# Space-grid approximations of hybrid stochastic differential equations and
first passage properties

Hansjoerg Albrecher  Faculty of Business and Economics, University of
Lausanne, Quartier de Chambronne, 1015 Lausanne, Switzerland
<EMAIL_ADDRESS>and  Oscar Peralta  Faculty of Business and
Economics, University of Lausanne, Quartier de Chambronne, 1015 Lausanne,
Switzerland<EMAIL_ADDRESS>
###### Abstract.

Hybrid stochastic differential equations are a useful tool to model
continuously varying stochastic systems which are modulated by a random
environment that may depend on the system state itself. In this paper, we
establish the pathwise convergence of the solutions to hybrid stochastic
differential equations via space-grid discretizations. While time-grid
discretizations are a classical approach for simulation purposes, our space-
grid discretization provides a link with multi-regime Markov modulated
Brownian motions, leading to computational tractability. We exploit our
convergence result to obtain efficient approximations to first passage
probabilities and expected occupation times of the solutions hybrid stochastic
differential equations, results which are the first of their kind for such a
robust framework. We finally illustrate the performanced of the resulting
approximations in numerical examples.

##  1\. Introduction

Stochastic systems that enjoy some sort of modulation have attracted
considerable attention in probability, both from applied and theoretical
perspectives. Examples of these are Markov-modulated Poisson processes [14],
Markov additive processes [5, Chapter XI] and regime-switching stochastic
differential equations [28, Section II.2], which have found applications in
risk theory [6, Chapter VII], queueing [27] and finance [13], just to name a
few. The main appeal behind this framework comes from the flexibility when
modelling phenomena whose parameters depend on random environmental factors.
For instance, modulation may be used to model correlated catastrophic events
in insurance, different workload regimes in a queue, or seasonal sentiment
changes in financial markets. In case the modulation is Markovian, a robust
toolbox of matrix-analytic methods has been developed to compute relevant
qualities; see e.g. [9] and references therein.

In this manuscript, we are interested in a class of processes whose dynamics
are described by a continuously varying component which arises as the solution
of a stochastic differential equation, and an environmental finite-state
component which is modelled after a jump process that may or may not be
Markovian. Such a class of processes is referred to in the literature as
_hybrid stochastic differential equations_ (hybrid SDEs), which generically
take the form

(1.1)
$\mathrm{d}X(t)=\mu(J(t),X(t))\mathrm{d}t+\sigma(J(t),X(t))\mathrm{d}B(t),$

where $B$ is a Brownian motion, $J$ is the environmental process, and $\mu$
and $\sigma$ are real functions which satisfy certain regularity conditions.

In particular, we will be interested in hybrid SDEs which exhibit the
following characteristics:

  * •

The process $X$ is continuous and one-dimensional.

  * •

At a given instant, the environmental process switches state with an intensity
that is dependent on the main component.

In simpler words, the class of hybrid SDEs we are interested in has components
whose evolution is interlaced and dependent on each other. This contrasts with
the Markov-modulated case, where one is able to draw a path of the
environmental component without any knowledge of the main one (see [25]). In
order to highlight this interlacing feature, in the literature such a model is
referred to as hybrid SDEs with state-dependent switching; here we will simply
call them hybrid SDEs for brevity.

Existence, uniqueness and stability properties for hybrid SDEs have been
extensively studied in recent years [33, 24, 34]. Another stream of research
lies in investigating efficient simulation methods of hybrid SDEs, most of
them relying on adapating well-known convergence results into this
considerably more challenging scenario [32, 25]. However, computing explicit
probabilistic descriptors for such a class of processes has been proven
challenging, even in simple scenarios. For instance, several descriptors for
Markov-modulated Brownian motion have been explicitly obtained in the
literature (see e.g. [4, 20, 11, 12, 26]), but virtually none of these results
have been extended to hybrid SDEs, even in the Markovian case. Such a task
seems intractable simply because the toolbox for general diffusions is
considerably more limited than the one available for the Brownian motion.

Our contributions to the literature of hybrid SDEs are the following. In a
first instance, we provide a novel pathwise approximation technique by means
of a space-grid discretization, resulting in approximations which belong to
the class of multi-regime Markov-modulated Brownian motions. The latter are
processes that, when restricted to each band of the space-grid, behave like a
Markov-modulated Brownian motion. While Wong-Zakai or Euler-Maruyama are the
most common approximation methods for simulation of hybrid SDEs in the
literature, our proposed approximation sets the course to exploit aspects that
are currently only knowm for multi-regime Markov-modulated Brownian motions.
As an example, we employ our pathwise approximation result to provide
approximations for the descriptors for first exit times of a hybrid SDEs over
a band $[0,a]$ , $a>0$, using recent results on the stationary measures of
multi-regime Markov-modulated Brownian motion in queueing theory [18, 1]. We
remark that, to the best of our knowledge, this is the first attempt to
compute first passage probabilities and expected occupation times for hybrid
SDEs, even when reduced to the case in which the environmental process is
Markovian.

The structure of this paper is as follows. In Section 2, we provide the proper
framework to construct strong solutions to hybrid SDEs via the uniformization
method and concatenation of paths. Later, in Section 3, we construct the
proposed multi-regime Markov-modulated Brownian motion approximation, and
prove its uniform convergence in probability to the original hybrid SDE over
increasing compact intervals. This approximation result is carried over in two
steps by considering an auxiliary process, which serves as a middle point to
prove our main result in Theorem 3.6. In Section 4 we show how our convergence
result can be applied to approximate first passage probabilities of hybrid
SDEs, as well as its expected occupation times; some numerical examples of
such approximations are explored as well. Finally, in Section 5 we provide a
brief summary of our findings, along with a discussion on some avenues of
further research.

##  2\. Hybrid stochastic differential equations

Let us provide a precise description of a hybrid SDE and its solution
$(J,X)=\\{(J(t),X(t))\\}_{t\geq 0}$, for which we require a complete
probability space $(\Omega,\mathbb{P},\mathcal{F})$ that supports the
following independent components:

  * •

A standard Brownian motion ${B}=\\{B(t)\\}_{t\geq 0}$; this will dictate the
continuously-varying nature of $X$.

  * •

A Poisson process ${N}=\\{N(t)\\}_{t\geq 0}$ of a sufficiently large parameter
$\gamma>0$, and a sequence $\\{U_{\ell}\\}_{\ell\geq 1}$ of $\mbox{Unif}(0,1)$
i.i.d. random variables: these will be used to describe the jump dynamics of
$J$, with $N$ marking the epochs at which jumps occur, and $U_{\ell}$
dictating where the $\ell$th jump leads to.

Our first goal is to rigorously construct a pair $(J,X)$ that solves the
hybrid SDE

(2.1)
$\mathrm{d}X(t)=\mu(J(t),X(t))\mathrm{d}t+\sigma(J(t),X(t))\mathrm{d}B(t),\quad
J(0)=i_{0},\;X(0)=x_{0},$

where $X$ is an a.s. continuous real process, ${J}$ is a càdlàg jump process
with finite state space $\mathcal{E}=\\{1,\dots,p\\}$,
$\mu:\mathcal{E}\times\mathds{R}\mapsto\mathds{R}$,
$\sigma:\mathcal{E}\times\mathds{R}\mapsto\mathds{R}$, $i_{0}\in\mathcal{E}$
and $x_{0}\in\mathds{R}$. Furthermore, we require that $(J,X)$ is adapted to
the $\mathds{P}$-completed filtration $\mathcal{F}_{t}$ generated by
$\big{\\{}B(s),N(s),U_{N(s)};s\leq t\big{\\}}$. A pair $(J,X)$ satisfying the
aforementioned characteristics is called a _strong solution_ of (2.1).

######  Remark 2.1.

In the context of classic Itô diffusions, a strong solution to an SDE needs to
be adapted to the filtration generated by the driving noise component $B$
only. In the case of hybrid stochastic differential equations, there is an
additional stochastic component, the environmental process $J$, which is why
the term ‘strong solution’ needs to be modified in the hybrid SDE framework.

In this paper we are interested in hybrid SDEs whose environmental process
${J}$ switches at a state-dependent rate. Specifically, we let $J$ evolve
according to

(2.2)
$\displaystyle\mathbb{P}(J(t+h)=j|X(t),J(t)=i)=\delta_{ij}+\Lambda_{ij}(X(t))h+o(h)$

for some family of intensity matrices $\\{\bm{\Lambda}(x)\\}_{x\in\mathds{R}}$
which, at this stage, we assume to be continuous w.r.t. $x$.

Note that (2.1) characterises _part_ of the pathwise construction of ${X}$.
Indeed if ${J}$ was given, then we could solve ${X}$ during sojourn times of
${J}$ with fixed drift and diffusion coefficients, updating them at each jump
time of ${J}$. However, (2.2) does _not_ tell us how ${J}$ is meant to be
constructed, it only provides a distributional property, implying that one
needs a construction whose characteristics coincide with (2.2). This would not
be a problem if, for instance, $\bm{\Lambda}(x)\equiv\bm{\Lambda}$,
$x\in\mathds{R}$, for some intensity matrix $\bm{\Lambda}$; in such a case of
a _Markov-modulated diffusion_ , the evolution of ${J}$ does not depend on the
values of ${X}$, and thus can be constructed via uniformization arguments
using the Poisson process ${N}$ and the sequence $\\{U_{\ell}\\}_{\ell\geq 1}$
only (see [25] for more details). In the general case of hybrid SDEs, the
construction is more involved since ${J}$ evolves using information provided
by ${X}$. Below we discuss a construction developed in [3] which relies on
using a related inhomogeneous uniformization argument. First, we state some
assumptions needed.

######  Assumption 1.

For all $i\in\mathcal{E}$, $\mu(i,\cdot)$ and $\sigma(i,\cdot)$ are Lipschitz-
continuous. i.e. there exists some $K>0$ such that

$\displaystyle|\mu(i,x)-\mu(i,y)|\vee|\sigma(i,x)-\sigma(i,y)|\leq
K|x-y|\quad\mbox{for all}\quad x,y\in\mathds{R}.$

######  Assumption 2.

The family $\\{\bm{\Lambda}(x)\\}_{x\in\mathds{R}}$ is uniformly bounded, i.e.

$\gamma:=\sup_{x\in\mathds{R}}|\Lambda_{ii}(x)|<\infty.$

Under Assumption 1, we can guarantee the existence of a unique and strong
solution to the (ordinary) stochastic differential equation (SDE)

(2.3)
$X^{[i,x,v]}(t)=x+\int_{0}^{t}\mu(i,X^{[i,x,v]}(t))\mathrm{d}r+\int_{0}^{t}\sigma(i,X^{[i,x,v]}(t))\mathrm{d}B^{[v]}(r),$

for all $i\in\mathcal{E}$, $x\in\mathbb{R}$ and $v,t\geq 0$, where
$B^{[v]}(t):=B(v+t)-B(v)$ is a time-shifted version of ${B}$. In other words,
${X}^{[i,x,v]}$ corresponds to the solution of the SDE driven by ${B}^{[v]}$,
with coefficients $\mu(i,\cdot)$ and $\sigma(i,\cdot)$, and starting point
$x$. On the other hand, Assumption 2 tells us that the jumps of ${J}$ are
_dominated_ by those of a Poisson process of intensity $\gamma$, meaning that
we can use the arrival points of ${N}$ as the (possible) jump epochs of ${J}$.

The precise construction of $(J,X)$ is as follows. Let
$\\{\theta_{\ell}\\}_{\ell\geq 0}$ denote the arrival times of ${N}$ (with
$\theta_{0}=0$). Define $X(0)=x_{0}$ and $J(0)=i_{0}$, and for
$t\in(0,\theta_{1})$, let $J(t)=i_{0}$ and $X(t)=X^{[i_{0},x_{0},0]}(t)$: we
have defined the processes ${X}$ and ${J}$ in $[0,\theta_{1})$. Since we want
${X}$ to be continuous, we ought to define $X(\theta_{1})=X(\theta_{1}-)$. For
${J}$, we decide if it jumps or not at time $\theta_{1}$ employing the values
of $U_{1}$ and $X(\theta_{1})$ as follows:

(2.4) $J(\theta_{1})=k\quad\mbox{if}\quad
U_{1}\in\left[\sum_{j=1}^{k-1}\delta_{i_{0}j}+\frac{\Lambda_{i_{0}j}(X(\theta_{1}))}{\gamma},\sum_{j=1}^{k}\delta_{i_{0}j}+\frac{\Lambda_{i_{0}j}(X(\theta_{1}))}{\gamma}\right).$

Although at a first look (2.4) may look complex, we are simply using $U_{1}$
and $X(\theta_{1})$ in such a way that ${J}$ jumps to State $k$ at time
$\theta_{1}$ with a mass given by the $(i_{0},k)$-th entry of the probability
matrix $\bm{I}+\bm{\Lambda}(X(\theta_{1}))/\gamma$, the uniformized version of
$\bm{\Lambda}(X(\theta_{1}))$ (which we rigorously verify in Lemma 2.1). After
having defined $(J,X)$ in $[0,\theta_{1}]$, we construct ${X}$ in subsequent
intervals $(\theta_{\ell},\theta_{\ell+1}]$ by _concatenating_ strong
solutions of the type (2.3) with appropriate chosen values for $(i,x,v)$, as
well as using a decision rule similar to (2.4) to establish which states ${J}$
visits. More specifically, in a recursive manner, for $\ell=1,2,3,\dots$ and
$t\in(0,\theta_{\ell+1}-\theta_{\ell})$ let

(2.5) $\displaystyle X(\theta_{\ell}+t)$
$\displaystyle=X^{[i_{\ell},x_{\ell},\theta_{\ell}]}(t),$ (2.6) $\displaystyle
X(\theta_{\ell+1})$ $\displaystyle=X(\theta_{\ell+1}-),$ (2.7) $\displaystyle
J(\theta_{\ell}+t)$ $\displaystyle=i_{\ell},$ (2.8) $\displaystyle
J(\theta_{\ell+1})$ $\displaystyle=k\quad\mbox{if}\quad
U_{\ell}\in\left[{C}_{\ell}(k),{C}_{\ell}(k)+{D}_{\ell}(k)\right),$

where $x_{\ell}=X(\theta_{\ell})$, $i_{\ell}=J(\theta_{\ell})$ and

${C}_{\ell}(k)=\sum_{j=1}^{k-1}\delta_{i_{\ell}j}+\frac{{\Lambda}_{i_{\ell}j}({X}(\theta_{\ell+1}))}{\gamma},\quad{D}_{\ell}(k)=\delta_{i_{\ell}k}+\frac{{\Lambda}_{i_{\ell}k}({X}(\theta_{\ell+1}))}{\gamma}.$

A few aspects that are straightforward to verify about this construction:

  * •

${X}$ and ${J}$ are $\mathcal{F}_{t}$-adapted,

  * •

${X}$ is continuous at $\theta_{1},\theta_{2},\dots$, and thus, has a.s.
continuous paths,

  * •

${J}$ is càdlàg,

  * •

${X}$ solves the hybrid SDE (2.1) on every interval
$(\theta_{\ell},\theta_{\ell+1})$, $\ell=0,1,2,\dots$.

Thus, $(J,X)$ is indeed a strong solution to (2.1).

######  Remark 2.2.

Note that ${X}$ is the unique solution to (2.1) under this _particular_
construction of ${J}$. Indeed, if we choose a different construction of ${J}$,
the process ${X}$ may take another form. For instance, use the r.v.’s
$U_{\ell}^{\prime}=1-U_{\ell}$ instead of $U_{\ell}$ in (2.8); the process
${J}$ will then be different and so will ${X}$.

We still need to verify that (2.2) holds, which we prove in a slightly more
general scenario next. Recall that (2.2) only makes sense if $\bm{\Lambda}(x)$
is continuous w.r.t. $x$. A generalization of this property for discontinuous
$\bm{\Lambda}(\cdot)$ is given by

$\displaystyle\mathbb{P}\left(J(t+v)=J(t)\mbox{ for all
}v\in[0,s]\;\mid\;\mathcal{F}_{t}\otimes\mathcal{F}^{{B}}_{t+s},X(t)=x,J(t)=i\right)$
(2.9)
$\displaystyle\quad=\exp\left(\int_{0}^{s}\Lambda_{ii}(X^{[i,x,t]}(v))\mathrm{d}v\right),$
(2.10)
$\displaystyle\mathbb{P}\big{(}J(\theta_{\ell})=j\;\mid\;J(\theta_{\ell}-)=i,J(\theta_{\ell}-)\neq
J(\theta_{\ell}),X(\theta_{\ell})\big{)}=\frac{\Lambda_{ij}(X(\theta_{\ell}))}{|\Lambda_{ii}(X(\theta_{\ell}))|},\quad
i\neq j,$

where $\mathcal{F}^{{B}}_{t}$ is the $\mathbb{P}$-completion of the
$\sigma$-algebra generated by $\\{B(s):0\leq s\leq t\\}$. In essence, (2.9)
and (2.10) correspond to how inhomogeneous Markov jump processes are
classically constructed through their integrated jump intensities/hazard rates
(see e.g. [29, Ch. 13]). That (2.9) and (2.10) imply (2.2) for continuous
$\bm{\Lambda}(\cdot)$ is readily obtained by noting that

$\exp\left(\int_{0}^{h}\Lambda_{ii}(X^{[i,x,t]}(v))\mathrm{d}v\right)=1+\Lambda_{ii}(x)+o(h),$

where we employed the continuity of $X^{[i,x,t]}$ too. For sake of
completeness, we now show that (2.9) and (2.10) indeed hold; this proof is a
simplified version of an analogous result found in [3].

######  Lemma 2.1.

Let $(J,X)$ be constructed via (2.5)-(2.8). Then (2.9) and (2.10) hold.

###### Proof.

Integrating with respect to the number and position of arrivals ${N}$ in
$[t,t+s]$, say $\\{\theta^{*}_{\ell}\\}_{\ell}$,

$\displaystyle\mathbb{P}\left(J(t+v)=J(t)\mbox{ for all
}v\in[0,s]\;\mid\;\mathcal{F}_{t}\otimes\mathcal{F}^{{B}}_{t+s},X(t)=x,J(t)=i\right)$
$\displaystyle\quad=\sum_{k=0}^{\infty}\frac{(\gamma s)^{k}}{k!}e^{-\gamma
s}\int_{0}^{s}\cdots\int_{0}^{s}\frac{\prod_{j=1}^{k}\mathds{P}(J(\theta^{*}_{j})=J(\theta^{*}_{j}-)\mid\theta^{*}_{j}=t+v_{j},X^{[i,x,t]}(v_{j}),J(\theta^{*}_{j})=i)}{s^{k}}\mathrm{d}v_{1}\dots\mathrm{d}v_{k},$
$\displaystyle\quad=\sum_{k=0}^{\infty}\frac{(\gamma s)^{k}}{k!}e^{-\gamma
s}\int_{0}^{s}\cdots\int_{0}^{s}\frac{\prod_{j=1}^{k}\left(1+\Lambda_{ii}(X^{[i,x,t]}(v_{j}))/\gamma\right)}{s^{k}}\mathrm{d}v_{1}\dots\mathrm{d}v_{k}$
$\displaystyle\quad=\sum_{k=0}^{\infty}\frac{(\gamma s)^{k}}{k!}e^{-\gamma
s}\left(1+\frac{\int_{0}^{s}\Lambda_{ii}(X^{[i,x,t]}(v))\mathrm{d}v}{\gamma
s}\right)^{k}=\exp\left(\int_{0}^{s}\Lambda_{ii}(X^{[i,x,t]}(v))\mathrm{d}v\right),$

where in the first equality we used that on the event $\\{J(t+v)=J(t)\mbox{
for all }v\in[0,s],J(t)=i,X(t)=t\\}$, $X(t+v)=X^{[i,x,t]}(v)$ for all
$v\in[0,s]$ by construction. Finally, for $i\neq j\in\mathcal{E}$

$\displaystyle\mathbb{P}\big{(}J(\theta_{\ell})=j\;\mid\;J(\theta_{\ell}-)=i,J(\theta_{\ell}-)\neq
J(\theta_{\ell}),X(\theta_{\ell})\big{)}$
$\displaystyle\quad=\frac{\mathbb{P}\big{(}J(\theta_{\ell})=j\;\mid\;J(\theta_{\ell}-)=i,X(\theta_{\ell})\big{)}}{\mathbb{P}\big{(}J(\theta_{\ell}-)\neq
J(\theta_{\ell})\;\mid\;J(\theta_{\ell}-)=i,X(\theta_{\ell})\big{)}}=\frac{\Lambda_{ij}(X(\theta_{\ell}))}{|\Lambda_{ii}(X(\theta_{\ell}))|},$

completing the proof. ∎

We note that the construction of strong solutions for hybrid SDEs can be
carried on in more general scenarios. For instance, Brownian-driven
multidimensional hybrid SDEs with unbounded jump intensities are considered in
[24], while the construction discussed in [3] concerns multidimensional past-
dependent hybrid SDEs driven by Lévy processes.

##  3\. Space-grid approximation of hybrid SDEs

Although solutions to state-dependent hybrid SDEs are easy to simulate
following the steps in (2.5)-(2.8), studying their distributional properties
is a challenging task (see [33] and references therein). Here we advocate to
study hybrid SDEs using an approximation method via discretizing the space-
grid.

Let $(\Omega,\mathbb{P},\mathcal{F})$ be as in Section 2. For each
$i\in\mathcal{E}$, let $\widehat{\mu}(i,\cdot)$ and
$\widehat{\sigma}(i,\cdot)$ be a piecewise constant approximation of
${\mu}(i,\cdot)$ and ${\sigma}(i,\cdot)$ on a space-grid $\\{\zeta_{m}\\}$ of
$\mathds{R}$. In this paper, we assume the following properties.

######  Assumption 3.

For each $i\in\mathcal{E}$:

  * •

$\widehat{\mu}(i,\cdot)$ and $\widehat{\sigma}(i,\cdot)$ are right-continuous
with left limits,

  * •

$|\widehat{\mu}(i,\cdot)|\leq|\mu(i,\cdot)|$ and
$|\widehat{\sigma}(i,\cdot)|\leq|\sigma(i,\cdot)|$,

  * •

$\widehat{\mu}$ and $\widehat{\sigma}$ are such that the stochastic
differential equation

(3.1)
$\widehat{X}^{[i,x,v]}(t)=x+\int_{0}^{t}\widehat{\mu}(i,\widehat{X}^{[i,x,v]}(t))\mathrm{d}r+\int_{0}^{t}\widehat{\sigma}(i,\widehat{X}^{[i,x,v]}(t))\mathrm{d}B^{[v]}(r)$

admits a strong and unique solution for all $x\in\mathds{R}$ and $v\geq 0$,

  * •

the space-grid $\\{\zeta_{m}\\}$ has a finite cardinality.

Below we present an elementary stability condition that is useful for our
forthcoming developments.

######  Lemma 3.1.

Let $\widehat{X}^{[i,x,v]}$ and $\widehat{X}^{[i,x,v]}$ be the solutions of
(2.3) and (3.1), respectively, where $\mu$, $\sigma$, $\widehat{{\mu}}$ and
$\widehat{{\sigma}}$ attain Assumptions 1 and 3. Then,

$\mathbb{E}\left(\sup_{s\leq
t}|X^{[i,x,v]}(s)|^{2}\right)<\infty\quad\mbox{and}\quad\mathbb{E}\left(\sup_{s\leq
t}|\widehat{{X}}^{[i,x,v]}(s)|^{2}\right)<\infty.$

###### Proof.

The $L_{2}$-uniform boundedness over compact intervals of ${X}^{[i,x,v]}$ is a
standard result, which we briefly replicate below for the sake of
completeness. Let $\rho_{m}:=\inf\\{s>0\;:\;|{X}^{[i,x,v]}(s)|>m\\}$, so that

$\frac{1}{3}\left|{X}^{[i,x,v]}(s\wedge\rho_{m})\right|^{2}\leq|x|^{2}+\left|\int_{0}^{s\wedge\rho_{m}}\mu\left(i,{X}^{[i,x,v]}(r)\right)\mathrm{d}r\right|^{2}+\left|\int_{0}^{s\wedge\rho_{m}}\sigma\left(i,{X}^{[i,x,v]}(r)\right)\mathrm{d}B^{[v]}(r)\right|^{2}.$

By Cauchy-Schwartz,

$\displaystyle\left|\int_{0}^{s\wedge\rho_{m}}\mu\left(i,{X}^{[i,x,v]}(r)\right)\mathrm{d}r\right|^{2}\leq(s\wedge\rho_{m})\int_{0}^{s\wedge\rho_{m}}\left|\mu\left(i,{X}^{[i,x,v]}(r)\right)\right|^{2}\mathrm{d}r.$

By Doob’s inequality and Itô isometry,

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t}\left|\int_{0}^{s\wedge\rho_{m}}\sigma\left(i,{X}^{[i,x,v]}(r)\right)\mathrm{d}B^{[v]}(r)\right|^{2}\right)\leq
2^{2}\mathbb{E}\left(\int_{0}^{s\wedge\rho_{m}}\left|\sigma\left(i,{X}^{[i,x,v]}(r)\right)\right|^{2}\mathrm{d}B^{[v]}(r)\right).$

By the Assumption 1, $\mu(i,\cdot)$ and $\sigma(i,\cdot)$ must be at most
linearly increasing, meaning that there exists some $K^{\prime}>0$ such that

$|\mu(i,x)\vee\sigma(i,x)|\leq K^{\prime}(1+|x|).$

Then,

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t\wedge\rho_{m}}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)$ $\displaystyle\leq
3|x|^{2}+3t\int_{0}^{t}K^{\prime}\left(\mathbb{E}\left(\sup_{s\leq
r\wedge\rho_{m}}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)+1\right)\mathrm{d}r$
$\displaystyle\quad+3\times
2^{2}\int_{0}^{t}K^{\prime}\left(\mathbb{E}\left(\sup_{s\leq
r\wedge\rho_{m}}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)+1\right)\mathrm{d}r$
$\displaystyle=K_{1}^{\prime}+K_{2}^{\prime}\int_{0}^{t}\mathbb{E}\left(\sup_{s\leq
r\wedge\rho_{m}}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)\mathrm{d}r,$

where $K_{1}^{\prime}=3|x|^{2}+3t^{2}+3\times 2^{2}t$ and
$K_{2}^{\prime}=K^{\prime}(3t+2^{2})$. Gronwall’s lemma then yields

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t\wedge\rho_{m}}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)\leq
K_{1}^{\prime}e^{K_{2}^{\prime}t};$

by letting $m\rightarrow\infty$, we get that $\mathbb{E}\left(\sup_{s\leq
t}\left|{X}^{[i,x,v]}(s)\right|^{2}\right)<\infty$. Now, since
$|\widehat{\mu}|\leq|\mu|$ and $|\widehat{\sigma}|\leq|\sigma|$, then
$\widehat{\mu}$ and $\widehat{\sigma}$ are at most linearly increasing as
well. Given this, the statement concerning $\widehat{X}^{[i,x,v]}$ follows by
analogous arguments to the previously presented. ∎

######  Remark 3.1.

Finding conditions under which a stochastic differential equation with
discontinuous coefficients has a unique strong solution is an active area of
research in recent years. For instance, in [21] it is established that (3.1)
has a unique and strong solution for $\widehat{\sigma}(i,\cdot)=1$ and general
measurable function $\widehat{\mu}(i,\cdot)$ that attains certain
integrability properties. In [22], the same result is established for
Lipschitz-continuous functions $\widehat{\sigma}(i,\cdot)$ and piecewise
Lipschitz-continuous functions $\widehat{\mu}(i,\cdot)$, both assumed to be
bounded. Given the nature of $\widehat{\mu}(i,\cdot)$ and
$\widehat{\sigma}(i,\cdot)$ as piecewise constant approximations of
$\mu(i,\cdot)$ and $\sigma(i,\cdot)$, and taking into account the existing
results in [21] and [22], our setup allows for the case in which
$\widehat{\sigma}(i,\cdot)$ and $\sigma(i,\cdot)$ are constant, and
$\widehat{\mu}(i,\cdot)$ and $\mu(i,\cdot)$ are bounded functions. Other
possible setups that produce strong and unique solutions to (3.1) ought to be
studied on a case by case basis, a topic which goes beyond the scope of this
manuscript.

Now, let $\widehat{{{J}}}$ be a càdlàg $\mathcal{F}_{t}$-adapted jump process
whose jump intensities are of the form

(3.2)
$\mathbb{P}(\widehat{{J}}(t+h)=j|\widehat{{X}}(t),\widehat{{J}}(t)=i)=\delta_{ij}+\widehat{{\Lambda}}_{ij}(\widehat{{X}}(t))h+o(h)$

with $\widehat{{\bm{\Lambda}}}(\cdot)$ being a piecewise constant
approximation of $\bm{\Lambda}(\cdot)$ over the space-grid $\\{\zeta_{m}\\}$.
Under Assumption 3 and supposing that the path $\widehat{J}$ is _somehow_
known, the solution $\widehat{X}$ to the hybrid SDE

(3.3)
$\mathrm{d}\widehat{{X}}(t)=\widehat{{\mu}}(\widehat{{J}}(t),\widehat{{X}}(t))\mathrm{d}t+\widehat{{\sigma}}(\widehat{{J}}(t),\widehat{{X}}(t))\mathrm{d}B(t),\quad\widehat{{J}}(0)=i_{0},\;\widehat{{X}}(0)=x_{0},$

can be constructed by a piecewise concatenation of paths, analogously to
(2.5)-(2.6). The resulting process is such that, when restricted to a space
interval $(\zeta_{m},\zeta_{m+1})$ and to the time intervals for which
$\widehat{J}$ is equal to $i$, $\widehat{X}$ behaves like a Brownian motion
with drift $\widehat{\mu}(i,\zeta_{m})$ and noise coefficient
$\widehat{\sigma}(i,\zeta_{m})$. In fact, the process $\widehat{X}$ falls
within the class of _multi-regime Markov-modulated Brownian motions_ studied
in [18] (see [23] for an earlier reference in the case $\widehat{\sigma}\equiv
0$). Heuristically speaking, since the functions $\widehat{\mu}$,
$\widehat{\sigma}$ and $\widehat{{\bm{\Lambda}}}$ are space-grid
approximations of $\mu$, $\sigma$ and $\bm{\Lambda}$, one can expect that
$\widehat{{X}}$ approximates the original solution $X$. A main purpose of this
paper is to formalize this statement.

######  Remark 3.2.

In the previous paragraph we did not specify how $\widehat{{J}}$ is
constructed. This was on purpose: while it is straightforward to follow
analogous steps to those in (2.7)-(2.8) to define $\widehat{{J}}$, here we
need to follow slightly different steps in order to guarantee the pathwise
convergence of $(\widehat{J},\widehat{X})$ to $(J,X)$. The specifics of such a
construction are contained in Subsection 3.2.

To prove pathwise convergence of $(\widehat{{J}},\widehat{{X}})$ to
$({J},{X})$ as some of its parameters go to $\infty$, we need to proceed with
care. First we study the unique continuous strong solution $\tilde{{X}}$ to

(3.4)
$\mathrm{d}\tilde{X}(t)=\widehat{{\mu}}({J}(t),\tilde{X}(t))\mathrm{d}t+\widehat{{\sigma}}({J}(t),\tilde{X}(t))\mathrm{d}B(t),\quad{J}(0)=i_{0},\;\tilde{X}(0)=\tilde{x}_{0},$

where ${J}$ is the jump process constructed through (2.5)-(2.8), while $X$ is
recursively defined by

(3.5)
$\displaystyle\tilde{X}(\theta_{\ell}+t)=\widehat{X}^{[i_{\ell},\tilde{x}_{\ell},\theta_{\ell}]}(t)\quad\mbox{and}\quad\tilde{X}(\theta_{\ell+1})=\widehat{X}(\theta_{\ell+1}-)\quad\mbox{where}\quad\tilde{x}_{\ell}=\tilde{X}(\theta_{\ell}).$

The _practicality_ of the process $\tilde{{X}}$ is essentially null.
$\tilde{{X}}$ is an approximation to ${X}$; however, it needs ${J}$ and ${X}$
to be constructed. This does not mean that it is ill-defined; in fact, once
$(J,X)$ is constructed (which can be done without issues following
(2.5)-(2.8)), then $\tilde{{X}}$ simply arises as a concatenation of solution
paths. Instead, the real usefulness of $\tilde{{X}}$ is in providing an
intermediate step towards quantifying the approximation error
$|{X}-\widehat{{{X}}}|$: we will first measure $|{X}-\tilde{{X}}|$
(discretization error brought by $\widehat{\mu}$ and $\widehat{\sigma}$) and
then the discrepancy between $\tilde{{X}}$ and $\widehat{{{X}}}$
(discretization error brought by $\widehat{{\bm{\Lambda}}}$).

###  3.1. Measuring $|{X}-\tilde{{X}}|$.

In order to quantify the approximation error of $\tilde{{X}}$ to $X$, let us
first measure the uniform mean-square distance over compact time intervals
between ${X}$ and $\tilde{{X}}$ in the case $\mathcal{E}=\\{1\\}$ (that is, no
switching occurs). Abusing notation for a brief moment, let us write $\mu$,
$\sigma$, $\widehat{\mu}$ and $\widehat{\sigma}$ instead of $\mu(1,\cdot)$,
$\sigma(1,\cdot)$, $\widehat{\mu}(1,\cdot)$ and $\widehat{\sigma}(1,\cdot)$,
respectively.

######  Theorem 3.2.

Suppose that $\mathcal{E}=\\{1\\}$ and let $t\geq 0$. Then,

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t}\left|X(s)-\tilde{X}(s)\right|^{2}\right)\leq
C(t)\left(\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2}+\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}+|x_{0}-\tilde{x_{0}}|^{2}\right),$

where $C(t)$ takes the form (3.12).

###### Proof.

For $t\geq 0$, define

$Z(t)=\mathbb{E}\left(\sup_{s\leq t}|X(s)-\tilde{X}(s)|^{2}\right).$

By Lemma 3.1, $Z(t)$ is finite for all $t\geq 0$. Using the inequality
$(a+b+c)^{2}\leq 3a^{2}+3b^{2}+3c^{2}$,

(3.6) $\displaystyle Z(t)$ $\displaystyle\leq 3\mathbb{E}\left(\sup_{s\leq
t}\left|\int_{0}^{s}\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\mathrm{d}r\right|^{2}\right)$
(3.7) $\displaystyle\quad+3\mathbb{E}\left(\sup_{s\leq
t}\left|\int_{0}^{s}\sigma(X(r))-\widehat{\sigma}(\tilde{X}(r))\mathrm{d}B(r)\right|^{2}\right)+3|x_{0}-\tilde{x_{0}}|^{2}.$

Employing the Cauchy-Schwarz inequality we get that

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t}\left|\int_{0}^{s}\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\mathrm{d}r\right|^{2}\right)$
$\displaystyle\leq\mathbb{E}\left(\left(\int_{0}^{t}1\times\left|\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\right|\mathrm{d}r\right)^{2}\right)$
$\displaystyle\leq\mathbb{E}\left(\left(\int_{0}^{t}1^{2}\mathrm{d}r\right)\left(\int_{0}^{t}\left|\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\right|^{2}\mathrm{d}r\right)\right)$
(3.8)
$\displaystyle=t\int_{0}^{t}\mathbb{E}\left(\left|\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\right|^{2}\right)\mathrm{d}r,$

while the Burkholder-Davis-Gundy inequality leads to

$\displaystyle\mathbb{E}\left(\sup_{s\leq
t}\left|\int_{0}^{s}\sigma(X(r))-\widehat{\sigma}(\tilde{X}(r))\mathrm{d}B(r)\right|^{2}\right)$
$\displaystyle\leq
C_{*}\mathds{E}\left(\int_{0}^{t}\left|\sigma(X(r))-\widehat{\sigma}(\tilde{X}(r))\right|^{2}\mathrm{d}r\right)$
(3.9)
$\displaystyle=C_{*}\int_{0}^{t}\mathds{E}\left(\left|\sigma(X(r))-\widehat{\sigma}(\tilde{X}(r))\right|^{2}\right)\mathrm{d}r$

for some universal constant $C_{*}>0$. Furthermore, the Lipschitz-continuity
of $\mu$ implies

$\displaystyle\left|\mu(X(r))-\widehat{\mu}(\tilde{X}(r))\right|^{2}$
$\displaystyle\leq
2\left|\mu(X(r))-\mu(\tilde{X}(r))\right|^{2}+2\left|\mu(\tilde{X}(r))-\widehat{\mu}(\tilde{X}(r))\right|^{2}$
(3.10) $\displaystyle\leq
2K^{2}\left|X(r)-\tilde{X}(r)\right|^{2}+2\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2};$

similarly,

(3.11)
$\displaystyle\left|\sigma(X(r))-\widehat{\sigma}(\tilde{X}(r))\right|^{2}$
$\displaystyle\leq
2K^{2}\left|X(r)-\tilde{X}(r)\right|^{2}+2\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}.$

By virtue of (3.7)-(3.11), we get

$\displaystyle Z(t)$ $\displaystyle\leq
6K^{2}(t+C_{*})\int_{0}^{t}\mathbb{E}\left(\left|X(r)-\tilde{X}(r)\right|^{2}\right)\mathrm{d}r+6t\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2}$
$\displaystyle\quad\quad+6t\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}+3|x_{0}-\tilde{x_{0}}|^{2}$
$\displaystyle\leq
6K^{2}(t+C_{*})\int_{0}^{t}Z(r)\mathrm{d}r+6t\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2}+6t\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}+3|x_{0}-\tilde{x_{0}}|^{2}.$

Finally, Gronwall’s lemma implies that

$\displaystyle Z(t)$ $\displaystyle\leq
C(t)\left(\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2}+\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}+|x_{0}-\tilde{x_{0}}|^{2}\right)$

where

(3.12) $C(t)=(6t\vee 3)e^{6K^{2}(t+C_{*})t}.$

∎

As usual, mean-square convergence implies convergence in probability, which in
our case takes the following explicit form.

######  Corollary 3.3.

Suppose that $\mathcal{E}=\\{1\\}$ and that there exists some $\alpha>0$ such
that

(3.13)
$\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|\vee\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|\vee|x_{0}-\tilde{x}_{0}|\leq\alpha.$

Then, for any $n\geq 2$,

$\mathbb{P}\left(\sup_{s\leq
t}\left|X(s)-\tilde{X}(s)\right|\geq\sqrt{3C(t)(\log n)}\alpha\right)\leq(\log
n)^{-1}.$

###### Proof.

Markov’s inequality and Theorem 3.2 imply that for any
$\Delta_{n,t,\alpha}>0$,

$\displaystyle\mathbb{P}$ $\displaystyle\left(\sup_{s\leq
t}\left|X(s)-\tilde{X}(s)\right|\geq\Delta_{n,t,\alpha}\right)\leq\frac{\mathbb{E}\left(\sup_{s\leq
t}|X(s)-\tilde{X}(s)|^{2}\right)}{\Delta_{n,t,\alpha}^{2}}$
$\displaystyle\leq\frac{C(t)\left(\sup_{x\in\mathbb{R}}\left|\mu(x)-\widehat{\mu}(x)\right|^{2}+\sup_{x\in\mathbb{R}}\left|\sigma(x)-\widehat{\sigma}(x)\right|^{2}+|x_{0}-\tilde{x_{0}}|^{2}\right)}{\Delta_{n,t,\alpha}^{2}}=\frac{3C(t)\alpha^{2}}{\Delta_{n,t,\alpha}^{2}}.$

The result follows by choosing $\Delta_{n,t,\alpha}=\sqrt{3C(t)(\log
n)}\alpha$. ∎

Corollary 3.3 implies that we can choose appropriate values of $t$ and
$\alpha$ which are dependent on $n$, say $t_{n}$ and $\alpha_{n}$, such that

(3.14) $\displaystyle
t_{n}\rightarrow\infty\quad\mbox{and}\quad\sqrt{3C(t_{n})(\log
n)}\alpha_{n}\rightarrow 0,$

and then conclude that the path $\tilde{{X}}$ converges in probability to
${X}$ uniformly over increasing compact intervals. Here we choose
$t_{n}=\sqrt{\log\log n}$ and $\alpha_{n}=(\log n)^{\beta}n^{-\gamma}$ for
some $\beta,\gamma>0$. In such a case, there exists some $n_{0}\geq 1$ such
that for all $n\geq n_{0}$,

$\displaystyle\sqrt{3C(t_{n})(\log n)}\alpha_{n}$
$\displaystyle\leq(6\sqrt{\log\log n})e^{12K^{2}\log\log n}(\log
n)^{\beta}n^{-\gamma}$ $\displaystyle\leq(\log n)(\log n)^{12K^{2}}(\log
n)^{\beta}n^{-\gamma}$ (3.15) $\displaystyle=(\log
n)^{\beta_{*}+\beta}n^{-\gamma},$

where $\beta_{*}=1+12K^{2}$; note that $n_{0}$ does not depend on the choice
of $\beta$ or $\gamma$. In light of this, we assume the following.

######  Assumption 4.

There exists some $\beta\geq 0$ and $\gamma>0$ such that for each $n\geq 2$,
the approximation $\tilde{{X}}^{(n)}$ (constructed as $\tilde{{X}}$ in (3.4)
where the coefficients $\widehat{\mu}$ and $\widehat{\sigma}$ now show an
explicit dependence on $n$ via their superscript) satisfies

$\sup_{x\in\mathbb{R},i\in\mathcal{E}}\left|\mu(i,x)-\widehat{\mu}^{(n)}(i,x)\right|\vee\sup_{x\in\mathbb{R},i\in\mathcal{E}}\left|\sigma(i,x)-\widehat{\sigma}^{(n)}(i,x)\right|\vee|x_{0}-\tilde{x}^{(n)}_{0}|\leq(\log
n)^{\beta}n^{-\gamma}.$

While it might seem that the proof in Theorem 3.2 can be easily modified to
handle the general case $\mathcal{E}=\\{1,\dots,p\\}$, the switching caused by
${J}$ destroys the local martingale property of the involved stochastic
integrals, and thus, the Burkholder-Davis-Gundy inequality no longer applies.
For this reason, our approach is to study the discrepancy between ${X}$ and
$\tilde{{X}}^{(n)}$ by concatenating paths at the time epochs
$\\{\theta_{\ell}\\}_{\ell\geq 0}$ for which switching can occur. We
explicitly do this in the proof below.

######  Theorem 3.4.

Let $\mathcal{E}=\\{1,\dots,p\\}$ and fix some small $\epsilon_{1}>0$. As
$n\rightarrow\infty$,

$\displaystyle\mathbb{P}\left(\sup_{s\leq\xi_{n}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right)=o(1).$

where $\xi_{n}=\sqrt{\log\log\log n}$.

###### Proof.

First, we claim that

(3.16) $\mathds{P}(N(\xi_{n})>\lfloor\log\log
n\rfloor)=o(1)\quad\mbox{as}\quad n\rightarrow\infty.$

Indeed, using [16, Proposition 1], for sufficiently large $n$

(3.17) $\displaystyle\mathds{P}(N(\xi_{n})>\lfloor\log\log n\rfloor)\leq$
$\displaystyle\left(1-\frac{\gamma\xi_{n}}{\lfloor\log\log
n\rfloor+1}\right)^{-1}\left(\frac{(\gamma\xi_{n})^{\lfloor\log\log
n\rfloor}}{\lfloor\log\log n\rfloor!}e^{-\gamma\xi_{n}}\right):$

the first element in the product of the r.h.s. in (3.17) clearly converges to
$1$, while the second element is bounded by

$\displaystyle\frac{(\gamma\xi_{n})^{\lfloor\log\log
n\rfloor}}{\lfloor\log\log n\rfloor!}$
$\displaystyle\quad\leq\left((\gamma\xi_{n})^{\lfloor\log\log
n\rfloor}\right)\left(\sqrt{2\pi\lfloor\log\log
n\rfloor}\left(\frac{\lfloor\log\log n\rfloor}{e}\right)^{\lfloor\log\log
n\rfloor}e^{\tfrac{1}{12\lfloor\log\log n\rfloor+1}}\right)^{-1}$
$\displaystyle\quad=O\left(\left(\frac{e\gamma\xi_{n}}{\lfloor\log\log
n\rfloor}\right)^{\lfloor\log\log n\rfloor}\right)=o(1).$

Now, fix some large $n\geq 1$ and for all $m\geq 0$ let

$\displaystyle A_{m}$
$\displaystyle=\left\\{\sup_{s\leq\xi_{n}\wedge\theta_{m}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>(\log
n)^{m\beta_{*}+\beta}n^{\gamma}\right\\}.$

Note that

$\displaystyle\mathbb{P}\left(A_{m}\right)$
$\displaystyle=\mathbb{P}(A_{m-1}\cap A_{m})+\mathbb{P}(A_{m-1}^{c}\cap
A_{m})$ $\displaystyle\leq\mathbb{P}(A_{m-1})+\mathbb{P}(A_{m}\mid
A_{m-1}^{c})=\sum_{\ell=1}^{m}\mathbb{P}(A_{\ell}\mid A_{\ell-1}^{c})$
$\displaystyle=\sum_{\ell=1}^{m}\mathbb{P}\left(\sup_{s\leq\xi_{n}\wedge\theta_{\ell}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>(\log
n)^{\ell\beta_{*}+\beta}n^{\gamma}\mid A_{\ell-1}^{c}\right)\leq m(\log
n)^{-1},$

where the last inequality follows by using Corollary 3.3 and (3.15) for all
$\widehat{\mu}(i,\cdot)$ and $\widehat{\sigma}(i,\cdot)$, $i\in\mathcal{E}$.
Finally, for sufficiently large $n$ (large enough that
$(\log\ell)^{\lfloor\log\log\ell\rfloor\beta_{*}+\beta}\geq n^{\epsilon_{1}}$
for all $\ell\geq 1$),

$\displaystyle\left\\{\sup_{s\leq\xi_{n}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right\\}\subset\left\\{N(\xi_{n})>\lfloor\log\log
n\rfloor\right\\}\cup\bigcup_{\ell=1}^{\lfloor\log\log
n\rfloor}A^{(n)}_{\ell},$

so that

$\displaystyle\mathbb{P}\left(\sup_{s\leq\xi_{n}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right)\leq
o(1)+(\lfloor\log\log n\rfloor)^{2}(\log n)^{-1}=o(1).$

∎

Now that we have assessed the convergence of $\tilde{X}$ to $X$, we will
continue with the discrepancy between $\tilde{X}$ and $\widehat{{X}}$ in the
following section, which will ultimately lead to measuring the final error
$|{X}-\widehat{{{X}}}|$.

###  3.2. Measuring $|\tilde{X}-\widehat{{X}}|$.

Assessing the discrepancy between $\tilde{X}$ and $\widehat{{X}}$ relies on
the following key observation: $\widehat{X}$ is _identical_ to $\tilde{X}$ in
a compact interval $[0,t]$ if $\widehat{J}$ coincides with $J$ in that
interval. Thus, the question of convergence of $\widehat{X}$ to $\tilde{X}$
may be posed as a question of if/when the process $\widehat{J}$ differs from
$J$. In order to answer this, we borrow the _decoupling_ idea introduced in
[3]. This concept relies on constructing the approximation $\widehat{J}^{(n)}$
(understood as the process $\widehat{J}$ with an explicit dependence on some
parameter $n$; we omit such dependence in some places for notational
convenience) in such a way that this process decouples or stops being
identical to $J$ in an interval $[0,t]$ with a probability that tends to $0$
as $n\rightarrow\infty$. As announced in Remark 3.2, instead of constructing
$\widehat{J}$ via (2.7)-(2.8), below we describe an alternative (but similar)
technique that yields the distributional property (3.2) as well.

First, let us add an extra process $\\{\widehat{H}_{\ell}\\}_{\ell\geq 0}$
which tracks the _decoupling_ event of $\widehat{J}$ w.r.t. $J$ at the time
epochs $\\{\theta_{\ell}\\}_{\ell\geq 0}$. Roughly speaking, we say a
decoupling occurs at time $\theta_{\ell}$ if we can no longer guarantee that
$\widehat{J}(\theta_{\ell})=J(\theta_{\ell})$, but we can still guarantee that
$\widehat{J}$ is identical to $J$ on $[0,\theta_{\ell})$. To properly define
this, we consider a process $\\{\widehat{H}_{\ell}\\}_{\ell\geq 0}$ taking
values in $\\{0,1,2\\}$, where $\\{\widehat{H}_{\ell}=0\\}$ will signify that
the decoupling has not occurred by time $\theta_{\ell}$,
$\\{\widehat{H}_{\ell}=1\\}$ will signify that the decoupling occured exactly
at time $\theta_{\ell}$, and $\\{\widehat{H}_{\ell}=2\\}$ will signify that a
decoupling occurred before time $\theta_{\ell}$. More specifically, let
$\widehat{X}=x_{0}$, $\widehat{J}(\theta_{0})=i_{0}$, $\widehat{H}_{0}=0$, and
recursively define for $\ell=0,1,2,\dots$ and
$t\in(0,\theta_{\ell+1}-\theta_{\ell})$,

(3.18) $\displaystyle\widehat{X}(\theta_{\ell}+t)$
$\displaystyle=\widehat{X}^{[\widehat{i}_{\ell},\widehat{x}_{\ell},\theta_{\ell}]}(t),$
(3.19) $\displaystyle\widehat{X}(\theta_{\ell+1})$
$\displaystyle=\widehat{X}(\theta_{\ell+1}-),$ (3.20)
$\displaystyle\widehat{J}(\theta_{\ell}+t)$
$\displaystyle=\widehat{i}_{\ell},$
$\displaystyle\widehat{J}(\theta_{\ell+1})$
$\displaystyle=k,\;\widehat{H}_{\ell+1}=0$ (3.21)
$\displaystyle\quad\mbox{if}\quad
U_{\ell}\in\left[{C}_{\ell}(k),{C}_{\ell}(k)+{D}_{\ell}(k)\wedge\widehat{D}_{\ell}(k)\right),\widehat{H}_{\ell}=0,$
$\displaystyle\widehat{J}(\theta_{\ell+1})$
$\displaystyle\sim(\widehat{D}_{\ell}(k)-{D}_{\ell}(k)\wedge\widehat{D}_{\ell}(k))_{k\in\mathcal{E}},\;\widehat{H}_{\ell+1}=1$
(3.22) $\displaystyle\quad\mbox{if}\quad
U_{\ell}\notin\cup_{k^{\prime}\in\mathcal{E}}\left[{C}_{\ell}(k^{\prime}),{C}_{\ell}(k^{\prime})+{D}_{\ell}(k^{\prime})\wedge\widehat{D}_{\ell}(k^{\prime})\right),\widehat{H}_{\ell}=0,$
(3.23) $\displaystyle\widehat{J}(\theta_{\ell+1})$
$\displaystyle\sim(\widehat{D}_{\ell}(k))_{k\in\mathcal{E}},\;\widehat{H}_{\ell+1}=2\quad\mbox{if}\quad\widehat{H}_{\ell}\in\\{1,2\\},$

where $\widehat{x}_{\ell}=\widehat{X}(\theta_{\ell})$,
$\widehat{i}_{\ell}=\widehat{{J}}(\theta_{\ell})$ and
$\widehat{D}_{\ell}(k)=\delta_{i_{\ell}k}+\frac{\widehat{\Lambda}_{i_{\ell}k}(\widehat{X}(\theta_{\ell+1}))}{\gamma}$.
Here the notation ‘$Y\sim\bm{\alpha}$’ for $\bm{\alpha}\neq\bm{0}$ denotes
‘$Y$ is sampled from $\bm{\alpha}/|\bm{\alpha}|$’.

While Equations (3.18)-(3.20) are analogous to their counterpart (2.5)-(2.7),
note that (3.21)-(3.23) are slightly more involved than (2.8). Nevertheless,
the idea is similar: $\widehat{J}$ is constructed in such a way that at the
uniformization epoch $\theta_{\ell+1}$, it lands in State $k$ with probability
$\widehat{D}_{\ell}(k)$. Indeed, if $\widehat{H}_{\ell}=0$, then in order to
land in $k$, either

  * •

$U_{\ell}$ is in
$\left[{C}_{\ell}(k),{C}_{\ell}(k)+{D}_{\ell}(k)\wedge\widehat{D}_{\ell}(k)\right)$
(with probability ${D}_{\ell}(k)\wedge\widehat{D}_{\ell}(k)$), or

  * •

$U_{\ell}$ lands in
$[0,1]\setminus\left(\cup_{k^{\prime}\in\mathcal{E}}\left[{C}_{\ell}(k^{\prime}),{C}_{\ell}(k^{\prime})+{D}_{\ell}(k^{\prime})\wedge\widehat{D}_{\ell}(k^{\prime})\right)\right)$
and then $k$ gets sampled from the vector
$\left(\widehat{D}_{\ell}(k^{\prime})-{D}_{\ell}(k^{\prime})\wedge\widehat{D}_{\ell}(k^{\prime})\right)_{k^{\prime}\in\mathcal{E}}$
(with probability
$\widehat{D}_{\ell}(k)-{D}_{\ell}(k)\wedge\widehat{D}_{\ell}(k)$).

In the second case, a decoupling is declared at that step and we let
$\widehat{H}_{\ell+1}=1$; under these circumstances, by construction we can
still guarantee that
$\\{\widehat{J}(t)\\}_{t<\theta_{\ell+1}}=\\{J(t)\\}_{t<\theta_{\ell+1}}$
(compare (2.8) and (3.21)-(3.23)) and
$\\{\widehat{X}(t)\\}_{t\leq\theta_{\ell+1}}=\\{\tilde{X}(t)\\}_{t\leq\theta_{\ell+1}}$
(compare (3.5) and (3.18)-(3.19)). Once a decoupling occurs, further jumps of
$J$ at the uniformization epochs occur by sampling directly from the vector
$(\widehat{D}_{\ell}(k^{\prime}))_{k^{\prime}\in\mathcal{E}}$, which are
points in time at which we can no longer guarantee that $\widehat{J}$ and $J$
are identical (ditto $\widehat{X}$ and $\tilde{X}$). Fortunately, since
$\widehat{\Lambda}$ is thought to be close to $\bm{\Lambda}$, so is
$\widehat{D}_{\ell}(k)$ to ${D}_{\ell}(k)$, suggesting that the probability of
a decoupling at a given step must be low. In the rest of the section, we
investigate a stronger statement: decoupling occurs with low probability in
any compact time interval.

Similarly to Assumption 4, let us now make explicit the dependence of
$\widehat{\bm{\Lambda}}$, $\widehat{X}$ and $\widehat{J}$ on $n$ via a
superscript. In order to obtain convergence rates of the decoupling of
$\widehat{J}^{(n)}$ w.r.t. $J$, we impose the following regularity condition
on $\widehat{\bm{\Lambda}}^{(n)}$.

######  Assumption 5.

For a square matrix $A=\\{a_{ij}\\}$, let $\|\cdot\|$ denote the norm defined
by

$\|A\|=\sup_{i}\left\\{\sum_{j}a_{ij}\right\\}.$

The intensity matrix $\bm{\Lambda}(\cdot)$ is $\log$-Hölder continuous, in the
sense that there exists a constant $G>0$ such that for any $x,y\in\mathds{R}$,

$\left\|\bm{\Lambda}(x)-\bm{\Lambda}(y)\right\|\leq\frac{G}{-\log|x-y|}.$

Furthermore, $\widehat{\bm{\Lambda}}^{(n)}=\widehat{\bm{\Lambda}}$ is right-
continuous with left limits and converges uniformly to $\bm{\Lambda}$ at a
$\log$-rate, more specifically,

$\sup_{z\in\mathbb{R}}\left\|\widehat{\bm{\Lambda}}^{(n)}(z)-\bm{\Lambda}(z)\right\|\leq\frac{G}{\log
n}.$

######  Remark 3.3.

Under Assumption 5, $\bm{\Lambda}$ does not admit any discontinuities,
however, recall that $\log$-Hölder continuity is considerably less restrictive
than Hölder- or Lipschitz-continuity. This means that we can _mimic_ a
discontinuous behaviour for $\bm{\Lambda}$ at a point $z_{0}$ by considering
instead some steep function which behaves like $(\log(z-z_{0}))^{-1}$ for
sufficiently close $z>z_{0}$.

######  Lemma 3.5.

Fix some small $\epsilon_{1}>0$. Then,

(3.24) $\mathbb{P}\left(N(\xi_{n})\leq\lfloor\log\log
n\rfloor,\widehat{H}_{N({\xi_{n}})}\neq
0,\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}^{(n)}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right)=o(1)\quad\mbox{as}\quad n\rightarrow\infty.$

Additionally, Equation (3.24) also holds when $\widehat{X}^{(n)}$ is replaced
by $\tilde{X}^{(n)}$.

###### Proof.

For notational convenience, let us write $\widehat{X}$ instead of
$\widehat{X}^{(n)}$, and $\widehat{\bm{\Lambda}}$ instead of
$\widehat{\bm{\Lambda}}^{(n)}$. Note that

$\displaystyle\left\\{N(\xi_{n})\leq\lfloor\log\log
n\rfloor,\widehat{H}_{N({\xi_{n}})}\neq
0,\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right\\}$
$\displaystyle\quad=\bigcup_{\ell=1}^{\lfloor\log\log
n\rfloor}\bigcup_{k=1}^{\ell}\left\\{N(\xi_{n})=\ell,\widehat{H}_{k}=1,\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right\\}$ (3.25)
$\displaystyle\quad\subseteq\bigcup_{\ell=1}^{\lfloor\log\log
n\rfloor}\bigcup_{k=1}^{\ell}\left\\{N(\xi_{n})=\ell,\;\widehat{H}_{k}=1,\;\sup_{s\leq\theta_{k}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right\\}.$

By conditioning on the history of the processes up to time $\theta_{k}$ and
letting

$F=\\{{N(\xi_{n})=\ell,\;\widehat{H}_{k-1}=0,\;\sup_{s\leq\theta_{k}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}}\\},$

we get

$\displaystyle\mathbb{P}\left(N(\xi_{n})=\ell,\widehat{H}_{k}=1,\sup_{s\leq\theta_{k}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\;|\;N(\xi_{n}),\widehat{H}_{k-1},\\{X(s)\\}_{s\leq\theta_{k}},\\{\widehat{X}(s)\\}_{s\leq\theta_{k}}\right)$
$\displaystyle\quad=\mathbb{P}\left(\widehat{H}_{k}=1\;|\;N(\xi_{n}),\\{X(s)\\}_{s\leq\theta_{k}},\\{\widehat{X}(s)\\}_{s\leq\theta_{k}}\right)\times\mathds{1}_{F}$
$\displaystyle\quad=\mathbb{P}\left(U_{\ell-1}\notin\cup_{k^{\prime}\in\mathcal{E}}\left[{C}_{\ell-1}(k^{\prime}),{C}_{\ell-1}(k^{\prime})+{D}_{\ell-1}(k^{\prime})\wedge\widehat{D}_{\ell-1}(k^{\prime})\right)\;|\;N(\xi_{n}),\\{X(s)\\}_{s\leq\theta_{k}},\\{\widehat{X}(s)\\}_{s\leq\theta_{k}}\right)\times\mathds{1}_{F}$
$\displaystyle\quad=\left(\sum_{k^{\prime}\in\mathcal{E}}(\widehat{D}_{\ell-1}(k^{\prime})-{D}_{\ell-1}(k^{\prime})\wedge\widehat{D}_{\ell-1}(k^{\prime}))\right)\times\mathds{1}_{F}\leq\left\|\frac{{\bm{\Lambda}}({X}(\theta_{k}))}{\gamma}-\frac{\widehat{\bm{\Lambda}}(\widehat{X}(\theta_{k}))}{\gamma}\right\|\times\mathds{1}_{F}$
$\displaystyle\quad\leq\frac{1}{\gamma}\left[\left\|{{\bm{\Lambda}}({X}(\theta_{k}))}-{{\bm{\Lambda}}(\widehat{X}(\theta_{k}))}\right\|+\left\|{{\bm{\Lambda}}(\widehat{X}(\theta_{k}))}-{\widehat{\bm{\Lambda}}(\widehat{X}(\theta_{k}))}\right\|\right]\times\mathds{1}_{F}$
$\displaystyle\quad\leq\frac{1}{\gamma}\left[\frac{G}{-\log|{X}(\theta_{k})-\widehat{X}(\theta_{k})|}+\frac{G}{\log
n}\right]\times\mathds{1}_{F}\leq\frac{G}{\gamma}\left[\frac{1}{\gamma-\epsilon_{1}}+1\right](\log
n)^{-1}.$

From (3.25) we get

$\displaystyle\mathbb{P}\left(N(\xi_{n})\leq\lfloor\log\log
n\rfloor,\widehat{H}_{N({\xi_{n}})}\neq
0,\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}^{(n)}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right)$
$\displaystyle\quad\leq\sum_{\ell=1}^{\lfloor\log\log
n\rfloor}\sum_{k=1}^{\ell}\mathbb{P}\left(N(\xi_{n})=\ell,\;\widehat{H}_{k}=1,\;\sup_{s\leq\theta_{k}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right)$ (3.26)
$\displaystyle\quad\leq\sum_{\ell=1}^{\lfloor\log\log
n\rfloor}\sum_{k=1}^{\ell}\frac{G}{\gamma}\left[\frac{1}{\gamma-\epsilon_{1}}+1\right](\log
n)^{-1}\leq\frac{G}{\gamma}\left[\frac{1}{\gamma-\epsilon_{1}}+1\right]\frac{\lfloor\log\log
n\rfloor^{2}}{\log n}=o(1),$

which proves (3.24). Finally, following the same set inclusions in (3.25), we
get

$\displaystyle\mathbb{P}\left(N(\xi_{n})\leq\lfloor\log\log
n\rfloor,\widehat{H}_{N({\xi_{n}})}\neq
0,\sup_{s\leq\xi_{n}}\left|X(s)-\tilde{X}^{(n)}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right)$
$\displaystyle\quad\leq\sum_{\ell=1}^{\lfloor\log\log
n\rfloor}\sum_{k=1}^{\ell}\mathbb{P}\left(N(\xi_{n})=\ell,\;\widehat{H}_{k}=1,\;\sup_{s\leq\theta_{k}}\left|X(s)-\tilde{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right);$

since on $\\{\widehat{H}_{k}=1\\}$ the paths
$\\{\tilde{X}\\}_{s\leq\theta_{k}}$ and $\\{\widehat{X}\\}_{s\leq\theta_{k}}$
are the same, then

$\displaystyle\mathbb{P}\left(N(\xi_{n})=\ell,\;\widehat{H}_{k}=1,\;\sup_{s\leq\theta_{k}}\left|X(s)-\tilde{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right)$
$\displaystyle=\mathbb{P}\left(N(\xi_{n})=\ell,\;\widehat{H}_{k}=1,\;\sup_{s\leq\theta_{k}}\left|X(s)-\widehat{X}(s)\right|\leq
n^{-\gamma+\epsilon_{1}}\right),$

so the result for the case $\tilde{X}$ follows by identical steps to those in
(3.26).

∎

Now we are ready to prove the main result of the paper.

######  Theorem 3.6.

For any fixed $\epsilon_{1}>0$,

(3.27)
$\displaystyle\mathbb{P}\left(\left\\{\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right\\}\cup\left\\{J(s)\neq\widehat{J}^{(n)}(s)\mbox{
for some }s\leq\xi_{n}\right\\}\right)=o(1)\quad\mbox{as}\quad
n\rightarrow\infty.$

###### Proof.

Define the events

$\displaystyle\tilde{F}_{1}:=\left\\{\sup_{s\leq\xi_{n}}\left|X(s)-\tilde{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right\\},\quad\widehat{F}_{1}:=\left\\{\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\right\\}$
$\displaystyle F_{2}:=\left\\{\widehat{H}_{N(\xi_{n})}\neq 0\right\\},\quad
F_{3}:=\left\\{N(\xi_{n})>\lfloor\log\log n\rfloor\right\\}.$

Given that on the event $F_{2}^{c}$ the paths
$\\{\widehat{X}^{(n)}(s)\\}_{s\leq\xi_{n}}$ and
$\\{\tilde{X}^{(n)}(s)\\}_{s\leq\xi_{n}}$ are identical, then

$\tilde{F}_{1}\cap F_{2}^{c}=\widehat{F}_{1}\cap F_{2}^{c}.$

Taking this into account, by standard set inclusion/exclusion operations we
get

(3.28) $\displaystyle\widehat{F}_{1}\cup F_{2}=(\widehat{F}_{1}\cap
F_{2}^{c})\cup F_{2}=(\tilde{F}_{1}\cap F_{2}^{c})\cup F_{2}=\tilde{F}_{1}\cup
F_{2}=\tilde{F}_{1}\cup(\tilde{F}_{1}^{c}\cap F_{2}).$

Employing (3.28) and additional standard set operations lead to the inclusions

$\displaystyle\widehat{F}_{1}\cup F_{2}$
$\displaystyle=\\{(\widehat{F}_{1}\cup F_{2})\cap
F_{3}^{c}\\}\cup\\{(\widehat{F}_{1}\cup F_{2})\cap F_{3}\\}$
$\displaystyle\subseteq\\{(\widehat{F}_{1}\cup F_{2})\cap F_{3}^{c}\\}\cup
F_{3}$ $\displaystyle=\\{(\tilde{F}_{1}\cup(\tilde{F}_{1}^{c}\cap F_{2}))\cap
F_{3}^{c}\\}\cup F_{3}$ $\displaystyle=(\tilde{F}_{1}\cap
F_{3}^{c})\cup(\tilde{F}_{1}^{c}\cap F_{2}\cap F_{3}^{c})\cup F_{3}$
$\displaystyle\subseteq\tilde{F}_{1}\cup(\tilde{F}_{1}^{c}\cap F_{2}\cap
F_{3}^{c})\cup F_{3}$

which in turn implies that

$\displaystyle\mathbb{P}$
$\displaystyle\left(\sup_{s\leq\xi_{n}}\left|X(s)-\widehat{X}^{(n)}(s)\right|>n^{-\gamma+\epsilon_{1}}\cup\left\\{J(s)\neq\widehat{J}^{(n)}(s)\mbox{
for some }s\leq\xi_{n}\right\\}\right)$ (3.29) $\displaystyle\hskip
128.0374pt\leq\mathbb{P}(\widehat{F}_{1}\cup
F_{2})\leq\mathds{P}(\tilde{F}_{1})+\mathds{P}(\tilde{F}_{1}^{c}\cap F_{2}\cap
F_{3}^{c})+\mathds{P}(F_{3}).$

Since each one of the summands on the r.h.s. of (3.29) are shown to be $o(1)$
as $n\rightarrow\infty$ in Theorem 3.4, Lemma 3.5 and (3.29), respectively,
the result follows. ∎

Theorem 3.6 essentially tells us that under Assumptions 1-5, there exists a
sequence of multi-regime Markov-modulated Brownian motions $\widehat{X}^{(n)}$
that converge in probability to the solution $X$ of a hybrid stochastic
differential equation, as well as their respective underlying processes
$\widehat{J}^{(n)}$ and $J$; such a convergence occurs in a uniform sense over
increasing compact intervals. In particular, this implies that any first
passage times of $X$, say into the set $(-\infty,a)$ or $(a,+\infty)$, can be
approximated by the corresponding first passage times of $\widehat{X}^{(n)}$.
In the next section, we exploit this to provide efficient approximations to
certain first passage probabilities for hybrid stochastic differential
equations.

##  4\. First passage probabilities and expected occupation times for
solutions of hybrid SDEs

Define the first passage times $\tau_{0}^{-}=\inf\\{t>0:X(t)<0\\}$,
$\tau_{a}^{+}=\inf\\{t>0:X(t)>a\\}$ ($a>0$). For $q\geq 0$, we are interested
in providing approximations for the Laplace transform of the first passage
times,

$m^{-}_{ij}(u,q,a)=\mathds{E}(e^{-q(\tau_{0}^{-}\wedge\tau_{a}^{+})}\mathds{1}\left\\{J(\tau_{0}^{-}\wedge\tau_{a}^{+})=j,X(\tau_{0}^{-}\wedge\tau_{a}^{+})=0\right\\}\,\mid\,J(0)=i,X(0)=u),$
$m^{+}_{ij}(u,q,a)=\mathds{E}(e^{-q(\tau_{0}^{-}\wedge\tau_{a}^{+})}\mathds{1}\left\\{J(\tau_{0}^{-}\wedge\tau_{a}^{+})=j,X(\tau_{0}^{-}\wedge\tau_{a}^{+})=a\right\\}\,\mid\,J(0)=i,X(0)=u),$

for $0<u<a\leq\infty$ and $i,j\in\mathcal{E}$. In essence, the functions
$m^{+}_{ij}$ and $m^{-}_{ij}$ characterize the distributional law of the exit
times of $X$ from the band $[0,a]$. Employing the law of total probability, it
is straightforward to reinterpret $m^{-}_{ij}$ and $m^{-}_{ij}$ as

$m^{-}_{ij}(u,q,a)=\mathds{P}(\tau_{0}^{-}\wedge\tau_{a}^{+}<e_{q},J(\tau_{0}^{-}\wedge\tau_{a}^{+})=j,X(\tau_{0}^{-}\wedge\tau_{a}^{+})=0\,\mid\,J(0)=i,X(0)=u),$
$m^{+}_{ij}(u,q,a)=\mathds{P}(\tau_{0}^{-}\wedge\tau_{a}^{+}<e_{q},J(\tau_{0}^{-}\wedge\tau_{a}^{+})=j,X(\tau_{0}^{-}\wedge\tau_{a}^{+})=a\,\mid\,J(0)=i,X(0)=u),$

where $e_{q}$ is an $\mbox{Exp}(q)$ random variable independent of everything
else; from now on, we adopt this reinterpretation.

In the existing literature, explicit solutions to $m^{+}_{ij}$ and
$m^{-}_{ij}$ only exist in very simple scenarios: either $q=0$ and
$\mathcal{E}=\\{1\\}$ (see e.g. [19, Chapter 4]), or $\mu_{i}$ and
$\sigma_{i}$ are constant for all $i\in\mathcal{E}$ (see e.g. [20]). Moreover,
we are as well interested in the present value of the expected ocupation times
in State $j\in\mathcal{E}$ while below level $b\in(0,a)$, defined as

$O_{ij}(u,q,a,b)=\mathds{E}\left(\int_{0}^{\tau_{0}^{-}\wedge\tau_{a}^{+}}e^{-qs}\mathds{1}\left\\{J(s)=j,\,0<X(s)\leq
b\right\\}\mathrm{d}s\,\mid\,J(0)=i,X(0)=u\right).$

Similarly to $m^{-}_{ij}$ and $m^{+}_{ij}$, here we reinterpret $O_{ij}$ as

$O_{ij}(u,q,a,b)=\mathds{E}\left(\int_{0}^{\tau_{0}^{-}\wedge\tau_{a}^{+}\wedge
e_{q}}\mathds{1}\left\\{J(s)=j,\,0<X(s)\leq
b\right\\}\mathrm{d}s\,\mid\,J(0)=i,X(0)=u\right).$

In this section we propose to exploit the approximations laid out in Theorem
3.6, along with the existing theory for multi-regime Markov-modulated Brownian
motions, to provide an efficient approximation scheme based on computing

$\widehat{m}^{-}_{ij}(u,q,a)=\mathds{P}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}^{+}_{a}<e_{q},\widehat{J}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}_{a}^{+})=j,\widehat{X}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}_{a}^{+})=0\,\mid\,\widehat{J}(0)=i,\widehat{X}(0)=u),$
$\widehat{m}^{+}_{ij}(u,q,a)=\mathds{P}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}^{+}_{a}<e_{q},\widehat{J}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}_{a}^{+})=j,\widehat{X}(\widehat{\tau}^{-}_{0}\wedge\widehat{\tau}_{a}^{+})=a\,\mid\,\widehat{J}(0)=i,\widehat{X}(0)=u),$
$\widehat{O}_{ij}(u,q,a,b)=\mathds{E}\left(\int_{0}^{\widehat{\tau}_{0}^{-}\wedge\widehat{\tau}_{a}^{+}\wedge
e_{q}}\mathds{1}\left\\{\widehat{J}(s)=j,\,0<\widehat{X}(s)\leq
b\right\\}\mathrm{d}s\,\mid\,\widehat{J}(0)=i,\widehat{X}(0)=u\right),$

where $\widehat{\tau}_{0}^{-}=\inf\\{t>0:\widehat{X}(t)<0\\}$ and
$\widehat{\tau}_{a}^{+}=\inf\\{t>0:\widehat{X}(t)>a\\}$. Indeed, the
convergence in probability over increasing compact intervals of Theorem 3.6
yields the convergence in probability of $\widehat{\tau}_{0}^{-}$ and
$\widehat{\tau}_{a}^{+}$ to $\tau_{0}^{-}$ and $\tau_{a}^{+}$, respectively.
Due to the continuity of paths of $\widehat{X}$ and $X$, this readily
translates to the pointwise convergence of the functions
$\widehat{m}^{-}_{ij}$ and $\widehat{m}^{+}_{ij}$ to ${m}^{-}_{ij}$ and
${m}^{+}_{ij}$. Additionally, under the assumption $q>0$, the dominated
convergence theorem implies the convergence of $\widehat{O}_{ij}$ to $O_{ij}$.

While the functions $\widehat{m}^{-}_{ij}$ and $\widehat{m}^{+}_{ij}$ are not
readily available in the existing literature, we can compute them by embedding
independent copies of the excursion
$(\widehat{J}_{*},\widehat{X}_{*}):=\\{(\widehat{J}(t),\widehat{X}(t)):t\leq\widehat{\tau}_{0}^{-}\wedge\widehat{\tau}_{a}^{+}\wedge
e_{q}\\}$ into a certain recurrent _queue_ on the strip $[0,a]$ (that is, a
process reflected on the level boundaries $0$ and $a$). This procedure is
similar to the ones exploited in [30, 31, 1] to provide finite-time ruin
probabilities for multi-regime Markov-modulated risk processes or its
subclasses. Below we spell out the details of our construction.

Let
$\\{(\widehat{J}^{\\{\ell\\}}_{*},\widehat{X}^{\\{\ell\\}}_{*})\\}_{\ell\geq
1}$ be a sequence of independent copies of
$(\widehat{J}_{*},\widehat{X}_{*})$. Here we assume that the total length of
each excursion $\widehat{X}^{\\{\ell\\}}_{*}$, say $T^{\\{\ell\\}}$, has
finite mean; this trivially holds if $q>0$, or if either
$\widehat{\tau}_{0}^{-}$ or $\widehat{\tau}_{a}^{+}$ have a finite mean.
Additionally, let our probability space support three independent sequences of
$\mbox{Exp}(1)$-distributed i.i.d. random variables
$\\{\rho_{-M}^{\\{\ell\\}}\\}$, $\\{\rho_{0}^{\\{\ell\\}}\\}$ and
$\\{\rho_{M}^{\\{\ell\\}}\\}$. Based on the construction of multi-regime
queues in [18, 1], here we define $Y$ on a space grid $\\{\zeta_{m}\\}_{-M\leq
m\leq M}$ (with $\zeta_{-M}=0$, $\zeta_{0}=u$ and $\zeta_{M}=a$) modulated by
the jump process $L$ that has a state space
$\mathcal{E}\cup\\{\partial_{0}\\}$, both of which evolve as follows:

  1. (1)

On the time interval $[0,\nu_{1}^{\\{1\\}})$ with
$\nu_{1}^{\\{1\\}}=T^{\\{1\\}}$, let $(L,Y)$ coincide with
$(\widehat{J}^{\\{\ell\\}}_{*},\widehat{X}^{\\{1\\}}_{*})$.

  2. (2)

At time $\nu_{1}^{\\{1\\}}$, one of three events happens:

     * •

If $\widehat{X}^{\\{1\\}}(\nu_{1}^{\\{1\\}})=0$, let
$\nu_{2}^{\\{1\\}}=\nu_{1}^{\\{1\\}}+\rho_{-M}^{\\{1\\}}$ and
$(Y(t)=0,L(t)=L(\nu_{1}^{\\{1\\}}-))$ for all
$t\in[\nu_{1}^{\\{1\\}},\nu_{2}^{\\{1\\}})$.

     * •

If $\widehat{X}^{\\{1\\}}(\nu_{1}^{\\{1\\}})=a$, let
$\nu_{2}^{\\{1\\}}=\nu_{1}^{\\{1\\}}+\rho_{M}^{\\{1\\}}$ and
$(Y(t)=a,L(t)=L(\nu_{1}^{\\{1\\}}-))$ for all
$t\in[\nu_{1}^{\\{1\\}},\nu_{2}^{\\{1\\}})$.

     * •

If $\widehat{X}^{\\{1\\}}(\nu_{1}^{\\{1\\}})\in(0,a)$, let
$\nu_{2}^{\\{1\\}}=\nu_{1}^{\\{1\\}}$.

  3. (3)

At time $\nu_{2}^{\\{1\\}}$, one of two events happens:

     * •

If $Y(\nu_{2}^{\\{1\\}})\leq u$, let $Y$ increase at unit rate up to reaching
level $u$, say, at time $\nu_{3}^{\\{1\\}}$. Let $L(t)=\partial_{0}$ for all
$t\in[\nu_{2}^{\\{1\\}},\nu_{3}^{\\{1\\}})$.

     * •

If $Y(\nu_{2}^{\\{1\\}})>u$, let $Y$ decrease at unit rate up to reaching
level $u$, say, at time $\nu_{3}^{\\{1\\}}$. Let $L(t)=\partial_{0}$ for all
$t\in[\nu_{2}^{\\{1\\}},\nu_{3}^{\\{1\\}})$.

  4. (4)

Let $P^{\\{1\\}}_{*}=\nu_{3}^{\\{1\\}}+\rho_{0}{\\{1\\}}_{u}$ and
$(Y(t)=u,L(t)=\partial_{0})$ for all
$t\in[\nu_{3}^{\\{1\\}},\nu_{4}^{\\{1\\}})$.

  5. (5)

Repeat Steps (1)-(4) for $\ell=2,3,\dots$ shifting the time accordingly in
order to concatenate the excursions along with the increasing sequences
$\\{\nu_{i}^{\\{\ell\\}}\\}_{\ell}$, $i\in\\{1,2,3,4\\}$. This produces a
process $(L,Y)$ that regenerates at the epochs
$\\{\nu_{4}^{\\{\ell\\}}\\}_{\ell\geq 1}$.

See Figure 4.1 for a graphic description of the aforementioned construction.

Figure 4.1. Queue model resulting from concatenating
$\\{\widehat{X}^{\\{\ell\\}}_{*}\\}_{\ell\geq 1}$ using Steps (1)-(4), which
are shown in colors blue, mustard, green and red, respectively.

As pointed out earlier, the process $(L,Y)$ falls within the class of multi-
regime Markov-modulated Brownian motion queueing models (see [18, 1] for
details), which has a law characterized by the level-dependent matrices
$(Q(x),R(x),S(x))_{0\leq x\leq a}$ with the following interpretation:

  * •

$Q_{ij}(x)$ is the jump intensity of $L$ from $i$ to $j$ while $Y$ is in level
$x$. We further specify the dependence on the level $x$ by employing intensity
matrices $\\{Q^{(m)}\\}_{-M+1\leq m\leq M}$ and $\\{\tilde{Q}^{(m)}\\}_{-M\leq
m\leq M}$ where

$Q(x)=\left\\{\begin{array}[]{ccc}Q^{(m)}&\mbox{for}&\zeta_{m-1}<x<\zeta_{m},\\\
\tilde{Q}^{(m)}&\mbox{for}&x=\zeta_{m}.\end{array}\right.$

  * •

$R_{ii}(x)$ is the drift of $Y$ at level $x$ while $L$ is in State $i$ (by
convention we let $R_{ij}(x)=0$ for all $i\neq j$). We further specify this
dependence on the level $x$ by employing diagonal matrices
$\\{R^{(m)}\\}_{-M+1\leq m\leq M}$ and $\\{\tilde{R}^{(m)}\\}_{-M\leq m\leq
M}$ where

$R(x)=\left\\{\begin{array}[]{ccc}R^{(m)}&\mbox{for}&\zeta_{m-1}<x<\zeta_{m},\\\
\tilde{R}^{(m)}&\mbox{for}&x=\zeta_{m}.\end{array}\right.$

  * •

$S_{ii}(x)$ is the diffusion coefficient of $Y$ at level $x$ while $L$ is in
State $i$ (by convention we let $S_{ij}(x)=0$ for all $i\neq j$). We further
specify the dependence on the level $x$ by employing nonnegative diagonal
matrices $\\{S^{(m)}\\}_{-M+1\leq m\leq M}$ where

$S(x)=S^{(m)}\quad\mbox{for}\quad\zeta_{m-1}\leq x<\zeta_{m}.$

Under the previous considerations, the corresponding matrices for our
particular model with state space $\mathcal{E}\cup\\{\partial_{0}\\}$ take the
form

$\displaystyle Q^{(m)}$
$\displaystyle=\begin{pmatrix}\widehat{\Lambda}(\zeta_{m-1})-q\bm{I}&q\bm{1}\\\
\bm{0}&0\end{pmatrix},\quad m\in\\{-M+1,-M+2,\dots,M\\},$
$\displaystyle\tilde{Q}^{(m)}$
$\displaystyle=\begin{pmatrix}\widehat{\Lambda}(\zeta_{m})-q\bm{I}&q\bm{1}\\\
\bm{0}&0\end{pmatrix},\quad m\in\\{-M+1,-M+2,\dots,M-1\\}\setminus\\{0\\},$
$\displaystyle\tilde{Q}^{(-M)}$
$\displaystyle=\tilde{Q}^{(M)}=\begin{pmatrix}-\bm{I}&\bm{1}\\\
\bm{0}&0\end{pmatrix},\quad\tilde{Q}^{(0)}=\begin{pmatrix}\widehat{\Lambda}(\zeta_{0})-q\bm{I}&q\bm{1}\\\
\bm{e}^{\intercal}_{i}&-1\end{pmatrix},$ $\displaystyle R^{(m)}$
$\displaystyle=\begin{pmatrix}\mbox{diag}\\{\widehat{\mu}_{i}(\zeta_{m-1})\\}&\bm{0}\\\
\bm{0}&1\end{pmatrix},\quad m\in\\{-M+1,-M+2,\dots,0\\},$ $\displaystyle
R^{(m)}$
$\displaystyle=\begin{pmatrix}\mbox{diag}\\{\widehat{\mu}_{i}(\zeta_{m-1})\\}&\bm{0}\\\
\bm{0}&-1\end{pmatrix},\quad m\in\\{1,2,\dots,M\\},$
$\displaystyle\tilde{R}^{(m)}$
$\displaystyle=\begin{pmatrix}\mbox{diag}\\{\widehat{\mu}_{i}(\zeta_{m})\\}&\bm{0}\\\
\bm{0}&1\end{pmatrix},\quad m\in\\{-M+1,-M+2,\dots,-1\\},$
$\displaystyle\tilde{R}^{(m)}$
$\displaystyle=\begin{pmatrix}\mbox{diag}\\{\widehat{\mu}_{i}(\zeta_{m})\\}&\bm{0}\\\
\bm{0}&-1\end{pmatrix},\quad m\in\\{1,-M+2,\dots,M-1\\},$
$\displaystyle\tilde{R}^{(-M)}$ $\displaystyle=\begin{pmatrix}\bm{0}&\bm{0}\\\
\bm{0}&1\end{pmatrix},\quad\tilde{R}^{(M)}=\begin{pmatrix}\bm{0}&\bm{0}\\\
\bm{0}&-1\end{pmatrix},\quad\tilde{R}^{(0)}=\begin{pmatrix}\mbox{diag}\\{\widehat{\mu}_{i}(\zeta_{0})\\}&\bm{0}\\\
\bm{0}&0\end{pmatrix},$ $\displaystyle S^{(m)}$
$\displaystyle=\begin{pmatrix}\mbox{diag}\\{\widehat{\sigma}_{i}(\zeta_{m-1})\\}&\bm{0}\\\
\bm{0}&0\end{pmatrix},\quad m\in\\{-M+1,-M+2,\dots,M\\},$

where $\bm{e}^{\intercal}_{i}$ denotes the $i$th canonical row vector, and
$\mbox{diag}\\{a_{i}\\}$ denotes the diagonal matrix with the elements
$\\{a_{1},\dots,a_{p}\\}$ filling the diagonal. In [18], the authors provide
an efficient algorithm to compute the steady state distribution of $(L,Y)$. In
particular, using their method we are able obtain:

  * •

the steady state probability atoms for levels $\zeta_{-M}$ and $\zeta_{M}$
while on the states $\mathcal{E}$ in row vector form, say $\bm{p}_{-M}$ and
$\bm{p}_{M}$ where

$\displaystyle(\bm{p}_{-M})_{j}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathds{P}(L(t)=j,Y(t)=0),$
$\displaystyle(\bm{p}_{M})_{j}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathds{P}(L(t)=j,Y(t)=a).$

  * •

the steady state probability atoms for level $\zeta_{0}$ while on the State
$\partial_{0}$, say $p_{0}$ where

$\displaystyle p_{0}$
$\displaystyle=\lim_{t\rightarrow\infty}\mathds{P}(L(t)=\partial_{0},Y(t)=u).$

  * •

the steady state probability distribution function while on State
$j\in\mathcal{E}$ over $(0,a)$, say $F_{j}$ where

$F_{j}(b)=\lim_{t\rightarrow\infty}\mathds{P}(L(t)=j,0<Y(t)\leq b),\quad
b\in(0,a).$

Below we link these quantities with the first passage probabilities
$\widehat{m}^{+}_{ij}$ and $\widehat{m}^{-}_{ij}$, as well as with the
expected occupation times $\widehat{O}_{ij}$.

######  Theorem 4.1.

Suppose that
$\mathds{E}(\widehat{\tau}_{0}^{-}\wedge\widehat{\tau}_{a}^{+}\wedge
e_{q}\,\mid\,\widehat{J}(0)=i,\widehat{X}(0)=u)<\infty$. Then,

(4.1)
$\widehat{m}^{-}_{ij}(u,q,a)=\frac{(\bm{p}_{-M})_{j}}{p_{0}}\quad\mbox{and}\quad\widehat{m}^{+}_{ij}(u,q,a)=\frac{(\bm{p}_{M})_{j}}{p_{0}}.$

Moreover,

(4.2) $\widehat{O}_{ij}(u,q,a,b)=\frac{F_{j}(b)}{p_{0}},\quad b\in(0,a).$

###### Proof.

The process $(L,Y)$ regenerates at the epochs
$\\{\nu_{4}^{\\{\ell\\}}\\}_{\ell\geq 1}$, all of which have a (common) finite
first moment. This in turn implies that $(L,Y)$ is positive recurrent, so by
[17, Theorem 1],

$\displaystyle\bm{p}_{M}\bm{1}$
$\displaystyle=\frac{\mathds{E}\left(\int_{0}^{\nu_{4}^{\\{1\\}}}\mathds{1}\\{Y(s)=u,L(s)\in\mathcal{E}\\}\mathrm{d}s\right)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)}$
$\displaystyle=\frac{\mathds{P}(Y(\nu_{1}^{\\{1\\}})=u,L(\nu_{1}^{\\{1\\}})\in\mathcal{E})\mathds{E}\left(\rho_{0}^{\\{1\\}}\right)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)}=\frac{\widehat{m}^{+}(u,q,a)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)},$

where we used that $\mathds{E}\left(\rho_{0}^{\\{1\\}}\right)=1$. Employing
similar arguments we get

$\displaystyle\bm{p}_{-M}\bm{1}$
$\displaystyle=\frac{\widehat{m}^{-}(u,q,a)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)}\quad\mbox{and}\quad
p_{0}=\frac{1}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)},$

so that (4.1) follows. Additionally,

$\displaystyle F_{j}(b)$
$\displaystyle=\frac{\mathds{E}\left(\int_{0}^{\nu_{4}^{\\{1\\}}}\mathds{1}\\{L(s)=j,\,0<Y(s)\leq
b\\}\mathrm{d}s\right)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)}$
$\displaystyle=\frac{\mathds{E}\left(\int_{0}^{\widehat{\tau}_{0}^{-}\wedge\widehat{\tau}_{a}^{+}\wedge
e_{q}}\mathds{1}\left\\{\widehat{J}(s)=j,\,0<\widehat{X}(s)\leq
b\right\\}\mathrm{d}s\right)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)}=\frac{\widehat{O}_{j,b}(u,q,a)}{\mathds{E}\left(\nu_{4}^{\\{1\\}}\right)},$

which in turn implies (4.2). ∎

In short, with the help of the algorithms developed in [18], one is able to
efficiently compute first passage probabilities and their expected occupation
times for $(\widehat{J},\widehat{X})$ using Theorem 4.1, which in turn
approximates the first passage probabilities and their expected occupation
times for $(J,X)$ by virtue of Theorem 3.6. Below we explore a couple of
synthetic examples.

######  Example 1.

Consider a three-state hybrid SDE $(X,J)$ evolving in the space-band $(0,1)$
with parameters

$\displaystyle\mu_{1}(x)=0.5,\quad\mu_{2}(x)=0.5(1-x),\quad\mu_{3}(x)=0.5(1-x)^{2},$
$\displaystyle\sigma_{1}=\sigma_{2}=\sigma_{3}=1.$

In essence, the process $X$ has a force pushing it upwards while $J$ is in any
of the three states. The force is weakest while in State $3$ and strongest
while in $1$; note that the difference between regimes is accentuated as $X$
gets closer to level $1$. In the context of actuarial science, this may e.g.
describe the case of the surplus process of an insurance company with three
possibly occurring regimes, all with positive drift, but in some being more
responsive with premium reductions when the surplus approaches $1$. This can
be described, for instance, by an intensity matrix function of the form

$\bm{\Lambda}(x)=10\times\begin{pmatrix}-x&x&0\\\ (1-x)&-1&x\\\
0&1-x&-(1-x)\end{pmatrix}.$

Employing the Matlab coded provided in http://www.hit.bme.hu/~ghorvath based
on [18] together with Theorem 4.1, we are able to compute the first passage
probabilities and expected occupation times of the multi-regime Markov-
modulated Brownian motion $(\widehat{J},\widehat{X})$ that approximates
$(J,X)$. We perform such computations for $q=0$, $a=1$, $i=2$ and $M=50$, with
results shown in Figure 4.2.

Figure 4.2. Left: Plots of $\widehat{m}^{-}_{2j}$ as a function of $u$. Right:
Plots of $\widehat{O}_{2j}$ as a function of $b$ with $u=0.5$. Each plot
contains the cases $j=1,2,3$.

For small values of $u$, downcrossings of level $0$ are most likely while in
State $2$; this can be explained by noting that the random oscillations while
in State $2$ (the initial state of $J$) can produce a downcrossing before a
switching even occurs. For medium values of $u$, the probabilities of
downcrossing level $0$ while in States $1$ and $2$ are comparable, both of
which are considerably larger than that corresponding to State $3$; this is
due to the state-dependent switching behaviour of $\Lambda$ which favours
States $1$ and $2$ while the level is low, and $2$ and $3$ while the level is
high. For high values of $u$, the probability of downcrossing $0$ is uniformly
low for all states. On the other hand, we can observe that the expected
occupation time while in State $2$ is larger than that in State $1$ or $2$
uniformly over all $u$; this is a consequence of State $2$ being the initial
state of the system.

While Theorem 3.6 guarantees that $\widehat{m}^{-}_{ij}$ converges to
$m^{-}_{ij}$ as the space-grid becomes denser, in Figure 4.3 we (empirically)
show how fast this convergence happens.

Figure 4.3. Plots of $\widehat{m}^{-}_{2j}$ as a function of $M$ for the cases
$j=1,2,3$.

As we can appreciate from the plot, convergence is quickly achieved, with
differences between the cases being essentially negligible for $M\geq 40$.

######  Example 2.

Consider now the same parameters as in Example 1, but take
$\mu_{3}(x)=-0.5x^{2}$ and $\sigma_{3}=0$. While this alternative scenario
does not necessarily have an immediate practical interpretation, it does help
to investigate some of the consequences of having a state which lacks any
random noise behaviour. From Figure 4.4, we note that the downcrossing
probabilities for States $1$ and $3$ are larger than those in Example 1; this
is because now we have one of the states pushing the level process downwards.
However, note that the probability of downcrossing $0$ while in State $3$ is
null; this is because $\mu_{3}$ approaches $0$ as the level gets closer to
$0$, meaning that the process simply cannot cross level $0$ while in State
$3$. Moreover, we note that for large values of $b$, the expected occupation
time while in State $3$ is larger than in States $1$ or $2$; this suggests
that before exiting the band $(0,1)$, the process switches to and stays in
State $3$ for a larger amount of time than in Example 1.

Figure 4.4. Left: Plots of $\widehat{m}^{-}_{2j}$ as a function of $u$. Right:
Plots of $\widehat{O}_{2j}$ as a function of $b$ with $u=0.5$. Each plot
contains the cases $j=1,2,3$.

##  5\. Summary and extensions

Under Assumptions 1-5, we considered the class of hybrid SDEs and their space-
grid approximations, the latter belonging to a the family of multi-regime
Markov-modulated Brownian motions. We provided a rigorous proof of their
pathwise convergence, which holds in a probability sense uniformly over
increasing compact intervals. As an application of our convergence result, we
considered the challenging problem of identifying first passage probabilities
and expected occupation times for solutions of hybrid SDEs, for which we
suggest an approximate and computationally efficient answer which employs
developments in the area of multi-regime Markov-modulated queues.

We point out that applications of our convergence result may also be used to
approximate other (and possibly more complex) descriptors of hybrid SDEs. For
instance, one can easily build hybrid SDEs with phase-type jumps by means of
the fluidization method [8]. Other modifications and extensions which are
straightforward to implement in our framework are the Omega model [2, 15], the
Erlangian approximations for finite-time probabilities of ruin [7], and
Parisian ruin problems with phase-type clocks [10].

Acknowledgement. The authors would like to acknowledge financial support from
the Swiss National Science Foundation Project 200021_191984.

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# The integrality of the Genocchi numbers obtained through a new identity and
other results

Bakir FARHI
Laboratoire de Mathématiques appliquées
Faculté des Sciences Exactes
Université de Bejaia, 06000 Bejaia, Algeria
<EMAIL_ADDRESS>
http://farhi.bakir.free.fr/

###### Abstract

In this note, we investigate some properties of the integer sequence of
general term $a_{n}:=\sum_{k=0}^{n-1}k!(n-k-1)!$ ($\forall n\geq 1$) to derive
a new identity of the Genocchi numbers $G_{n}$ ($n\in{\mathbb{N}}$), which
immediately shows that $G_{n}\in{\mathbb{Z}}$ for any $n\in{\mathbb{N}}$. In
another direction, we obtain nontrivial lower bounds for the $2$-adic
valuations of the rational numbers $\sum_{k=1}^{n}\frac{2^{k}}{k}$.

This note replaces and improves the previous one, entitled “On a curious
integer sequence” and referenced by: arXiv:2204.10136v1 [math.NT] 21 Apr 2022.

MSC 2010: Primary 11B65, 11B68, 11B73.
Keywords: Genocchi numbers, Stirling numbers, binomial coefficients, $p$-adic
valuations.

##  1 Introduction and Notation

Throughout this note, we let ${\mathbb{N}}^{*}$ denote the set of positive
integers. For $x\in{\mathbb{R}}$, we let $\lfloor x\rfloor$ denote the integer
part of $x$. For a given prime number $p$ and a given positive integer $n$, we
let $\vartheta_{p}(n)$ and $s_{p}(n)$ respectively denote the usual $p$-adic
valuation of $n$ and the sum of base-$p$ digits of $n$. A well-known formula
of Legendre (see e.g., [6, Theorem 2.6.4, page 77]) states that for any prime
number $p$ and any positive integer $n$, we have

$\vartheta_{p}(n!)=\frac{n-s_{p}(n)}{p-1}.$ (1.1)

Next, we let $s(n,k)$ and $S(n,k)$ (with $n,k\in{\mathbb{N}}$, $n\geq k$)
respectively denote the Stirling numbers of the first and second kinds, which
can be defined as the integer coefficients appearing in the polynomial
identities:

$\begin{split}X(X-1)\cdots(X-n+1)&=\sum_{k=0}^{n}s(n,k)X^{k},\\\
X^{n}&=\sum_{k=0}^{n}S(n,k)X(X-1)\cdots(X-k+1)\end{split}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$

This immediately implies the orthogonality relations:

$\sum_{k\leq i\leq n}s(n,i)S(i,k)=\sum_{k\leq i\leq
n}S(n,i)s(i,k)=\delta_{nk}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n,k\in{\mathbb{N}},n\geq k),$ (1.2)

where $\delta_{nk}$ is the Kronecker delta. Among the many formulas related to
the Stirling numbers, we mention the following (see e.g., [1, §1.14, page
51]):

$\frac{\log^{k}(1+x)}{k!}=\sum_{n=k}^{+\infty}s(n,k)\frac{x^{n}}{n!}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
k\in{\mathbb{N}}),$ (1.3)

which is needed later on. We let finally $B_{n}$ and $G_{n}$
($n\in{\mathbb{N}}$) respectively denote the Bernoulli and the Genocchi
numbers which can be defined by their respective exponential generating
functions:

$\frac{x}{e^{x}-1}=\sum_{n=0}^{+\infty}B_{n}\frac{x^{n}}{n!}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\frac{2x}{e^{x}+1}=\sum_{n=0}^{+\infty}G_{n}\frac{x^{n}}{n!}.$
(1.4)

The famous Genocchi theorem [4] states that the $G_{n}$’s are all integers.
There are at least two beautiful proofs of the Genocchi theorem: the first one
uses the formula $G_{n}=2(1-2^{n})B_{n}$ (see e.g., [1]) together with the
Fermat little theorem and the von Staudt-Clausen theorem, while the second one
uses the remarkable Seidel formula [9]:

$\sum_{k=0}^{n}\binom{n}{k}G_{n+k}=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$

In this note, we give a new proof of the integrality of the $G_{n}$’s by
expressing them in terms of the Stirling numbers of the second kind. The
starting point of this research is the study of the integer sequence
${(a_{n})}_{n\in{\mathbb{N}}}$, defined by:

$a_{0}=0~{}~{}\text{and}~{}~{}a_{n}:=\sum_{k=0}^{n-1}k!(n-k-1)!~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}^{*}).$ (1.5)

This sequence is closely related to the sum of the inverses of binomial
coefficients which is studied by several authors (see [5, 8, 12, 13, 14, 16,
17]). It must be noted that both Stirling numbers, Genocchi numbers and the
numbers $a_{n}$ ($n\in{\mathbb{N}}$) have combinatorial interpretations (see
e.g., [1, 11] for the Stirling numbers, [2, 15] for the Genocchi numbers and
the sequence A003149 of [10] for the $a_{n}$’s).

In another direction, by leaning on Legendre’s formula (1.1), an identity due
to Rockett [8], and another identity due to the author [3], we obtain
nontrivial lower bounds for the $2$-adic valuations of the rational numbers
$\sum_{k=1}^{n}\frac{2^{k}}{k}$ ($n\in{\mathbb{N}}^{*}$).

##  2 The results and the proofs

Our main result is the following:

######  Theorem 2.1.

For all natural number $n$, we have

$G_{n}=\sum_{1\leq\ell\leq k\leq n}(-1)^{k-1}(\ell-1)!(k-\ell)!S(n,k).$ (2.1)

In particular, $G_{n}$ is an integer for any $n\in{\mathbb{N}}$.

To prove this theorem, we need some intermediary results. The first one
(Proposotion 2.2 below) can be immediately derived from the following identity
of Rockett [8]:

$\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}=\frac{n+1}{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^{k}}{k}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$ (2.2)

But for convenience, we prefer reproduce its proof here.

######  Proposition 2.2.

For all natural number $n$, we have

$a_{n}=\frac{n!}{2^{n}}\sum_{k=1}^{n}\frac{2^{k}}{k}.$ (2.3)

###### Proof.

We begin by establishing a recurrent formula for the sequence ${(a_{n})}_{n}$.
For any integer $n\geq 2$, we have

$\displaystyle a_{n}$ $\displaystyle:=\sum_{k=0}^{n-1}k!(n-k-1)!$
$\displaystyle=\sum_{k=0}^{n-2}k!(n-k-1)!+(n-1)!$
$\displaystyle=\sum_{k=0}^{n-2}k!(n-k-2)!(n-k-1)+(n-1)!$
$\displaystyle=n\sum_{k=0}^{n-2}k!(n-k-2)!-\sum_{k=0}^{n-2}(k+1)!(n-k-2)!+(n-1)!.$

But since

$\sum_{k=0}^{n-2}k!(n-k-2)!=a_{n-1}$

and

$\displaystyle\sum_{k=0}^{n-2}(k+1)!(n-k-2)!$
$\displaystyle=\sum_{\ell=1}^{n-1}\ell!(n-\ell-1)!~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by
putting }\ell=k+1)$ $\displaystyle=\sum_{\ell=0}^{n-1}\ell!(n-\ell-1)!-(n-1)!$
$\displaystyle=a_{n}-(n-1)!,$

it follows that:

$a_{n}=na_{n-1}-a_{n}+2\cdot(n-1)!.$

Hence

$a_{n}=\frac{n}{2}a_{n-1}+(n-1)!.$ (2.4)

Further, we remark that Formula (2.4) also holds for $n=1$. Now, according to
Formula (2.4), we have for any positive integer $k$:

$\frac{2^{k}}{k!}a_{k}-\frac{2^{k-1}}{(k-1)!}a_{k-1}=\frac{2^{k}}{k}.$

Then by summing both sides of the last equality from $k=1$ to $n$, we obtain
(because the sum on the left is telescopic and $a_{0}=0$) that:

$\frac{2^{n}}{n!}a_{n}=\sum_{k=1}^{n}\frac{2^{k}}{k},$

which gives the required formula. The proof is achieved. ∎

######  Corollary 2.3.

The exponential generating function of the sequence ${(a_{n})}_{n}$ is given
by:

$\sum_{n=0}^{+\infty}a_{n}\frac{x^{n}}{n!}=\frac{-2\log(1-x)}{2-x}.$ (2.5)

###### Proof.

Using Formula (2.3) of Proposition 2.2, we have

$\sum_{n=0}^{+\infty}a_{n}\frac{x^{n}}{n!}=\sum_{n=1}^{+\infty}\left(\frac{1}{2^{n}}\sum_{k=1}^{n}\frac{2^{k}}{k}\right)x^{n}=\sum_{k=1}^{+\infty}\sum_{n=k}^{+\infty}\frac{1}{2^{n}}\frac{2^{k}}{k}x^{n}=\sum_{k=1}^{+\infty}\frac{2^{k}}{k}\left(\sum_{n=k}^{+\infty}\left(\frac{x}{2}\right)^{n}\right).$

But since
$\sum_{n=k}^{+\infty}\left(\frac{x}{2}\right)^{n}=\left(\frac{x}{2}\right)^{k}\frac{1}{1-\frac{x}{2}}=\frac{x^{k}}{2^{k}}\cdot\frac{2}{2-x}$,
we get

$\sum_{n=0}^{+\infty}a_{n}\frac{x^{n}}{n!}=\frac{2}{2-x}\sum_{k=1}^{+\infty}\frac{x^{k}}{k}=\frac{2}{2-x}\left(-\log(1-x)\right),$

as required. This achieves the proof. ∎

Next, from Corollary 2.3 and Formula (1.3), we derive the following corollary:

######  Corollary 2.4.

For any natural number $n$, we have

$a_{n}=(-1)^{n-1}\sum_{k=0}^{n}G_{k}s(n,k).$ (2.6)

###### Proof.

Let us consider the following three functions (which are analytic on the
neighborhood of zero):

$f(x):=\frac{-2\log(1-x)}{2-x}~{},~{}g(x):=\frac{2x}{e^{x}+1}~{},~{}\text{ and
}h(x):=\log(1-x).$

We easily check that $f=-g\circ h$. Since in addition $h(0)=0$ then the power
series expansion of $f$ about the origin can be obtained by substituting $h$
in the power series expansion of $g$ about the origin (which is given by
(1.4)) and multiplying by $(-1)$. Doing so, we get

$f(x)=-\sum_{k=0}^{+\infty}G_{k}\frac{(h(x))^{k}}{k!}=-\sum_{k=0}^{+\infty}G_{k}\frac{\log^{k}(1-x)}{k!}.$
(2.7)

Further, by substituting in (1.3) $x$ by $(-x)$, we have for any
$k\in{\mathbb{N}}$:

$\frac{\log^{k}(1-x)}{k!}=\sum_{n=k}^{+\infty}(-1)^{n}s(n,k)\frac{x^{n}}{n!}.$

So, by inserting this last into (2.7), we get

$\displaystyle f(x)$
$\displaystyle=-\sum_{k=0}^{+\infty}G_{k}\sum_{n=k}^{+\infty}(-1)^{n}s(n,k)\frac{x^{n}}{n!}$
$\displaystyle=-\sum_{n=0}^{+\infty}\sum_{k=0}^{n}(-1)^{n}G_{k}s(n,k)\frac{x^{n}}{n!}$
$\displaystyle=\sum_{n=0}^{+\infty}\left[(-1)^{n-1}\sum_{k=0}^{n}G_{k}s(n,k)\right]\frac{x^{n}}{n!}.$

Comparing this with Formula (2.5) of Corollary 2.3, we conclude that:

$a_{n}=(-1)^{n-1}\sum_{k=0}^{n}G_{k}s(n,k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}),$

as required. ∎

We finally derive our main result from Corollary 2.4 above by applying the
well-known inversion formula recalled in the following proposition:

######  Proposition 1.

Let ${(u_{n})}_{n\in{\mathbb{N}}}$ and ${(v_{n})}_{n\in{\mathbb{N}}}$ be two
real sequences. Then the two following identities $(I)$ and $(II)$ are
equivalent:

$\displaystyle u_{n}$
$\displaystyle=\sum_{k=0}^{n}v_{k}s(n,k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}),$ ($I$) $\displaystyle v_{n}$
$\displaystyle=\sum_{k=0}^{n}u_{k}S(n,k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$ ($II$)

###### Proof.

Use the orthogonality relations (1.2) (see e.g., [1] or [7] for the details).
∎

###### Proof of Theorem 2.1.

It suffices to apply Proposition 1 for $u_{n}=(-1)^{n-1}a_{n}$ and
$v_{n}=G_{n}$ ($\forall n\in{\mathbb{N}}$). In view of (2.6), Identity $(I)$
holds; so $(II)$ also, that is

$G_{n}=\sum_{k=0}^{n}(-1)^{k-1}a_{k}S(n,k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
n\in{\mathbb{N}}).$

Finally, by substituting in this last equality $a_{k}$ by its expression given
by (1.5), we get for any $n\in{\mathbb{N}}$:

$\displaystyle G_{n}$
$\displaystyle=\sum_{k=1}^{n}(-1)^{k-1}\left(\sum_{i=0}^{k-1}i!(k-i-1)!\right)S(n,k)$
$\displaystyle=\sum_{k=1}^{n}(-1)^{k-1}\left(\sum_{\ell=1}^{k}(\ell-1)!(k-\ell)!\right)S(n,k)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\text{by
putting }\ell=i+1)$ $\displaystyle=\sum_{1\leq\ell\leq k\leq
n}(-1)^{k-1}(\ell-1)!(k-\ell)!S(n,k),$

as required. ∎

Now, we turn to present another type of results providing nontrivial lower
bounds for the $2$-adic valuations of the rational numbers
$\sum_{k=1}^{n}\frac{2^{k}}{k}$ ($n\in{\mathbb{N}}^{*}$).

######  Theorem 2.5.

For any positive integer $n$, we have

$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq s_{2}(n)$ (2.8)

and (more strongly):

$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq
n-\left\lfloor\frac{\log{n}}{\log{2}}\right\rfloor.$ (2.9)

###### Proof.

Let $n$ be a fixed positive integer. Since $a_{n}\in{\mathbb{Z}}$ then we have
$\vartheta_{2}(a_{n})\geq 0$. But, by using Formula (2.3) of Proposition 2.2,
this is equivalent to:

$\vartheta_{2}(n!)-n+\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq
0.$

Then, using the Legendre formula (1.1) for the prime number $p=2$, which says
that $\vartheta_{2}(n!)=\frac{n-s_{2}(n)}{2-1}=n-s_{2}(n)$, we get

$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq s_{2}(n),$

confirming (2.8). To establish the stronger lower bound (2.9), we use the
Rockett formula (2.2) together with the identity:

$\mathrm{lcm}\left\\{\binom{m}{0},\binom{m}{1},\dots,\binom{m}{m}\right\\}=\frac{\mathrm{lcm}(1,2,\dots,m,m+1)}{m+1}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\forall
m\in{\mathbb{N}}),$ (2.10)

established by the author in [3]. According to (2.2) and (2.10), we have that

$\sum_{k=1}^{n}\frac{2^{k}}{k}=\frac{2^{n}}{n}\sum_{k=0}^{n-1}\frac{1}{\binom{n-1}{k}}~{}~{}\text{and}~{}~{}1=\frac{n}{\mathrm{lcm}(1,2,\dots,n)}\cdot\mathrm{lcm}\left\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\right\\}.$

By multiplying side by side these last equalities, we get

$\sum_{k=1}^{n}\frac{2^{k}}{k}=\frac{2^{n}}{\mathrm{lcm}(1,2,\dots,n)}\cdot\mathrm{lcm}\left\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\right\\}\sum_{k=0}^{n-1}\frac{1}{\binom{n-1}{k}}.$

But since the rational number
$\mathrm{lcm}\\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{n-1}\\}\sum_{k=0}^{n-1}\frac{1}{\binom{n-1}{k}}$
is obviously an integer, then we derive that:

$\vartheta_{2}\left(\sum_{k=1}^{n}\frac{2^{k}}{k}\right)\geq\vartheta_{2}\left(\frac{2^{n}}{\mathrm{lcm}(1,2,\dots,n)}\right)=n-\left\lfloor\frac{\log
n}{\log 2}\right\rfloor,$

confirming (2.9) and completes the proof. ∎

##  3 Two open problems

  1. 1.

Find a generalization of Theorem 2.5 to other prime numbers $p$ other than
$p=2$. Notice that the generalization that might immediately come to mind:

$\vartheta_{p}\left(\sum_{k=1}^{n}\frac{p^{k}}{k}\right)\geq s_{p}(n)$

is false for $p>2$ (take for example $n=2$).

  2. 2.

Find a combinatorial interpretation for Formula (2.1) of Theorem 2.1.

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