t}-B_{t}|\geq\alpha\delta(\lambda)\right)=o(\lambda^{-q}),$

where $\alpha=\alpha(q,T)>0$ is some constant dependent on $q$ and $T$ only.
For $\lambda>0$, let $\mathcal{X}^{\lambda}$ be the unique solution
$\mathcal{X}^{\lambda}$ to the regime-switching SDE

$\displaystyle\mathrm{d}X_{t}^{\lambda}$
$\displaystyle=\left(\mu_{J_{t}}(t,X_{t}^{\lambda})-\frac{1}{2}\sigma_{J_{t}}(t,X^{\lambda}_{t})\partial_{2}\left[\sigma_{J_{t}}(t,X^{\lambda}_{t})\right]\right)\mathrm{d}t+\sigma_{J_{t}}\left(t,X_{t}^{\lambda}\right)\mathrm{d}F_{t}^{\lambda},$
(1.3) $\displaystyle X^{\lambda}_{0}$ $\displaystyle=x_{0}.$ (1.4)

Then,

  * (i)

the family of processes $\\{\mathcal{X}^{\lambda}\\}_{\lambda>0}$ converges
strongly to the unique solution $\mathcal{X}$ of the regime–switching
time–inhomogeneous SDE (1.1)

  * (ii)

moreover, for all $q,T>0$ there exists $\gamma=\gamma(q,T)>0$ such that for
all $\varepsilon>0$

$\displaystyle\mathds{P}\left(\sup_{t\in[0,T]}|X^{\lambda}_{t}-X_{t}|\geq\gamma\delta(\lambda)\lambda^{\varepsilon}\right)=o(\lambda^{-q}).$
(1.5)

Thus, Theorem 1.1 states that if $\mathcal{F}^{\lambda}$ strongly converges to
$\mathcal{B}$ at rate $\delta(\lambda)$, then the Wong–Zakai approximation
$\mathcal{X}^{\lambda}$ strongly converges to $\mathcal{X}$ at rate
$\delta(\lambda)\lambda^{\varepsilon}$, for any $\varepsilon>0$.

Details of Assumptions 1–4 can be found in later sections. Briefly speaking,
Assumption 1 is about linear growth and local Lipschitz continuity, and is
standard for ensuring existence and uniqueness of solutions of SDEs.
Assumption 2 provides bounds for the diffusion coefficient $\sigma$ and its
derivative, and is standard for Wong–Zakai approximations. Assumption 3 is
needed specifically because of the time–inhomogeneity feature, and does not
automatically follow from the properties of $\mu$ and $\sigma$ described in
Assumptions 1 and 2. Assumption 4 arises due to the regime switching feature,
and bounds the tail of the jump distribution of $\mathcal{J}$ over a compact
time interval. This assumption allows us to obtain the convergence rate in the
regime–switching case, without being overly restrictive: it holds whenever
$\mathcal{J}$ has a bounded jump intensity. Cases where Assumption 4 is
satisfied include, but are not limited to, time–homogeneous Markovian,
time–inhomogeneous Markovian, and semi–Markovian processes.

Most related to this paper is the work of Römisch and Wakolbinger [24], which
determined the rate of strong convergence of the Wong–Zakai approximation for
time-homogeneous SDEs without regime switching, under Assumptions 1 and 2
reduced to the time–homogeneous setting. While our strong approximation is an
extension of [24], our methods significantly differ from theirs in order to
handle challenges that the regime–switching and time–inhomongeneity
generalizations bring to the problem. A key technique in our analysis is
applying the Lamperti transform [16] to regime–switching SDEs (1.1). This was
previously applied in Pettersson [22] and Zhang [32] in the simpler case of
time–homogeneous SDEs without regime switching. The Lamperti transformation
leads to clear expressions in the approximation and an insight on how to bound
approximation errors at jump epochs of $\mathcal{J}$.

###  1.1 Related literature

Wong and Zakai [28, 29] were the first to consider almost-sure approximations
to solutions of SDEs. Let $\pi$ be a finite partition of $[a,b]$, where
$\pi=\\{t_{1},\ldots,t_{k+1}\\}$, $a=t_{1}<t_{2}<\ldots<t_{k+1}=b$, and let
$\mathcal{B}^{n}=\\{B^{n}_{t}\\}_{t\geq 0}$ be a process whose the sample
paths coincide with those of $\mathcal{B}$ at $t_{j}$ and are linear between
them. Then, as the mesh of $\pi$, i.e. $\max_{j}|t_{j+1}-t_{j}|$, tends to
$0$, the solutions $\mathcal{Y}^{n}=\\{Y^{n}_{t}\\}_{t\geq 0}$ of the equation

$\displaystyle
Y^{n}_{t}=y_{0}+\int_{a}^{t}\mu(s,Y^{n}_{s})\,\mathrm{d}s+\int_{a}^{t}\sigma(s,Y^{n}_{s})\,\mathrm{d}B^{n}_{s},\quad
t\in[a,b],$ (1.6)

converge in quadratic mean to $\mathcal{Y}=\\{Y_{t}\\}_{t\geq 0}$ [29], which
is a solution of

$\displaystyle
Y_{t}=y_{0}+\int_{a}^{t}\mu(s,Y_{s})\,\mathrm{d}s+\int_{a}^{t}\sigma(s,Y_{s})\,\mathrm{d}B_{s}+\frac{1}{2}\int_{a}^{t}\sigma(s,Y_{s})\partial_{2}[\sigma(s,Y_{s})]\,\mathrm{d}s.$
(1.7)

The last term on the RHS of (1.7) is known as the _Wong-Zakai correction term_
, and the solution to (1.7) is also known as _the Stratonovich solution to the
SDE_. In [28], Wong and Zakai extended their previous result to show that,
given any piece-wise smooth approximation $\mathcal{B}^{n}$ for $\mathcal{B}$,
the convergence from $\mathcal{Y}^{n}$ to $\mathcal{Y}$ holds almost surely.

Subsequently, approximations following this framework are generally referred
to as of the _Wong-Zakai type_ ; see the comprehensive survey by Twardowska
[27] and references therein. We describe briefly three major directions of
generalization of Wong–Zakai results. First, the Wong–Zakai framework has been
extended to SDEs driven by higher–dimensional Brownian motion processes.
Stroock and Varadhan [26] proposed piecewise–linear approximations and showed
convergence in probability; Ikeda _et al._ [13] and Ikeda and Watanabe [14]
proved, respectively, convergence in meansquare and uniform convergence over
finite time interval. Second, Wong–Zakai approximations have also been
obtained for reflected SDEs. Pettersson [22] analysed SDEs reflected on the
boundary of a convex set, replacing the Brownian motion process with its
polygonal approximation, and obtained uniform convergence. Evans and Stroock
[6] approximated SDEs reflected in general domains and obtained convergence in
law. Zhang [31] showed strong convergence of reflected SDE in
multi–dimensional general domains. Third, Wong–Zakai approximations have been
generalized to stochastic partial differential equations, such as the Wong-
Zakai theorem obtained by Hairer and Pardoux [12] for one–dimensional
parabolic nonlinear stochastic PDEs driven by space–time white noise.

Results on convergence rates in Wong-Zakai approximations are much rarer; we
briefly highlight a few. Brzeniak and Flandoli [3] proved almost sure
convergence for parabolic and hyperbolic evolution equations and provided
estimates for the rate of convergence. Later, Brzeniak and Carroll [2]
obtained convergence rate for approximations of SDEs in M–type $2$ Banach
spaces. More recently, Nakayama and Tappe [19] obtained convergence rate for
semilinear stochastic partial differential equation, and Ammou and Lanconelli
[1] for Stratonovich and Itô quasi–linear SDEs.

###  1.2 Paper structure

We outline in Section 2 some results of stochastic calculus for regime-
switching processes which will be useful later. In 3.1, we introduce the
Lamperti transformation, $\mathcal{L}$, of a solution of a regime-switching
SDE, $\mathcal{X}$, and prove its properties. In 3.2, we construct the
transformation pathwise and prove its uniqueness. In Section 4, we apply the
Lamperti transformation to the approximating process $\mathcal{X}^{\lambda}$,
given in (1.2), to obtain $\mathcal{L}^{\lambda}$ and compute its convergence
rate to $\mathcal{L}$. We show that, in fact, $\mathcal{L}^{\lambda}$ is the
unique càdlàg solution to a regime-switching SDE driven by
$\mathcal{F}^{\lambda}$ with _jumps_. Lipschitz continuity helps us obtain the
almost sure convergence, and subsequently the convergence rate, from
$\mathcal{X}^{\lambda}$ to $\mathcal{X}$, giving rise to Theorem 1.1. Finally,
in Section 6 we state a direct implication of Theorem 1.1, with respect to
recent approximations of Markov-modulated Brownian motion [17, 21].

##  2 Preliminaries and notation

Here, we briefly summarize key results in stochastic calculus for regime-
switching processes needed for this paper. We write
$\partial_{k\ell}\equiv\partial_{k}\partial_{\ell}$ to denote the partial
differential operator first w.r.t. to the $k$th entry and then w.r.t. the
$\ell$th entry. Furthermore, $\mathcal{C}^{i,j}$ denotes the class of
continuous $\mathds{R}^{2}\mapsto\mathds{R}$ functions with continuous $i$th
derivative with respect to the first parameter and continuous $j$th derivative
with respect to the second parameter.

Let $\mathcal{Z}=\\{Z_{t}\\}_{t\geq 0}$ be an almost surely continuous
stochastic process; we write $\\{[Z]_{t}\\}_{t\geq 0}$ to denote the quadratic
variation process of $\mathcal{Z}$.

######  Theorem 2.1 (Itô’s formula).

Let $f\in\mathcal{C}^{1,2}$. Then, for all $t\geq 0$

$\displaystyle f(t,Z_{t})$
$\displaystyle=f(0,Z_{0})+\int_{0}^{t}\partial_{1}[f(s,Z_{s})]\mathrm{d}s+\int_{0}^{t}\partial_{2}[f(s,Z_{s})]\mathrm{d}Z_{s}+\frac{1}{2}\int_{0}^{t}\partial_{22}[f(s,Z_{s})]\mathrm{d}[Z]_{s}\quad\mbox{a.s.}$

######  Remark 2.2.

_If $\mathcal{Z}$ is has finite variation, then $[Z]\equiv 0$. If
$\mathcal{Z}=\mathcal{B}$, then $[Z]\equiv\mbox{Id}$._

######  Theorem 2.3 (Itô’s formula for regime-switching processes).

For $i\in\mathcal{E}$, let $f_{i}\in\mathcal{C}^{1,2}$. Then, for all $t\geq
0$

$\displaystyle f_{J_{t}}(t,Z_{t})$
$\displaystyle=f_{J_{0}}(0,Z_{0})+\int_{0}^{t}\partial_{1}[f_{J_{s}}(s,Z_{s})]\mathrm{d}s+\int_{0}^{t}\partial_{2}[f_{J_{s}}(s,Z_{s})]\mathrm{d}Z_{s}+\frac{1}{2}\int_{0}^{t}\partial_{22}[f_{J_{s}}(s,Z_{s})]\mathrm{d}[Z]_{s}$
$\displaystyle\quad+\sum_{s\leq t:J_{s}\neq
J_{s-}}\left(f_{J_{s}}(s,Z_{s})-f_{J_{s-}}(s,Z_{s})\right)\quad\mbox{a.s.}$
(2.1)

###### Proof.

This follows by concatenation. Let $t_{1}<t_{2}<\dots<t_{n}$ denote the jump
epochs of $\mathcal{J}$ on $(0,t]$, and let $t_{0}:=0$, $t_{n+1}:=t$. Let
$j_{i}=J_{t_{i}}$, $0\leq i\leq n$. Notice that

$\displaystyle
f_{J_{t}}(t,Z_{t})-f_{J_{0}}(0,Z_{0})=f_{J_{n}}(t_{n+1},Z_{t_{n+1}})-f_{j_{0}}(t_{0},Z_{t_{0}})$
$\displaystyle=\sum_{k=0}^{n}\left(f_{j_{k}}(t_{k+1},Z_{t_{k+1}})-f_{j_{k}}(t_{k},Z_{t_{k}})\right)+\sum_{k=1}^{n}\left(f_{j_{k}}(t_{k},Z_{t_{k}})-f_{j_{k-1}}(t_{k},Z_{t_{k}})\right)$
$\displaystyle=\sum_{k=0}^{n}\left(f_{j_{k}}(t_{k+1},Z_{t_{k+1}})-f_{j_{k}}(t_{k},Z_{t_{k}})\right)+\sum_{s\leq
t:J_{s}\neq J_{s-}}\left(f_{J_{s}}(s,Z_{s})-f_{J_{s-}}(s,Z_{s})\right).$ (2.2)

Then, (2.1) follows by applying Itô’s formula to each term in the first sum of
(2.2). ∎

Note that the concatenation method used in the proof of Theorem 2.3 holds for
both Markovian and non-Markovian jump processes. This technique can be found
in the proof of Lemma 3 in [25, Section II.2.1], where an alternative form of
Itô’s formula is provided for the particular case of regime-switching
diffusions with an underlying Markovian jump mechanism; see [30, Equation
(2.8)] for a related formula.

######  Remark 2.4.

_Since $\mathcal{F}^{\lambda}$ converges strongly to $\mathcal{B}$, for each
$t\geq 0$ we have_

$\displaystyle\lim_{\lambda\rightarrow\infty}f_{J_{t}}(t,F^{\lambda}_{t})-f_{J_{t}}(t,B_{t})=0\quad\mbox{a.s.}$
(2.3)

_Applying Theorem 2.3 to both terms in (2.3) we obtain_

$\lim_{\lambda\rightarrow\infty}\int_{0}^{t}\partial_{2}[f_{J_{s}}(s,F^{\lambda}_{s})]\mathrm{d}F^{\lambda}_{s}=\int_{0}^{t}\partial_{2}[f_{J_{s}}(s,B_{s})]\mathrm{d}B_{s}+\frac{1}{2}\int_{0}^{t}\partial_{22}[f_{J_{s}}(s,B_{s})]\mathrm{d}s\quad\mbox{a.s.}$

_In particular, if $\psi_{i}\in\mathcal{C}^{1,1}$, then by taking
$f_{i}(t,x):=\int_{0}^{x}\psi_{i}(t,y)\mathrm{d}y$ we have_

$\displaystyle\lim_{\lambda\rightarrow\infty}\int_{0}^{t}\psi_{J_{s}}(s,F^{\lambda}_{s})\mathrm{d}F^{\lambda}_{s}=\int_{0}^{t}\psi_{J_{s}}(s,B_{s})\mathrm{d}B_{s}+\frac{1}{2}\int_{0}^{t}\partial_{2}[\psi_{J_{s}}(s,B_{s})]\mathrm{d}s\quad\mbox{a.s.}$
(2.4)

_As seen in ( 2.4), whenever we compute the stochastic integral w.r.t. a
finite-variation process with regime switching, the limit as
$\lambda\rightarrow\infty$ is equal to the same stochastic integral w.r.t. the
Brownian motion _plus_ a correction term._

Now, consider the regime-switching SDE given by

$\displaystyle\mathrm{d}Y_{t}$
$\displaystyle=\mu_{J_{t}}(t,Y_{t})\mathrm{d}t+\sigma_{J_{t}}(t,Y_{t})\mathrm{d}Z_{t},\quad
Y_{0}=x_{0}.$ (2.5)

In order to guarantee the existence of a unique càdlàg solution of (2.5), we
need the following concepts. Let
$g:\mathds{R}_{+}\times\mathds{R}\mapsto\mathds{R}$. We say that the function
$g$ is:

  * •

_linearly growing_ if there exists $E>0$ such that

$g(t,x)\leq E(1+t+|x|)\quad\forall t\geq 0,x\in\mathds{R};$

  * •

_locally Lipschitz continuous_ if for each compact set
$\mathcal{K}\subset\mathds{R}_{+}\times\mathds{R}$, there exists
$M_{\mathcal{K}}>0$ such that

$\displaystyle\Big{|}g(t_{1},x_{1})-g(t_{2},x_{2})\Big{|}$ $\displaystyle\leq
M_{\mathcal{K}}\left(|t_{1}-t_{2}|+|x_{1}-x_{2}|\right)\quad\forall(t_{1},x_{1}),(t_{2},x_{2})\in\mathcal{K};$

  * •

_Lipschitz continuous on its second entry_ if there exists $M>0$ such that

$\displaystyle\Big{|}g(t,x_{1})-g(t,x_{2})\Big{|}$ $\displaystyle\leq
M|x_{1}-x_{2}|\quad\forall t\geq 0,x\in\mathds{R}.$

Note that a function being Lipschitz continuous on its second entry does not
necessarily imply linear growth nor local Lipschitz continuity (e.g.
$g(t,x)=\sqrt{t}+t^{2}$).

######  Assumption 1.

For all $i\in\mathcal{E}$, $t\geq 0$ and $x\in\mathds{R}$:

  1. _(i)_

$\mu_{i}$ is in $\mathcal{C}^{0,0}$, linearly growing and locally Lipschitz
continuous,

  2. _(ii)_

$\sigma_{i}$ is in $\mathcal{C}^{1,1}$, linearly growing, locally Lipschitz
continuous, and Lipschitz continuous on its second entry,

  3. _(iii)_

$\partial_{1}[\sigma_{i}]$ and $\partial_{2}[\sigma_{i}]$ are locally
Lipschitz continuous.

Assumption 1 corresponds to classic conditions of existence and uniqueness of
the solution of an SDE without regime switching [23, Section V.12]. For
regime-switching SDEs, existence and uniqueness of solutions between jump
times of $\mathcal{J}$, together with a continuous concatenation of paths,
yields the existence and uniqueness of $\mathcal{Y}=\\{Y_{t}\\}_{t\geq 0}$. In
Section 3.2, we make explicit this concatenation technique for the Lamperti
transform of the solution of the regime–switching SDEs (1.1).

##  3 Lamperti transform for regime-switching SDEs

In general, _Lamperti transform_ refers a class of transformations of
solutions of SDEs that lead to the diffusion term in the resulting processes
being independent of the state [16].

###  3.1 Functional construction

Here, we introduce a Lamperti transformation for a regime-switching SDE.
Denote by $\\{t_{k}\\}_{k\geq 0}$ the jump times of $\mathcal{J}$, with
$t_{0}:=0$, and we write $J_{k}$ for $J_{t_{k}}$ for $k\geq 0$ whenever there
is no ambiguity. Let $\mathcal{X}$ be the unique solution to (1.1). To
construct the transformation, we need an extra assumption which is also in
classic papers dealing with Wong-Zakai approximations [28, 24].

######  Assumption 2.

For $i\in\mathcal{E}$,

  * _(i)_

there exist constants $v$ and $V$ such that $0<v<\sigma_{i}<V<\infty$;

  * _(ii)_

there exists a constant $K>0$ such that
$|\partial_{1}[\sigma_{i}]|/\sigma_{i}^{2}\leq K$.

######  Theorem 3.1 (Lamperti transform for regime-switching SDEs).

Suppose Assumptions 1 and 2 hold. For $t\geq 0$ and $i\in\mathcal{E}$, define

$\displaystyle
h_{i}(t,x):=\int_{x_{0}}^{x}\frac{1}{\sigma_{i}(t,y)}\mathrm{d}y,\quad
x\in\mathds{R},$ (3.1)

and $L_{t}:=h_{J_{t}}(t,X_{t})$. Then, the process
$\mathcal{L}=\\{L_{t}\\}_{t\geq 0}$ satisfies the regime-switching stochastic
differential equation with jumps given by

$\displaystyle\mathrm{d}L_{t}$
$\displaystyle=\mu^{*}_{J_{t}}(t,L_{t})\mathrm{d}t+\mathrm{d}B_{t}+\sum_{s\leq
t:J_{s}\neq
J_{s^{-}}}\left(h_{J_{s}}\left(s,h^{-1}_{J_{s^{-}}}(s,L_{s^{-}})\right)-L_{s^{-}}\right),$
(3.2) $\displaystyle L_{0}$ $\displaystyle=h_{J_{0}}(0,x_{0}),$

where, for each fixed $t$, $h_{i}^{-1}(t,\cdot):\mathds{R}\mapsto\mathds{R}$
denotes the inverse of $h_{i}(t,\cdot):\mathds{R}\mapsto\mathds{R}$ and

$\displaystyle\mu^{*}_{i}(t,\ell):=\partial_{1}\left[h_{i}\left(t,h_{i}^{-1}(t,\ell)\right)\right]+\frac{\mu_{i}\left(t,h_{i}^{-1}(t,\ell)\right)}{\sigma_{i}\left(t,h_{i}^{-1}(t,\ell)\right)}-\frac{1}{2}\partial_{2}\left[\sigma_{i}\left(t,h_{i}^{-1}(t,\ell)\right)\right].$
(3.3)

Even though $\mathcal{L}$ has discontinuities at $\\{t_{k}\\}_{k>0}$, the
important characteristic is that it has a unit diffusion coefficient; it is
identically $1$ and independent of $J_{t}$, $t$, and $L_{t}$.

###### Proof.

Since $1/V<1/\sigma_{i}(t,y)<1/v$, the function $h_{i}(t,\cdot)$ is non-
decreasing and thus bijective with a continuous inverse. The boundedness and
$\mathcal{C}^{1,1}$ properties of $\sigma_{i}$ imply that
$h_{i}(\cdot,\cdot)\in\mathcal{C}^{1,2}$. Applying Theorem 2.3 gives

$\displaystyle L_{t}-L_{0}$
$\displaystyle=h_{J_{t}}(t,X_{t})-h_{J_{0}}(0,x_{0})$
$\displaystyle=\int_{0}^{t}\partial_{1}\left[h_{J_{s}}(s,X_{s})\right]\mathrm{d}s+\int_{0}^{t}\partial_{2}\left[h_{J_{s}}(s,X_{s})\right]\mathrm{d}X_{s}+\frac{1}{2}\int_{0}^{t}\partial_{22}\left[h_{J_{s}}(s,X_{s})\right]\mathrm{d}[X]_{s}$
$\displaystyle\quad+\sum_{s\leq t:J_{s}\neq
J_{s-}}\left(h_{J_{s}}(s,X_{s})-h_{J_{s-}}(s,X_{s})\right)$

Since
$\mathrm{d}[X]_{t}=\sigma_{J_{t}}^{2}(t,X_{t})\mathrm{d}[B]_{t}=\sigma_{J_{t}}^{2}(t,X_{t})\mathrm{d}t,$
we have

$\displaystyle L_{t}-L_{0}$
$\displaystyle=\int_{0}^{t}\left(\partial_{1}\left[h_{J_{s}}(s,X_{s})\right]+\partial_{2}\left[h_{{J_{s}}}(s,X_{s})\right]\mu_{J_{{s}}}({s},X_{s})+\frac{1}{2}\partial_{22}\left[h_{{J_{s}}}(s,X_{s})\right]\sigma_{J_{{s}}}^{2}({s},X_{{s}})\right)\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\partial_{2}[h_{{J_{s}}}(s,X_{s})]\sigma_{J_{{s}}}({s},X_{{s}})\mathrm{d}B_{s}+\sum_{s\leq
t:J_{s}\neq J_{s-}}\left(h_{J_{s}}(s,X_{s})-h_{J_{s-}}(s,X_{s})\right).$

Since
$\partial_{2}\left[h_{{J_{t}}}(t,X_{t})\right]=1/\sigma_{J_{t}}(t,X_{t}),$ we
have
$\partial_{2}\left[h_{{J_{s}}}(s,X_{s})\right]\sigma_{J_{{s}}}({s},X_{{s}})\mathrm{d}B_{s}=\mathrm{d}B_{s}.$
Furthermore, as $\mathcal{X}$ is continuous, it follows that

$\displaystyle h_{J_{s}}(s,X_{s})$
$\displaystyle=h_{J_{s}}(s,X_{s^{-}})=h_{J_{s}}\left(s,h^{-1}_{J_{s^{-}}}(s,L_{s^{-}})\right),\quad\mbox{and}\quad$
$\displaystyle h_{J_{s-}}(s,X_{s})$
$\displaystyle=h_{J_{s-}}(s,X_{s^{-}})=L_{s^{-}},$

Finally, (3.3) follows by noticing that

$\displaystyle\partial_{22}\left[h_{{J_{t}}}(t,X_{t})\right]=-\frac{\partial_{2}\left[\sigma_{J_{t}}(t,X_{t})\right]}{\sigma_{J_{t}}^{2}(t,X_{t})}\quad\mbox{and}\quad
X_{t}=h_{{J_{t}}}^{-1}(t,L_{t}),$

and thus $\mathcal{L}$ solves (3.2). ∎

######  Remark 3.2.

_Since $h_{i}(t,\cdot)$ is invertible, the Lamperti transform is injective.
Thus, if we find the solution $\mathcal{L}$ to (3.2) and prove its uniqueness,
then we can recover the solution $\mathcal{X}$ to (1.1) by taking
$X_{t}:=h_{J_{t}}^{-1}(t,L_{t})$ for all $t\geq 0$._

######  Lemma 3.3.

Under Assumptions 1 and 2, for each $i\in\mathcal{E}$ the function
$\mu_{i}^{*}$, defined as in (3.3), is locally Lipschitz continuous and has
linear growth.

###### Proof.

Note that $\mu^{*}_{i}$ is formed by sums and compositions of
$\partial_{1}[h_{i}]$, $h_{i}^{-1}(t,\cdot)$, $\mu_{i}/\sigma_{i}$ and
$\partial_{2}[\sigma_{i}]$, all locally Lipschitz continuous and linearly
growing: the first three by Lemma 7.1 and the last one by Assumption 1(ii,
iii); the function $\partial_{2}[\sigma_{i}]$ is in fact bounded by the
Lipschitz continuity on the second entry of $\sigma_{i}$. This implies that
$\mu^{*}_{i}$, too, is locally Lipschitz and linearly growing. ∎

###  3.2 Pathwise construction and uniqueness

By a concatenation argument, Lemma 3.3 implies that under Assumptions 1 and 2
a unique càdlàg solution $\mathcal{L}$ to the regime-switching SDE (3.2)
exists. Given its importance, we give a proof of this in this section.

Here, we explicitly consider $\mathcal{L}$ as a path functional of
$\mathcal{B}$ and $\mathcal{J}$. Since $\mathcal{L}$ has a unit diffusion, the
pathwise construction of $\mathcal{L}$ between jumps is considerably easier
than in [5, 8, 24], as shown next. For fixed
$i\in\mathcal{E},b\in\mathds{R},r\geq 0$ consider the process
$\mathcal{Y}_{i,b,r}=\\{Y_{i,b,r}(t)\\}_{t\geq 0}$ where

$\displaystyle Y_{i,b,r}(t)$
$\displaystyle:=b+\int_{0}^{t}\mu_{i}^{*}\left(r+u,Y_{i,b,r}(u)+B_{r+u}-B_{r}\right)\mathrm{d}u,$
(3.4)

where $\mu^{*}_{i}$ is given in (3.3). Here, the parameter $b$ acts as the
starting point, and $r$ a displacement in time with respect to $\mathcal{B}$.
By Theorem 3.3, for any given realization of $\mathcal{B}$, a unique solution
$\mathcal{Y}_{i,b,r}$ to the system exists [23, Section V.2].

Next, define the process $\mathcal{S}=\\{S_{t}\\}_{t\geq 0}$ as follows. Let
$S_{0}=x_{0}$, and recursively over $n\geq 0$ let

$\displaystyle S_{t}$
$\displaystyle:=Y_{J_{n},S_{t_{n}},t_{n}}(t-t_{n})+B_{t}-B_{t_{n}}\quad\mbox{for}\quad
t\in(t_{n},t_{n+1}),$ (3.5) $\displaystyle S_{t_{n}}$
$\displaystyle:=h_{J_{n}}\left(t_{n},h_{J_{n-1}}^{-1}(t_{n},S_{t_{n}^{-}})\right),\quad
n\geq 1.$ (3.6)

######  Theorem 3.4.

The process $\mathcal{S}$ is the unique solution to the regime-switching
stochastic differential equation with jumps given by (3.2).

###### Proof.

Clearly, the process $\mathcal{S}$ is continuous on
$[0,\infty)\setminus\\{t_{k}\\}_{k\geq 1}$. Furthermore, since
$\lim_{s\downarrow 0}Y_{i,b,r}(s)=b$ for $i\in\mathcal{E},b\in\mathds{R},r\geq
0$, we have $\lim_{s\downarrow t_{n}}S_{s}=S_{t_{n}}$ for $n\geq 0$. In other
words, $\mathcal{S}$ is càdlàg.

Next, for $t\in(t_{n},t_{n+1})$

$\displaystyle S_{t}-S_{t_{n}}$
$\displaystyle=\left(Y_{J_{n},S_{t_{n}},t_{n}}(t-t_{n})+B_{t}-B_{t_{n}}\right)-S_{t_{n}}$
$\displaystyle=\int_{0}^{t-t_{n}}\mu_{J_{n}}^{*}\left(u+t_{n},Y_{J_{n},S_{t_{n}},t_{n}}(u)+B_{u+t_{n}}-B_{t_{n}}\right)\mathrm{d}u+B_{t}-B_{t_{n}}$
$\displaystyle=\int_{t_{n}}^{t}\mu^{*}_{J_{n}}\left(s,Y_{J_{n},S_{t_{n}},t_{n}}(u-t_{n})+B_{u}-B_{t_{n}}\right)\mathrm{d}u+B_{t}-B_{t_{n}}$
$\displaystyle=\int_{t_{n}}^{t}\mu_{J_{n}}^{*}(u,S_{u})\mathrm{d}u+B_{t}-B_{t_{n}}.$

Thus, $\mathcal{S}$ indeed evolves according to (3.2) in its continuity
points. Since for $n\geq 1$

$\displaystyle
S_{t_{n}}-S_{t_{n}^{-}}=h_{J_{n}}\left(t_{n},h_{J_{n-1}}^{-1}(t_{n},S_{t_{n}^{-}})\right)-S_{t_{n}^{-}},$

the discontinuities of $\mathcal{S}$, too, coincide with those given in (3.2).

Finally, the uniqueness of $\mathcal{S}$ follows by the uniqueness of
$\mathcal{Y}_{i,b,r}$ for $i\in\mathcal{E},b\in\mathds{R},r\geq 0$, and by the
fact that each point of discontinuity, say $S_{t_{n}}-S_{t_{n}^{-}}$, is
completely determined by $\mathcal{J}$ and $\\{S_{t}:t\in[0,t_{n})\\}$. ∎

Note that the Lamperti transformation introduced in Theorem 3.1 can be applied
to solutions of general stochastic differential equations driven by continuous
processes that are not standard Brownian motion. In that case, the pathwise
construction above can be generalised to build, and show the uniqueness of,
those solutions. In fact, this is what we shall need in the next section.

##  4 Rate of convergence for transformed processes

Consider a Wong-Zakai approximation to $\mathcal{X}$, denoted by
$\mathcal{X}^{\lambda}$, as given in Eqn (1.2).

######  Proposition 4.1.

The process $\mathcal{X}^{\lambda}$ defined by (1.2) is the unique solution of

$\displaystyle\mathrm{d}X_{t}^{\lambda}$
$\displaystyle=\left(\mu_{J_{t}}(t,X_{t}^{\lambda})-\frac{1}{2}\sigma_{J_{t}}(t,X^{\lambda}_{t})\partial_{2}\left[\sigma_{J_{t}}(t,X^{\lambda}_{t})\right]\right)\mathrm{d}t+\sigma_{J_{t}}(t,X_{t}^{\lambda})\mathrm{d}F_{t}^{\lambda},$
(4.1) $\displaystyle X_{0}^{\lambda}$ $\displaystyle=x_{0}.$ (4.2)

###### Proof.

By Assumption 1, $\sigma_{i}\partial_{2}[\sigma_{i}]$ is a product of two
locally Lipschitz functions, so it is itself locally Lipschitz. Furthermore,
$\sigma_{i}\partial_{2}[\sigma_{i}]$ is the product of a bounded function and
one with linear growth, so that $\sigma_{i}\partial_{2}[\sigma_{i}]$ has
linear growth. This implies uniqueness of the solution to the RSSDE (4.1). ∎

To show strong convergence from $\mathcal{X}^{\lambda}$ to $\mathcal{X}$, we
first apply the Lamperti transformation given in Theorem 3.1 to
$\mathcal{X}^{\lambda}$, $\mathcal{L}^{\lambda}=\\{L^{\lambda}_{t}\\}_{t\geq
0}=\\{h_{J_{t}}(t,X^{\lambda}_{t})\\}_{t\geq 0}$, which satisfies the SDE

$\displaystyle\mathrm{d}L^{\lambda}_{t}$
$\displaystyle=\mu^{*}_{J_{t}}(t,L^{\lambda}_{t})\mathrm{d}t+\mathrm{d}F^{\lambda}_{t}+\sum_{s\leq
t:J_{s}\neq
J_{s^{-}}}\left(h_{J_{s}}\left(s,h^{-1}_{J_{s^{-}}}(s,L^{\lambda}_{s^{-}})\right)-L^{\lambda}_{s^{-}}\right),$
(4.3) $\displaystyle L^{\lambda}_{0}$ $\displaystyle=h_{J_{0}}(0,x_{0}).$

By similar arguments to those in Section 3.2, we have that the unique solution
to the above SDE is satisfied by $\mathcal{S}^{\lambda}$ constructed as
follows. Let
$\mathcal{Y}^{\lambda}_{i,b,r}=\\{Y^{\lambda}_{i,b,r}(t)\\}_{t\geq 0}$ be
given by

$\displaystyle Y^{\lambda}_{i,b,r}(t)$
$\displaystyle:=b+\int_{0}^{t}\mu_{i}^{*}\left(r+u,Y^{\lambda}_{i,b,r}(u)+F^{\lambda}_{r+u}-F^{\lambda}_{r}\right)\mathrm{d}u.$
(4.4)

Next, define $\mathcal{S}^{\lambda}=\\{S^{\lambda}_{t}\\}_{t\geq 0}$, with
$S^{\lambda}_{0}=x_{0}$, as follows

$\displaystyle S^{\lambda}_{t}$
$\displaystyle:=Y^{\lambda}_{J_{n},S^{\lambda}_{t_{n}},t_{n}}(t-t_{n})+F^{\lambda}_{t}-F^{\lambda}_{t_{n}}\quad\mbox{for
}t\in(t_{n},t_{n+1}),n\geq 0,$ (4.5) $\displaystyle S_{t_{n}}^{\lambda}$
$\displaystyle:=h_{J_{n}}\left(t_{n},h_{J_{n-1}}^{-1}\left(t_{n},S^{\lambda}_{t_{n}^{-}}\right)\right),\quad
n\geq 1.$ (4.6)

Because of uniqueness, $\mathcal{S}$ is the same as $\mathcal{L}$, and
similarly $\mathcal{S}^{\lambda}$ as $\mathcal{L}^{\lambda}$. To emphasize
their pathwise constructions, we use the notation $\mathcal{S}$ and
$\mathcal{S}^{\lambda}$ for the rest of the paper. Here, we assess the rate of
strong convergence of $\mathcal{S}^{\lambda}$ to $\mathcal{S}$; in Section 5,
we apply the inverse transformation to show strong convergence from
$\mathcal{L}^{\lambda}$ to $\mathcal{L}$.

To prove that

$\displaystyle\lim_{\lambda\rightarrow\infty}\sup_{t\in[0,T]}\left|S_{t}-S^{\lambda}_{t}\right|=0\quad\mbox{
a.s. }\quad\mbox{ for all }T>0,$

and to compute its rate of convergence, without loss of generality we assume
$T=1$.

###  4.1 Two additional assumptions

In addition to Assumptions 1 and 2, we need the following:

######  Assumption 3.

For $i\in\mathcal{E}$, the function $\mu_{i}^{*}$ defined in (3.3) is
Lipschitz continuous on its second entry:

$\displaystyle\Big{|}\mu_{i}^{*}(t,x_{1})-\mu_{i}^{*}(t,x_{2})\Big{|}$
$\displaystyle\leq M^{*}_{i}|x_{1}-x_{2}|\quad\forall t\geq 0,x\in\mathds{R}.$

The following is an important special case for which Assumption 3 holds, and
shows that the time–inhomogeneous case is thus not a “direct” extension of the
time–homogeneous case, in the sense that some extra conditions may be needed.

######  Lemma 4.2.

If $\mu_{i}/\sigma_{i}$ and $\partial_{2}[\sigma_{i}]$ are Lipschitz
continuous on its second entry, then $\mu_{i}^{*}$ is Lipschitz continuous on
its second entry.

###### Proof.

In the case $\mu_{i}/\sigma_{i}$ and $\partial_{2}[\sigma_{i}]$ are Lipschitz
continuous on its second entry, then $\mu^{*}$ is formed by sums and second-
entry compositions of functions which are Lipschitz continuous on its second
entry, implying that $\mu^{*}$ is Lipschitz continuous on its second entry. ∎

From here on, we assume Assumption 3, not necessarily in the context of Lemma
4.2. Next, define

$\displaystyle N:=\\#\\{s\in(0,1]:J_{s}\neq J_{s^{-}}\\},$ (4.7)

the number of jumps of $\mathcal{J}$ on the interval $(0,1]$. We also assume
the following.

######  Assumption 4.

There exists some $\gamma_{0}>0$ such that $\mathds{P}(N>n)=o\left(e^{-n(\log
n-\gamma_{0})}\right)$.

Assumption 4 trivially holds if $\mathcal{J}$ is deterministic. The following
lemma shows how Assumption 4 arises naturally in common scenarios for random
jump processes.

######  Lemma 4.3.

Suppose that $N$ is stochastically dominated by $N^{*}\sim$_Poisson_ $(c)$,
where $c<\infty$. Then, Assumption 4 holds with $\gamma_{0}=\log c+1$.

###### Proof.

By definition, stochastic domination means $\mathds{P}(N\geq
n)\leq\mathds{P}(N^{*}\geq n)$ for $n\geq 0$. The asymptotic tail behaviour of
Poisson distribution [9, Cor. 1] implies that $\mathds{P}(N^{*}\geq
n)=O\left(c^{n}/n!\right)$. By Stirling’s formula [7, p. 52],

$\mathds{P}(N^{*}\geq n)=O\left(e^{n\log c-(n+1/2)\log
n+n}\right)=O\left(n^{-1/2}e^{-n(\log n-(\log c+1))}\right),$

which completes the proof. ∎

Lemma 4.3 implies that Assumption 4 holds whenever $\mathcal{J}$ has a bounded
jump intensity. This follows by choosing a sufficiently large $c>0$ such that
each holding time of $\mathcal{J}$ stochastically dominates an exponential
variable with rate $c$, implying that $N^{*}\sim\mbox{Poisson}(c)$
stochastically dominates $N$. Thus, Lemma 4.3 allows us to work with jump
processes that are not necessarily time–homogeneous Markovian, such as
time–inhomogeneous Markovian or semi–Markovian processes.

###  4.2 From $\mathcal{S}^{\lambda}$ to $\mathcal{S}$

We write $\|\mathcal{U}\|_{\infty}$ to denote $\sup_{t\in[0,1]}|U_{t}|$ for a
process $\mathcal{U}=\\{U_{t}\\}_{t\geq 0}\in\mathcal{D}(0,\infty)$. The
following lemma bounds the difference between two processes
$\mathcal{Y}^{\lambda}_{i,a,r}$ and $\mathcal{Y}_{i,b,r}$ over the time
interval $[0,1]$, given that the two processes have different starting points.

######  Lemma 4.4.

Let $a,b\in\mathds{R}$ and $r\in[0,1]$. Then, for all $i\in\mathcal{E}$,

$\displaystyle\Big{|}Y^{\lambda}_{i,a,r}(t)-Y_{i,b,r}(t)\Big{|}\leq
K_{1}\left(\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}+|a-b|\right)\quad\forall
t\in[0,1-r],$ (4.8)

where $K_{1}:=(2\overline{M}\vee 1)\exp(\overline{M})$ and
$\overline{M}:=\max_{i\in\mathcal{E}}M^{*}_{i}$.

###### Proof.

For $t\in[0,1-r]$,

$\displaystyle\Big{|}Y^{\lambda}_{i,a,r}(t)-Y_{i,b,r}(t)\Big{|}$
$\displaystyle\leq\int_{0}^{t}\Big{|}\mu_{i}^{*}\left(r+s,Y^{\lambda}_{i,a,r}(s)+F^{\lambda}_{r+s}-F^{\lambda}_{r}\right)-\mu_{i}^{*}\left(r+s,Y^{\lambda}_{i,b,r}(s)+B_{r+s}-B_{r}\right)\Big{|}\mathrm{d}s+|a-b|$
$\displaystyle\leq\overline{M}\left(\int_{0}^{t}\left|Y^{\lambda}_{i,a,r}(s)-Y_{i,b,r}(s)\right|+\left|\left(F^{\lambda}_{r+s}-F^{\lambda}_{r}\right)-\left(B_{r+s}-B_{r}\right)\right|\mathrm{d}s\right)+|a-b|$
$\displaystyle\leq\overline{M}\int_{0}^{t}\left|Y^{\lambda}_{i,a,r}(s)-Y_{i,b,r}(s)\right|\mathrm{d}s+2\overline{M}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}+|a-b|.$

Gronwall’s Inequality implies

$\displaystyle\Big{|}Y^{\lambda}_{i,a,r}(t)-Y_{i,b,r}(t)\Big{|}\leq\left(2\overline{M}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}+|a-b|\right)\exp(\overline{M}).$

Take $K_{1}:=(2\overline{M}\vee 1)\exp(\overline{M})$ and (4.8) follows. ∎

In general, $\mathcal{S}$ and $\mathcal{S}^{\lambda}$ do not coincide at jump
epochs $\\{t_{k}\\}_{k\geq 1}$; Lemma 4.4 helps bounding the error
_accumulated_ at each jump time up to $T=1$, as it is shown next.

######  Theorem 4.5.

On the event $\\{N=u\\}$, for $u\in\mathds{N}$,

$\displaystyle\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty}\leq
K_{2}K_{3}^{u}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty},$ (4.9)

where $N$ is the number of jumps of $\mathcal{J}$ on the time interval $[0,1]$
given by (4.7), and $K_{2}:=K_{3}(2+K_{1})/(K_{3}-1)$ and $K_{3}:=K_{1}V/v$.

###### Proof.

Let $n\in\\{0,1,\dots,u\\}$ and fix $t\in[t_{n},t_{n+1}\wedge 1)$. Then,

$\displaystyle\left|S^{\lambda}_{t}-S_{t}\right|$
$\displaystyle=\left|Y^{\lambda}_{J_{n},S^{\lambda}_{t_{n}},t_{n}}(t-t_{n})+F^{\lambda}_{t}-F^{\lambda}_{t_{n}}-\left(Y_{J_{n},S_{t_{n}},t_{n}}(t-t_{n})+B_{t}-B_{t_{n}}\right)\right|$
$\displaystyle\leq\left|Y^{\lambda}_{J_{n},S^{\lambda}_{t_{n}},t_{n}}(t-t_{n})-Y_{J_{n},S_{t_{n}},t_{n}}(t-t_{n})\right|+2\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}$
$\displaystyle\leq
K_{1}\left(\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}+\left|S^{\lambda}_{t_{n}}-S_{t_{n}}\right|\right)+2\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty},$
(4.10)

by Lemma 4.4. By definition,

$\displaystyle\left|S^{\lambda}_{t_{n}}-S_{t_{n}}\right|$
$\displaystyle=\left|h_{J_{n}}\left(t_{n},h_{J_{n-1}}^{-1}(t_{n},S^{\lambda}_{t_{n}^{-}})\right)-h_{J_{n}}\left(t_{n},h_{J_{n-1}}^{-1}(t_{n},S_{t_{n}^{-}})\right)\right|$
$\displaystyle\leq\frac{1}{v}\left|h_{J_{n-1}}^{-1}\big{(}t_{n},S^{\lambda}_{t_{n}^{-}}\big{)}-h_{J_{n-1}}^{-1}\left(t_{n},S_{t_{n}^{-}}\right)\right|$
$\displaystyle\leq\frac{V}{v}\left|S^{\lambda}_{t_{n}^{-}}-S_{t_{n}^{-}}\right|,$
(4.11)

where for the last two inequalities we used the Lipschitz continuity on the
second entry of $h_{i}$ and $h_{i}^{-1}$, respectively, which is proven in
Lemma 7.1(i).

Let $\rho:=K_{1}V/v$. Substituting (4.11) into (4.10), applying the same steps
recursively for other jump epochs, and using $S^{\lambda}_{0}-S_{0}=0$, we
have

$\displaystyle\left|S^{\lambda}_{t}-{S}_{t}\right|$
$\displaystyle\leq\sum_{k=0}^{n}\rho^{k}(2+K_{1})\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}$
$\displaystyle=\frac{\rho^{u+1}-1}{\rho-1}(2+K_{1})\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}$
$\displaystyle\leq\left[\frac{\rho(2+K_{1})}{\rho-1}\right]\rho^{u}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty},$

so that (4.9) follows. ∎

Theorem 4.5 implies that the error bound increases geometrically with respect
to the number of jump occurrences in $[0,1]$. In general this would be
problematic because the error keeps building up with each jump of
$\mathcal{J}$—this is the main challenge when dealing with regime switching.
However, as will be seen in Theorem 4.6, if the tail distribution of $N$ is
_light enough_ , the error bound given by (4.9) will still be controllable.
Recall that Assumption 4 gives the property of the tail distribution being
light enough.

Now we are ready to assess the rate of convergence of $\mathcal{S}^{\lambda}$
to $\mathcal{S}$.

######  Theorem 4.6.

Under Assumptions 1–4, suppose that there exists
$\delta:\mathds{R}_{+}\mapsto\mathds{R}_{+}$ with
$\lim_{\lambda\rightarrow\infty}\delta(\lambda)=0$ such that for all $q>0$

$\mathds{P}\left(\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}\geq\alpha\delta(\lambda)\right)=o(\lambda^{-q}),$

where $\alpha=\alpha(q)>0$ is some constant dependent on $q$ only. Then, there
exists some constant $\beta=\beta(q)>0$ such that for all $\varepsilon>0$

$\displaystyle\mathds{P}\left(\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)=o\left(\lambda^{-q}\right).$
(4.12)

Theorem 4.6 implies that under Assumptions 1–4, as $\lambda\rightarrow\infty$
the unique solution $\mathcal{S}^{\lambda}$ of the regime-switching, time-
inhomogeneous SDE (4.3) converges strongly to $\mathcal{S}$.

###### Proof.

We will determine the value of $\beta=\beta(q)$ towards the end of the proof;
for now let it be arbitrary but fixed. Define $f_{\gamma_{0},q}:=\exp(x(\log
x-\gamma)/q)$ and let $A_{\lambda}:=f_{\gamma_{0},q}^{-1}(\lambda)$, where
$\gamma_{0}$ is from Assumption 4. Let $W$ denote _the Lambert $W$ function_,
the inverse of the mapping $z\mapsto ze^{z}$. By Lemma 7.9 in the Appendix,

$\displaystyle
A_{\lambda}=e^{\gamma_{0}}\exp\left\\{W(qe^{-\gamma_{0}}\log\lambda)\right\\},$

which implies $e^{A_{\lambda}((\log A_{\lambda})-\gamma_{0})/q}=\lambda$. By
Assumption 4,

$\displaystyle\mathds{P}(N>A_{\lambda})=o\left(e^{-A_{\lambda}((\log
A_{\lambda})-\gamma_{0})}\right)=o(\lambda^{-q}).$ (4.13)

Next, fix $\varepsilon>0$. By Theorem 4.5,

$\displaystyle\mathds{P}$
$\displaystyle\left(\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)$
$\displaystyle\leq\mathds{P}\left(N\leq
A_{\lambda},\;\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)+\mathds{P}\left(N>A_{\lambda}\right)$
$\displaystyle\leq\mathds{P}\left(K_{2}K_{3}^{A_{\lambda}}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)+\mathds{P}\left(N>A_{\lambda}\right)$
$\displaystyle=\mathds{P}\left(K_{2}K_{3}^{A_{\lambda}}\lambda^{-\varepsilon}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}\geq\beta\delta(\lambda)\right)+\mathds{P}\left(N>A_{\lambda}\right).$
(4.14)

Note that

$\displaystyle\lim_{\lambda\rightarrow\infty}K_{3}^{A_{\lambda}}\lambda^{-\varepsilon}$
$\displaystyle=\lim_{\lambda\rightarrow\infty}\exp\left(A_{\lambda}\log
K_{3}-\varepsilon\log\lambda\right).$ (4.15)

By the asymptotic decomposition of Lambert W functions [4, Eqn (4.18)] given
by $W(x)=\log x-\log(\log x)+o(1)$ as $x\rightarrow\infty,$ we obtain

$\displaystyle\lim_{\lambda\rightarrow\infty}A_{\lambda}$
$\displaystyle=\lim_{\lambda\rightarrow\infty}e^{\gamma_{0}}\exp\left\\{\log\left(qe^{-\gamma_{0}}\log\lambda\right)-\log\left(\log\left(qe^{-\gamma_{0}}\log\lambda\right)\right)+o(1)\right\\}$
$\displaystyle=\lim_{\lambda\rightarrow\infty}e^{\gamma_{0}}\frac{qe^{-\gamma_{0}}\log\lambda}{\log\left(qe^{-\gamma_{0}}\log\lambda\right)}.$
(4.16)

Substituting (4.16) into (4.15) gives

$\displaystyle\lim_{\lambda\rightarrow\infty}$ $\displaystyle
K_{3}^{A_{\lambda}}\lambda^{-\varepsilon}=\exp\left(e^{\gamma_{0}}\log
K_{3}\lim_{\lambda\rightarrow\infty}\left\\{\frac{q\log
K_{3}}{\log\left(qe^{-\gamma_{0}}\log\lambda\right)}-\varepsilon\right\\}\log\lambda\right)=0.$

Let $\lambda_{0}=\lambda_{0}(\varepsilon)>0$ be such that
$K_{3}^{A_{\lambda}}\lambda^{-\varepsilon}\leq 1$ for all
$\lambda\geq\lambda_{0}$. Take $\beta:=K_{2}K_{1}\alpha$, then for all
$\lambda\geq\lambda_{0}$,

$\displaystyle\mathds{P}\left(K_{2}K_{3}^{A_{\lambda}}\lambda^{-\varepsilon}\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}\geq\beta\delta(\lambda)\right)$
$\displaystyle\leq\mathds{P}\left(\|\mathcal{F}^{\lambda}-\mathcal{B}\|_{\infty}\geq\alpha\delta(\lambda)\right)$
$\displaystyle=o(\lambda^{-q})\quad\mbox{as}\quad\lambda\rightarrow\infty,$

and so by (4.13) and (4.14) we complete the proof. ∎

##  5 Rate of strong convergence from $\mathcal{X}^{\lambda}$ to $\mathcal{X}$

Now that we have obtained a rate of convergence of
$\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty}$, we return to the original
setting: approximating the solution $\mathcal{X}$ to the regime-switching
time-inhomogeneous stochastic differential equation (1.1).

In the following we assess the distance between $\mathcal{X}$ and
$\mathcal{X}^{\lambda}$ of (4.1) in terms of the distance between their
respective transformations, $\mathcal{S}$ and $\mathcal{S}^{\lambda}$.

######  Proposition 5.1.

$\displaystyle\|\mathcal{X}^{\lambda}-\mathcal{X}\|_{\infty}\leq
V\|\mathcal{S}^{\lambda}-\mathcal{S}\|_{\infty},$ (5.1)

where $V$ is the uniform upper bound for $\sigma_{i}$, $i\in\mathcal{E}$.

###### Proof.

As $X_{t}=h_{J_{t}}^{-1}(t,S_{t})$ and
$X_{t}^{\lambda}=h_{J_{t}}^{-1}(t,S^{\lambda}_{t})$, Inequality (5.1) follows
from the Lipschitz continuity on the second entry of $h_{i}$, as shown in
Lemma 7.1(i). ∎

Theorem 1.1 then follows directly from Theorems 4.6 and Proposition 5.1.

######  Remark 5.2.

_Even though Theorem 4.6 and Proposition 5.1 concern the case $T=1$, their
generalization to arbitrary $T>0$ is straightforward and thus Theorem 1.1
indeed follows._

##  6 Special case: Markov-modulated Brownian motion process

Here, we focus on the special case where the finite–variation process
$\mathcal{F}^{\lambda}=\\{F^{\lambda}_{t}\\}_{t\geq 0}$ is a _uniform
transport process_ [11] (also known as the _telegraph process_ [15] or _flip-
flop process_ [17]):

$F^{\lambda}_{t}:=\sqrt{\lambda}\int_{0}^{t}(-1)^{M_{s}^{\lambda}}\mathrm{d}s,$

where $\\{M_{t}^{\lambda}\\}_{t\geq 0}$ is a Poisson process of intensity
$\lambda>0$. Through two different constructions, Gorostiza and Griego [10]
and Nguyen and Peralta [21] showed that there exists a probability space on
which $\mathcal{F}^{\lambda}$ converges strongly to $\mathcal{B}$, at rates
$\delta_{1}(\lambda)=\lambda^{-1/4}(\log\lambda)^{5/2}$ and
$\delta_{2}(\lambda)=\lambda^{-1/4}\log\lambda$, respectively.

Now, for $i\in\mathcal{E}$, let $\mu_{i}$ and $\sigma_{i}$ be fixed constants.
Then, $\mathcal{X}$ and $\mathcal{X}^{\lambda}$ respectively take the form

$\displaystyle X_{t}$
$\displaystyle=x_{0}+\int_{0}^{t}\mu_{J_{s}}\mathrm{d}s+\int_{0}^{t}\sigma_{J_{s}}\mathrm{d}B_{s},\quad\mbox{and}$
$\displaystyle X_{t}^{\lambda}$
$\displaystyle=x_{0}+\int_{0}^{t}\mu_{J_{s}}\mathrm{d}s+\int_{0}^{t}\sigma_{J_{s}}\mathrm{d}F_{s}^{\lambda}.$

When $\mathcal{J}$ is a time-homogeneous Markovian jump process independent of
$\mathcal{B}$, the process $\mathcal{X}$ is called a _Markov-modulated
Brownian motion_. In this context, Latouche and Nguyen [17] proved that
$\mathcal{X}^{\lambda}$ converges weakly to $\mathcal{X}$. Theorem 1.1,
together with Theorem 3.1 in [20], implies that such convergence also holds in
a strong sense.

Significantly, Theorem 1.1 is applicable in much more general settings, not
only in terms of the drift and diffusion coefficients but also in terms of the
jump process $\mathcal{J}$; the latter can be, for example,
_time–inhomogeneous_ or _semi–Markov_.

## Acknowledgement

The authors acknowledge the funding of the Australian Research Council
Discovery Project DP180103106.

##  7 Appendix

###  7.1 Properties of the Lamperti transformation

The following are some standalone results needed throughout the main text.

######  Lemma 7.1.

Suppose Assumptions 1 and 2 hold. Then, for $i\in\mathcal{E}$,

  1. _(i)_

$h_{i}$, $h_{i}^{-1}$ and $\partial_{1}[h_{i}]$ are Lipschitz continuous on
their second entry,

  2. _(ii)_

$h_{i}$ is locally Lipschitz continuous and of linear growth,

  3. _(iii)_

$h^{-1}$ is locally Lipschitz continuous and of linear growth, where
$h^{-1}(t,y):=x$ such that $h(t,x)=y$.

  4. _(iv)_

$\partial_{1}[h_{i}]$ is locally Lipschitz continuous and of linear growth,

  5. _(v)_

$\mu_{i}/\sigma_{i}$ is locally Lipschitz continuous and of linear growth.

###### Proof.

(i) By Assumption 2(i), there exist $V<\infty,v>0$ such that $v<\sigma_{i}<V$.
Then,

$\displaystyle\Big{|}h_{i}(t,x_{1})-h_{i}(t,x_{2})\Big{|}=\left|\int_{x_{1}}^{x_{2}}\frac{1}{\sigma_{i}(t,y)}\mathrm{d}y\right|\leq\frac{1}{v}|x_{1}-x_{2}|,\quad
t\geq 0,\,x_{1},x_{2}\in\mathds{R},$

which implies $h_{i}$ is Lipschitz continuous on its second entry.

Next,

$\displaystyle\Big{|}h_{i}(t,x_{1})-h_{i}(t,x_{2})\Big{|}=\left|\int_{x_{1}}^{x_{2}}\frac{1}{\sigma_{i}(t,y)}\mathrm{d}y\right|\geq\frac{1}{V}|x_{1}-x_{2}|,\quad
t\geq 0,\,x_{1},x_{2}\in\mathds{R}.$ (7.1)

Substituting $x_{1}:=h_{i}^{-1}(t,y_{1})$ and $x_{2}:=h_{i}^{-1}(t,y_{2})$
into (7.1) implies that

$\displaystyle\Big{|}h^{-1}_{i}(t,y_{1})-h^{-1}_{i}(t,y_{2})\Big{|}\leq
V|y_{1}-y_{2}|,\quad t\geq 0,\,y_{1},\,y_{2}\in\mathds{R},$

meaning $h_{i}^{-1}$ is Lipschitz continuous on its second entry.

By Assumption 2(ii),

$\displaystyle\Big{|}\partial_{1}\left[h_{i}(t,x_{1})\right]-\partial_{1}\left[h_{i}(t,x_{2})\right]\Big{|}$
$\displaystyle=\left|\int_{x_{1}}^{x_{2}}\frac{\partial_{1}\left[\sigma_{i}(t,y)\right]}{\sigma_{i}^{2}(t,y)}\mathrm{d}y\right|$
$\displaystyle\leq K|x_{1}-x_{2}|,\quad t\geq 0,\,x_{1},x_{2}\in\mathds{R},$
(7.2)

implying that $\partial_{1}\left[h_{i}\right]$ is Lipschitz continuous on its
second entry.

For Parts (ii)–(v), without loss of generality we focus on a compact set
$\mathcal{K}=[t_{1},t_{2}]\times[y_{1},y_{2}]$, where $0\leq t_{1}\leq t_{2}$,
$y_{1}\leq x_{0}\leq y_{2}$, and let
$(s_{1},x_{1}),(s_{2},x_{2})\in\mathcal{K}$.

(ii) As $h_{i}$ is Lipschitz continuous on its second entry,

$\displaystyle\Big{|}h_{i}(s_{1},x_{1})-h_{i}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq\Big{|}h_{i}(s_{1},x_{1})-h_{i}(s_{1},x_{2})\Big{|}+\Big{|}h_{i}(s_{1},x_{2})-h_{i}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq\frac{1}{v}|x_{1}-x_{2}|+\Big{|}h_{i}(s_{1},x_{2})-h_{i}(s_{2},x_{2})\Big{|}.$

Since $|\partial_{1}[\sigma_{i}]|/\sigma_{i}^{2}\leq K$ by Assumption 2(ii),

$\displaystyle\Big{|}\partial_{1}[h_{i}(t,x)]\Big{|}=\left|\int_{x_{0}}^{x}\frac{\partial_{1}[\sigma_{i}(t,y)]}{\sigma_{i}^{2}(t,y)}\mathrm{d}y\right|\leq
K|x-x_{0}|.$ (7.3)

The mean value theorem gives $|h_{i}(s_{1},x_{2})-h_{i}(s_{2},x_{2})|\leq
K|x_{2}-x_{0}||s_{1}-s_{2}|,$ which leads to

$\displaystyle\Big{|}h_{i}(s_{1},x_{1})-h_{i}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq\frac{1}{v}|x_{1}-x_{2}|+K|x_{2}-x_{0}||s_{1}-s_{2}|$
$\displaystyle\leq\frac{1}{v}|x_{1}-x_{2}|+K|y_{1}-y_{2}||s_{1}-s_{2}|$ (7.4)

and thus $h_{i}$ is locally Lipschitz. The Lipschitz continuity on the second
entry of $h_{i}$ implies that

$\Big{|}h_{i}(t,x)\Big{|}\leq\Big{|}h_{i}(t,x_{0})\Big{|}+\frac{1}{v}|x-x_{0}|=\frac{1}{v}|x-x_{0}|,$

so that $h_{i}$ has linear growth.

(iii) We have

$\displaystyle\Big{|}h_{i}^{-1}(s_{1},x_{1})-h_{i}^{-1}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq\Big{|}h_{i}^{-1}(s_{1},x_{1})-h_{i}^{-1}(s_{1},x_{2})\Big{|}+\Big{|}h_{i}^{-1}(s_{1},x_{2})-h_{i}^{-1}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq
V\left|x_{1}-x_{2}\right|+\Big{|}h_{i}^{-1}(s_{1},x_{2})-h_{i}^{-1}(s_{2},x_{2})\Big{|}.$
(7.5)

Note that $h_{i}^{-1}(t,0)=x_{0}$; furthermore,

$\displaystyle\partial_{2}\left[h_{i}^{-1}(t,x)\right]$
$\displaystyle=\frac{1}{\partial_{2}\left[h_{i}\left(t,h_{i}^{-1}(t,x)\right)\right]}=\sigma_{i}\left(t,h_{i}^{-1}(t,x)\right),$

Thus,

$\displaystyle
h_{i}^{-1}(t,x)=x_{0}+\int_{0}^{x}\sigma_{i}\left(t,h_{i}^{-1}(t,y)\right)\mathrm{d}y.$
(7.6)

Without loss of generality, suppose that $x_{2}\geq 0$. Then

$\displaystyle\Big{|}h_{i}^{-1}(s_{1},x_{2})-h_{i}^{-1}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq\int_{0}^{x_{2}}\Big{|}\sigma_{i}\left(s_{1},h_{i}^{-1}(s_{1},y)\right)-\sigma_{i}\left(s_{2},h_{i}^{-1}(s_{2},y)\right)\Big{|}\mathrm{d}y$
$\displaystyle\leq\int_{0}^{x_{2}}G_{\mathcal{K}}\left(\left|s_{1}-s_{2}\right|+\Big{|}h_{i}^{-1}(s_{1},y)-h_{i}^{-1}(s_{2},y)\Big{|}\right)\mathrm{d}y$
$\displaystyle\leq
G_{\mathcal{K}}x_{2}|s_{1}-s_{2}|+G_{\mathcal{K}}\int_{0}^{x_{2}}\Big{|}h_{i}^{-1}(s_{1},y)-h_{i}^{-1}(s_{2},y)\Big{|}\mathrm{d}y,$

where, by Assumption 1(ii), $G_{\mathcal{K}}$ is the local Lipschitz constant
of $\sigma_{i}$ on $\mathcal{K}$.

Gronwall’s Inequality implies that

$\displaystyle\Big{|}h_{i}^{-1}(s_{1},x_{2})-h_{i}^{-1}(s_{2},x_{2})\Big{|}\leq
G_{\mathcal{K}}x_{2}|s_{1}-s_{2}|\exp(G_{\mathcal{K}}x_{2}).$ (7.7)

Substituting (7.7) into (7.5) gives

$\displaystyle\Big{|}h_{i}^{-1}(s_{1},x_{1})-h_{i}^{-1}(s_{2},x_{2})\Big{|}$
$\displaystyle\leq
V|x_{1}-x_{2}|+G_{\mathcal{K}}x_{2}|s_{1}-s_{2}|\exp(G_{\mathcal{K}}x_{2})$
$\displaystyle\leq
V|x_{1}-x_{2}|+G_{\mathcal{K}}y_{2}|s_{1}-s_{2}|\exp(G_{\mathcal{K}}y_{2})$
(7.8)

and so $h_{i}^{-1}$ is locally Lipschitz continuous.

The linear growth of $h_{i}^{-1}$ is implied by the inequality
$\Big{|}h_{i}^{-1}(t,x)\Big{|}\leq|x_{0}|+V|x|.$

(iv) Note that, by (7.2),

$\displaystyle\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{1})\right]-\partial_{1}\left[h_{i}(s_{2},x_{2})\right]\Big{|}$
$\displaystyle\leq\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{1})\right]-\partial_{1}\left[h_{i}(s_{1},x_{2})\right]\Big{|}+\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{2})\right]-\partial_{1}\left[h_{i}(s_{2},x_{2})\right]\Big{|}$
$\displaystyle\leq
K|x_{1}-x_{2}|+\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{2})\right]-\partial_{1}\left[h_{i}(s_{2},x_{2})\right]\Big{|}.$

As $v\leq\sigma_{i}\leq V$ and $\sigma_{i}$ is locally Lipschitz,
$1/\sigma_{i}$ is also locally Lipschitz. Since the product of locally
Lipschitz is locally Lipschitz, by Assumption 1(ii, iii) the function
$\partial_{1}[\sigma_{i}]/\sigma_{i}^{2}$, too, is locally Lipschitz. This
implies the existence of $Q_{\mathcal{K}}>0$ such that

$\displaystyle\left|\frac{\partial_{1}\left[\sigma_{i}(s_{1},x_{1})\right]}{\sigma_{i}^{2}(s_{1},x_{1})}-\frac{\partial_{1}\left[\sigma_{i}(s_{1},x_{1})\right]}{\sigma_{i}^{2}(s_{1},x_{1})}\right|\leq
Q_{\mathcal{K}}\left(|s_{1}-s_{2}|+|x_{1}-x_{2}|\right)$

Thus,

$\displaystyle\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{2})\right]-\partial_{1}\left[h_{i}(s_{2},x_{2})\right]\Big{|}$
$\displaystyle=\left|\int_{x_{0}}^{x_{2}}\left\\{\frac{\partial_{1}[\sigma_{i}(s_{1},y)]}{\sigma_{i}^{2}(s_{1},y)}-\frac{\partial_{1}[\sigma_{i}(s_{2},y)]}{\sigma_{i}^{2}(s_{2},y)}\right\\}\mathrm{d}y\right|$
$\displaystyle\leq\left|\int_{x_{0}}^{x_{2}}Q_{\mathcal{K}}|s_{1}-s_{2}|\mathrm{d}y\right|$
$\displaystyle=Q_{\mathcal{K}}|x_{0}-x_{2}||s_{1}-s_{2}|$ $\displaystyle\leq
Q_{\mathcal{K}}(y_{2}-y_{1})|s_{1}-s_{2}|.$

Thus,

$\displaystyle\Big{|}\partial_{1}\left[h_{i}(s_{1},x_{1})\right]-\partial_{1}\left[h_{i}(s_{2},x_{2})\right]\Big{|}$
$\displaystyle\leq K|x_{1}-x_{2}|+Q_{\mathcal{K}}(y_{2}-y_{1})|s_{1}-s_{2}|,$

which means $\partial_{1}h_{i}$ is locally Lipschitz continuous. The linear
growth condition follows from (7.3).

(v) As $\mu_{i}/\sigma_{i}$ is the product of two locally Lipschitz continuous
functions, it is locally Lipschitz. Its linear growth follows from the linear
growth of $\mu_{i}$ and the boundedness of $\sigma_{i}$. ∎

###  7.2 Inverse of function $f_{\gamma,q}$

######  Lemma 7.2.

For $\gamma,q>0$, let $f_{\gamma,q}:[e^{\gamma},\infty)\mapsto[1,\infty)$ be
defined by

$\displaystyle f_{\gamma,q}(x):=\exp(x(\log x-\gamma)/q).$ (7.9)

Then $f_{\gamma,q}$ is bijective with inverse given by

$\displaystyle f_{\gamma,q}^{-1}(y)=e^{\gamma}\exp\left(W(qe^{-\gamma}\log
y)\right),$ (7.10)

where $W$ denotes _the Lambert $W$ function_, the inverse of the mapping
$z\mapsto ze^{z}$.

###### Proof.

The bijectivity of $f_{\gamma,q}$ follows from the fact that it is strictly
increasing, with $\lim_{x\rightarrow\infty}f_{\gamma,q}(x)=\infty$. Let $y$ be
in the image of $f_{\gamma,q}$, so that it is of the form

$y=\exp(x(\log x-\gamma)/q)\quad\mbox{for some }x\geq e^{\gamma}.$

Taking the log of both sides and multiplying $qe^{-\gamma}$ across gives

$\displaystyle qe^{-\gamma}\log y=e^{-\gamma}x(\log x-\gamma).$ (7.11)

Note that $e^{-\gamma}x(\log x-\gamma)=e^{-\gamma}x(\log
x+\log(e^{-\gamma}))=\log(xe^{-\gamma})e^{\log(xe^{-\gamma})}$. Let
$z:=\log(xe^{-\gamma})$, we have that (7.11) is equivalent to
$qe^{-\gamma}\log y=ze^{z}.$ Thus,

$\displaystyle\log(xe^{-\gamma})=W\left(qe^{-\gamma}\log y\right).$ (7.12)

Solving (7.12) for $x$ gives (7.10). ∎

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