# Bi-Directional Grid Constrained Stochastic Processes’ Link to Multi-Skew
Brownian Motion

Aldo Taranto, Ron Addie, Shahjahan Khan  Aldo.Taranto@, Ron.Addie@,
Shahjahan.Khan@, @usq.edu.au
School of Sciences
University of Southern Queensland
Toowoomba, QLD 4350, Australia

###### Abstract.

Bi-Directional Grid Constrained (BGC) stochastic processes (BGCSPs) constrain
the random movement toward the origin steadily more and more, the further they
deviate from the origin, rather than all at once imposing reflective barriers,
as does the well-established theory of Itô diffusions with such reflective
barriers. We identify that BGCSPs are a variant rather than a special case of
the multi-skew Brownian motion (M-SBM). This is because they have their own
complexities, such as the barriers being hidden (not known in advance) and not
necessarily constant over time. We provide a M-SBM theoretical framework and
also a simulation framework to elaborate deeper properties of BGCSPs. The
simulation framework is then applied by generating numerous simulations of the
constrained paths and the results are analysed. BGCSPs have applications in
finance and indeed many other fields requiring graduated constraining, from
both above and below the initial position.

###### Key words and phrases:

Wiener Processes, Itô Processes, Reflecting Barriers, Stochastic Differential
Equation (SDE), Stopping Times, First Passage Time (FPT), Multi-Skew Brownian
Motion (M-SBM)

2021 Mathematics Subject Classification: Primary 60G40, Secondary 60J60,
65R20, 60J65.

The first author was supported by an Australian Government Research Training
Program (RTP) Scholarship.

We would like to thank the Editors of this journal for their positive feedback
and invaluable advice on refining this paper.

###### Contents

  1. 1 Introduction
  2. 2 Literature Review
    1. 2.1 Constraining Stochastic Processes by Reflective Barriers
    2. 2.2 Multi-Skew Brownian Motion
  3. 3 Methodology
    1. 3.1 Itô Diffusions Constrained by Two Reflective Barriers
    2. 3.2 Multi-Skew Brownian Motion
    3. 3.3 Constructing BGC Stochastic Processes
  4. 4 Results and Discussion
  5. 5 Conclusions

## Notation

Term | Description
---|---
BGC | Bi-Directional grid constrained
BGCSP | BGC stochastic process
M-SBM | Multi-skew Brownian motion
$X_{t}$ | Stochastic process over time $t$
$B_{t}$ | Brownian motion over time $t$
$W_{t}$ | Wiener process over time $t$
$T$ | Time
$f(X_{t},t)$, $\mu$ | Drift coefficient of $X$ over $t$
$g(X_{t},t)$, $\sigma$ | Diffusion coefficient of $X$ over $t$
$\Psi(X_{t},t)$ | BGC coefficient of $X$ over $t$
$\operatorname{sgn}[X_{t}]$ | Sign function of $X_{t}$
$\mathfrak{B}_{L}$ | Lower barrier
$\mathfrak{B}_{U}$ | Upper barrier
$\tau_{\mathfrak{B}}$ | Stopping time when $\mathfrak{B}$ is hit
$|x|$ | Absolute value of $x$

##  1\. Introduction

In Taranto et al., [33], the concept of Bi-Directional Grid Constrained (BGC)
stochastic processes (BGCSPs) was introduced, and the impact of BGC on the
iterated logarithm bounds of the corresponding stochastic differential
equation (SDE) was derived. In the subsequent paper (Taranto and Khan, 2021b
[32]), the hidden geometry of the BGC process was examined, in particular how
the hidden reflective barriers are formed, the various possible formulations
that can be formed, and an algorithm to simulate BGCSPs was provided. BGCSPs
were also compared to the Langevin equation, the Ornstein-Uhlenbeck process
(OUP), and it was shown how BGCSPs are more complex due to a more general
framework that does not always admit an exact solution. In this paper, we
examine in what respects a BGCSP resembles and in what respect it differs from
a type of multi-skew Brownian motion (M-SBM).

Before we do this, we recall that the BGC term
$\operatorname{sgn}[X_{t}]\Psi(X_{t},t)$, from now onwards simply
$\Psi(X_{t},t)$, unless required to be fully expressed, has been defined as
impacting the $dt$ term as in [33] or the $dW_{t}$ term to a lesser extent as
in [32]. Here, we notice that even if this term is placed as a factor outside
the sum of the $dt$ and $dW_{t}$ terms, it will still constrain the Itô
diffusion in a slightly different, yet fundamentally the same way. Since
$\Psi(x)$ is continuous and differentiable, we require $\Psi(x)$ to include
the infinitessimal $d$, giving $d\Psi(x)$. We thus provide the following third
alternative definition of the BGCSP.

######  Definition 1.1.

(Definition III of BGC Stochastic Processes). For a complete filtered
probability space $(\Omega,\mathcal{F},\\{\mathcal{F}\\}_{t\geqslant
0},\mathbb{P})$ and a (continuous) BGC function
$\Psi(x):\mathbb{R}\rightarrow\mathbb{R}$, $\forall x\in\mathbb{R}$,
$f(X_{t},t)$ is the drift coefficient, $g(X_{t},t)$ is the diffusion
coefficient, $\Psi(X_{t},t)$ is the BGC function. $f(X_{t},t)$, $g(X_{t},t)$
and $\Psi(X_{t},t)$ are convex functions and $\operatorname{sgn}[x]$ is the
sign function defined in the usual sense,

$\displaystyle\operatorname{sgn}[x]$ $\displaystyle=$
$\displaystyle{\begin{cases}\displaystyle\phantom{-}1&,\quad x>0\\\
\displaystyle\phantom{-}0&,\quad x=0\\\ \displaystyle-1&,\quad
x<0\end{cases}}.$

Then, the corresponding BGC Itô diffusion is defined as follows,

$dX_{t}=f(X_{t},t)\,dt+g(X_{t},t)\,dW_{t}-\underbrace{\operatorname{sgn}[X_{t}]\,d\Psi(X_{t},t)}_{\textbf{BGC}}.$
(1.1)

$\blacksquare$

By choosing $\Psi(X_{t},t)$ to be a ‘suitable’ parabolic cylinder function in
relation to the underlying Itô diffusion, as shown in [32], namely
$\Psi(X_{t},t)=\left(\frac{X_{t}}{10}\right)^{2}$, the hidden barriers become
visible when enough simulations hit them. Figure 1 shows that (1.1) is
simulated 1,000 times for both with and without BGC, together with the hidden
lower reflective barrier $a=\mathfrak{B}_{L}$ and the hidden reflective upper
barrier $b=\mathfrak{B}_{U}$ also being displayed.

Figure 1. Hidden Reflective Barriers due to Ideal $\Psi(X_{t},t)$

An even more specialized case that is of great interest, due to its
simplicity, is where drift function $f(x):\mathbb{R}\rightarrow\mathbb{R}$ is
set to $f(x)=\mu$, $\forall x\in\mathbb{R}$ and similarly, the diffusion
function $g(x):\mathbb{R}\rightarrow\mathbb{R}$ is set to $g(x)=\sigma$,
$\forall x\in\mathbb{R}$. One can even define $f(x)$ and $g(x)$ in a more
gradual manner such that in the limit they approach the typical constant
expressions for the drift and diffusion coefficients,

$\lim_{x\rightarrow\infty}f(x)=\mu\quad,\quad\lim_{x\rightarrow\infty}g(x)=\sigma.$
(1.2)

Depending on whether the generalized $f(x)$ and $g(x)$ are used, or whether
the simplified $\mu$ and $\sigma$ are used, then the resulting reflective
boundary theorems will either have more complexity, or less complexity,
respectively.

Before proceeding to the Methodology section, (1.1) is discussed within a
multi-Dimensional context with some examples, which will help explain the
multidimensionality of M-SBMs.

######  Remark 1.2.

By multi-Dimensional diffusion, we are not referring to the usage in papers
such as Sacerdote et al., [27] in which each dimension is reserved for each
possible path or simulation, as shown in Figure 2(a).

(a). (b).

Figure 2. Clarifying Subtle Differences for BGC Stochastic Processes

Notice that in Figure 2(b), the BGCSP exhibits a ‘skew’ and can be considered
as a 2-SBM -which will be elaborated in the Methodology section. Instead of
the usage in Figure 2(a), we use the standard interpretation and generally
accepted usage of the term ‘multi-Dimensional diffusion’ in which each
dimension is reserved for each co-ordinate of the multivariate Itô process, as
shown in Figure 3.

(a). 2-D Itô Diffusion as seen from $\bm{X}$ (b). 2-D Itô Diffusion as seen
from $\bm{Y}$

Figure 3. 2-Dimensional Itô Diffusion as seen from 2 Different Dimensions

We also note that the Methodology, Results and Discussion sections are either
directly multi-Dimensional or can be extended to multi-Dimensional
expressions.

We now have a more geometric understanding of how BGCSPs can be constrained
along multiple dimensions, to be in a better position to express (1.1) in a
more generalized $n$-Dimensional framework in (1.3).

######  Definition 1.3.

(Definition IV - Multi-Dimensional BGC Stochastic Processes). Let $\textstyle
X:[0,\infty)\times\Omega\rightarrow\mathbb{R}^{n}$ defined on a probability
space $(\Omega,\mathcal{F},\mathcal{\\{F\\}}_{t\geqslant 0},\mathbb{P})$ be an
Itô diffusion satisfying the conditions given in the definition of the
1-Dimensional Itô process for each $\\{1\leqslant i\leqslant n\\}$,
$\\{1\leqslant j\leqslant m\\}$, then we can form $n$ 1-Dimensional Itô
processes in an SDE of the form,

$\begin{array}[]{ccc}dX_{1}&=&f_{1}(X_{t},t)\,dt+\underbrace{g_{1,1}(X_{t},t)\,dW_{1}(t)+\cdots+g_{1,m}(X_{t},t)\,dW_{m}(t)}_{\textbf{m}}\\\
&&-\underbrace{\operatorname{sgn}[X_{t}]\,d\Psi_{1}(X_{t},t)-...-\operatorname{sgn}[X_{t}]\,d\Psi_{m}(X_{t},t)}_{\textbf{m
BGC}}\\\ \vdots&\vdots&\vdots\\\
dX_{n}&=&f_{n}(X_{t},t)\,dt+\underbrace{g_{n,1}(X_{t},t)\,dW_{1}(t)+\cdots+g_{n,m}(X_{t},t)\,dW_{m}(t)}_{\textbf{m}}\\\
&&-\underbrace{\operatorname{sgn}[X_{t}]\,d\Psi_{1}(X_{t},t)-...-\operatorname{sgn}[X_{t}]\,d\Psi_{m}(X_{t},t)}_{\textbf{m
BGC}}\end{array}$

(1.3)

where $W_{t}=(W_{1}(X_{t},t),...,W_{m}(X_{t},t))$ is an $m$-Dimensional Wiener
process and $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ and
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times m}$ are the drift and
diffusion fields respectively. For a point $\displaystyle x\in\mathbb{R}^{n}$,
let $\displaystyle\mathbb{P}^{x}$ denote the law of $X$ given initial datum
$\displaystyle X_{0}=x$, and let $\displaystyle\mathbb{E}^{x}$ denote
expectation with respect to $\displaystyle\mathbb{P}^{x}$. Now, (1.3) can also
be expressed in matrix notation as,

$d\mathbf{X}_{t}=\bm{f}_{t}\,dt+\bm{g}_{t}\,d\mathbf{W}_{t}-\bm{h}_{t},$ (1.4)

where,

$\mathbf{X}_{t}=\left(\begin{array}[]{c}dX_{1}(t)\\\ \vdots\\\
dX_{n}(t)\end{array}\right),\text{
}\bm{f}_{t}=\left(\begin{array}[]{c}f_{1}(X_{t},t)\\\ \vdots\\\
f_{n}(X_{t},t)\end{array}\right),\text{
}\bm{g}_{t}=\left(\begin{array}[]{ccc}g_{1,1}(X_{1}(t),t)&\cdots&g_{1,m}(X_{1}(t),t)\\\
\vdots&\ddots&\vdots\\\
g_{n,1}(X_{n}(t),t)&\cdots&g_{n,m}(X_{n}(t),t)\end{array}\right),$

$d\mathbf{W}_{t}=\left(\begin{array}[]{c}dW_{1}(t)\\\ \vdots\\\
dW_{n}(t)\end{array}\right),\text{
}\bm{h}_{t}=\left(\begin{array}[]{c}\operatorname{sgn}[X_{1}(t)]\,d\Psi_{1}(X_{1}(t),t)\\\
\vdots\\\
\operatorname{sgn}[X_{n}(t)]\,d\Psi_{n}(X_{n}(t),t)\end{array}\right),$

for vectors $\bm{f}_{t}$, $\bm{h}_{t}$, $\bm{W}_{t}$ and matrix $\bm{g}_{t}$.
$\blacksquare$

Having reviewed the multi-Dimensional nature of unconstrained and BGC Itô
diffusions, the paper is structured as follows; Section 2 provides the
Literature Review, Section 3 the Methodology, Section 4 the Results and
Discussion, and finally, Section 5 the Conclusion.

##  2\. Literature Review

###  2.1. Constraining Stochastic Processes by Reflective Barriers

The constraining of stochastic processes in the form of discrete random walks
and continuous Wiener processes has been researched for many decades. By
reviewing Weesakul, [34] and the references therein, we see an established and
rigorous analysis of random walks between a reflecting and an absorbing
barrier. Lehner, [16] extended this to 1-Dimensional random walks with a
partially reflecting (semipermeable) barrier. Gupta, [9] generalised this
concept further to random walks in the presence of a multiple function barrier
(MFB) where the barrier can either be partially reflective, absorptive,
transmissive or hold for a moment, but not terminating or killing the random
variable. Dua et al., [7] extended the work of [16] to random walks in the
presence of partially reflecting barriers in which the probability of a random
variable or datum reaching certain states was determined. Lions and Sznitman,
[20] extended the research on reflecting boundary conditions through the
refinement to SDEs. Percus, [24] considered an asymmetric random walk, with
one or two boundaries, on a 1-Dimensional lattice. At the boundaries, the
walker is either absorbed or reflected back to the system. Budhiraja and
Dupuis, [5] considered the large deviation properties of the empirical measure
for 1-Dimensional constrained processes, such as reflecting Wiener processes,
the M/M/1 queue, and discrete-time analogs. L’epingle, [18] examined
stochastic variational inequalities to provide a unified treatment for SDEs
existing in a closed domain with normal reflection and (or) singular repellent
drift. When the domain is a polyhedron, he proved that the reflected-repelled
Wiener process does not hit the non-smooth part of the boundary. Bramson et
al., [4] examined the positive recurrence (to the origin) of reflecting Wiener
processes in 3-Dimensional space. Ball and Roma, [3] examined the detection of
mean reversion within reflecting barriers with an application to the European
exchange rate mechanism (EERM).

###  2.2. Multi-Skew Brownian Motion

The concept of skew Brownian motion (SBM) was first introduced in the book by
Itô and McKean, [13] as a diffusion with a drift represented by a generalized
function, which solves an SDE involving its symmetric local time.
Specifically, an SBM $X=\\{X_{t}\\}_{t\in[0,T]}$ is the solution of,

$X_{t}=B_{t}+(2\alpha-1)L^{0}_{t}(X),\quad\alpha\in(0,1),$

where $B=\\{B_{t}\\}_{t\in[0,T]}$ is a standard Brownian motion (BM). However,
standard Brownian motion is oftentimes abbreviated as SBM, so to reduce any
possible confusion and to eliminate any reference to the original botanical
context of Robert Brown -to which the term ‘Brownian motion’ is attributed -we
will express $B_{t}$ in the rest of this paper in the more mathematically
precise context as a Wiener process $W=\\{W_{t}\\}_{t\in[0,T]}$.
$L^{0}_{t}(X)=\lim_{\epsilon\downarrow
0}\frac{1}{2\epsilon}\int^{t}_{0}\mathds{1}_{\\{|B_{X_{s}}|\leqslant\epsilon\\}}\,ds$
is the symmetric local time at $X_{0}$. Note that for $\alpha=\frac{1}{2}$,
the above equation is reduced to a Wiener process. Harrison and Shepp, [10]
then considered diffusions with a discontinuous local time. The literature on
SBMs was consolidated by Harrison and Shepp, [10] and later by Lejay, [17].
Applications of SMBs were extended by Ramirez, [26] by applying multi-SBM
(M-SBM) and diffusions in layered media that involve advection flows.
Appuhamillage and Sheldon, [1] linked SBMs to existing research by deriving
the first passage time (FPT) of SBM. In 2015, the multiple barrier research of
[26] was extended by Atar and Budhiraja, [2], Ouknine et al., [23] who
collapsed barriers to an accumulation point, and by Dereudre et al., [6] who
derived an explicit representation of the transition densities of SBM with
drift and two semipermeable barriers. Mazzonetto, [21] extended her prior
research [6] on SBMs by deriving exact simulations of SBMs and M-SBMs with
discontinuous drift in her Doctoral dissertation. Gairat and Shcherbakov, [8]
applied SBMs and their functionals to finance. Krykun, [14] also extended the
convergence of SBM with local times at several points that are contracted into
a single one. Mazzonetto, [22] has also recently examined the rates of
convergence to the local time of oscillating and SBMs.

For applications of BGC stochastic processes, the reader is referred to [30],
[28], [29], [31].

##  3\. Methodology

Before proceeding to the main result of this paper, it is instructive to
establish a theoretical foundation by considering the key research for Itô
diffusions constrained by two reflective barriers and then examining the
necessary extensions that need to be derived for M-SBM constrained Itô
diffusions.

###  3.1. Itô Diffusions Constrained by Two Reflective Barriers

Given a filtered probability space
$\Lambda:=(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geqslant
0},\mathbb{P})$ with the filtration $\\{\mathcal{F}_{t}\\}_{t\geqslant 0}$,
then the reflected diffusion $\\{X_{t}:t\geqslant 0\\}$ with two-sided
barriers $\mathfrak{B}_{L}$, $\mathfrak{B}_{U}$ at $a$, $b$ respectively can
be defined as,

$dX_{t}=f(X_{t})\,dt+g(X_{t})\,dW_{t}+\underbrace{\overbrace{d\mathcal{A}_{t}}^{a}-\overbrace{d\mathcal{B}_{t}}^{b}}_{\text{regulators}},\quad
X_{0}\in(a,b),\quad x\in[a,b].$ (3.1)

######  Remark 3.1.

The process $\mathcal{A}$ and $\mathcal{B}$ are known in the literature as
‘regulators’ for the points $a$ and $b$, however, we believe that a better
term is ‘detectors’ because they mainly detect or count how many times $X_{t}$
reaches $a$ and $b$.

Here, the drift $f(x)$ is Lipschitz continuous, the diffusion $g(x)$ is
strictly positive and Lipschitz continuous. $a$, $b$ with
$-\infty<a<b<+\infty$ are given real numbers, and $(W_{t},0\leqslant
t<\infty)$ is the 1-Dimensional standard Wiener process on $\Lambda$. The
processes $\mathcal{A}=\\{\mathcal{A}_{t}\\}_{t\geqslant 0}$ and
$\mathcal{B}=\\{\mathcal{B}_{t}\\}_{t\geqslant 0}$ are the minimal non-
decreasing and non-negative processes, which restrict the process
$X_{t}\in[a,b],\forall t\geqslant 0$. More precisely, the processes
$\\{\mathcal{A}_{t},t\geqslant 0\\}$ and $\\{\mathcal{B}_{t},t\geqslant 0\\}$
increase only when $X_{t}$ hits the boundary $a$ and $b$, respectively, so
that $\mathcal{A}_{0}=\mathcal{B}_{0}=0$, $\mathds{1}$ is the characteristic
function of the set and,

$\int^{\infty}_{0}\mathds{1}_{\\{X_{t}>a\\}}\,d\mathcal{A}_{t}=0,\quad\int^{\infty}_{0}\mathds{1}_{\\{X_{t}<b\\}}\,d\mathcal{B}_{t}=0.$
(3.2)

Furthermore, the processes $\mathcal{A}$ and $\mathcal{B}$ are uniquely
determined by the following properties (Harrison, [11]),

  1. (1)

both $t\rightarrow\mathcal{A}_{t}$ and $t\rightarrow\mathcal{B}_{t}$ are
nondecreasing processes,

  2. (2)

$\mathcal{A}$ and $\mathcal{B}$ increase only when $X=a$ and $X=b$,
respectively, that is
$\int^{t}_{0}\mathds{1}_{\\{X_{s}=a\\}}\,d\mathcal{A}_{s}=\mathcal{A}_{t}$ and
$\int^{t}_{0}\mathds{1}_{\\{X_{s}=b\\}}\,d\mathcal{B}_{s}=\mathcal{B}_{t}$,
for $t\geqslant 0$.

We can consider the two reflective barriers as forming a 2-SBM. Furthermore,
it is instructive for BGCSP to see the two barriers $a$ and $b$ in
$\mathbb{R}^{2}$, shown in Figure 4(a) as embedded in $\mathbb{R}^{3}$ by a
governing BGC surface $\Psi(X_{t},t)$, as shown in Figure 4(b).

(a). Barriers in $\bm{\mathbb{R}^{2}}$ \- Contour plot (b). Barriers in
$\bm{\mathbb{R}^{3}}$\- Contour & surface plot

Figure 4. Diffusion Between Two Constant Reflective Barriers

######  Remark 3.2.

Many papers such as [12] and Linetsky, [19] define the two barriers at the
boundaries of $[0,r]$ for some $r\in\mathbb{R}_{+}$. By applying a series of
transformations, one can map their findings to the context of BGCSP, as shown
in Figure 5.

Figure 5. Mapping Traditional Barriers to BGC Barriers

$\blacksquare$

###  3.2. Multi-Skew Brownian Motion

Having examined Itô diffusions constrained by two reflective barriers, we now
consider the so-called multi-skew Brownian motion, constrained by multiple
barriers of varying degrees of reflectiveness.

######  Definition 3.3.

(M-SBM). A multi-skew Brownian motion (M-SBM) represented (adapted from
Mazzonetto, [21]) by $(\beta_{1},...,\beta_{n})$-SBM, or more simply by
$\beta$-SBM with $n$ semipermeable barriers of varying permeability
coefficients, respectively $\beta=(\beta_{1},...,\beta_{n})$, $x_{0}$ is the
starting position, the coefficients $\beta_{j}\in[-1,1]$, barriers
$x_{1}<...<x_{n}$, local times $L^{x_{j}}_{t}$, and $\mathcal{E}$ is the set
of all parameters of the M-SBM, then the M-SBM is expressed as,

$\displaystyle{\begin{cases}\begin{array}[]{ccl}\displaystyle
dX_{t}&=&\mu\,dt+\sigma\,dW_{t}+\beta_{1}\,dL^{x_{1}}_{t}+...+\beta_{n}\,dL^{x_{n}}_{t}\\\
\displaystyle X_{0}&=&x_{0}\\\
\displaystyle\mathcal{E}&=&\big{\\{}\mu,\sigma,\left(\beta_{1},...,\beta_{n}\right)\big{\\}}\in\mathbb{R}\\\
\displaystyle
L^{x_{j}}_{t}&=&\int^{t}_{0}\mathds{1}_{\\{X_{s}=x_{j}\\}}\,dL^{x_{j}}_{s},\text{
}\forall j\in\\{1,...,n\\}\end{array}.\end{cases}}$

$\blacksquare$

######  Remark 3.4.

The term $\sigma$ has been added to the [21] definition so that the process
can fit a wider range of models. We require that $\beta_{i}\in[-1,1]$. The
cases when $\beta_{i}=\\{-1,1\\}$ are said to exhibit zero permeability (i.e.
impermeability or full reflectiveness), and when $\beta_{i}\in(-1,1)$ the
process is said to exhibit partial reflectiveness (i.e. semi permeability).
Note that a 0-SBM is simply a Wiener process and a $\pm$ $1$-SBM is a
positively/negatively reflected Wiener process. The definition of M-SBM is
illustrated in Figure 6, representing a typical example of M-SBM. The standard
definition of a skew Brownian motion has a drift term $\mu\in\mathbb{R}$
making it no longer, strictly speaking, Brownian motion.

Figure 6. Example Standard M-SBM Framework

The M-SBM of Definition 3.3 allows any barrier combination to be either fully
reflective or semipermeable.

######  Remark 3.5.

If the permeability is $\beta_{j}=1$ at the barrier $x_{j}$ for some
$j\in\\{1,...,j_{1},j,j+1,...,n\\}$ and the initial position is
$x_{0}\in(x_{j},+\infty)$, then the lower barriers $x_{1},...,x_{j-1}$ will
almost surely be never reached [21]. For this to happen, it must be that
$\beta_{j}=-1$, so that as the Itô diffusion descends (down) to $x_{j}$, it is
fully reflected back (up). This of course assumes that the diffusion
coefficient $\sigma$ is ‘relatively small’ and so allows the Itô diffusion to
be ‘well behaved’ and never ‘jump over’ and go below the $x_{j}$ barrier, as
also illustrated in Figure 6.

######  Theorem 3.6.

(Multiple Barriers of M-SBM Merging to One, adapted from Mazzonetto, [21]).
Before expressing the skewness parameter $\beta$ for a general number of
barriers $n$, we derive $\beta$ for the first two simplest scenarios.

If $n=2$, $\beta_{1},\beta_{2}\in[-1,1]$, $\mu\in\mathbb{R}$ and
$x^{(n)}_{2}=x_{1}+\frac{1}{n}$, $\forall n\in\mathbb{N}$. Let,

$\beta:=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}.$

Let us denote by $(X^{(n)}_{t})_{t}$ the $(\beta_{1},\beta_{2})$-SBM with
drift $\mu$, barriers $x_{1},...,x_{n}$, and denote by $(Y_{t})_{t}$ the 1-SBM
with drift $\mu$, and barrier $x_{1}$. Let us assume $X^{(n)}_{0}=Y_{0}$, then
$X^{(n)}$ converges to $Y$ in the following sense,

$\mathbb{E}\left[\sup_{s\in[0,t]}|X^{(m)}_{s}-Y_{s}|\right]\xrightarrow{m\to\infty}0,\text{
}\forall t\geqslant 0.$

The same holds in the case of $n>2$ barriers merging. In this case
$(X^{(n)}_{t})_{t}$ is the $(\beta_{1},...,\beta_{n})$-SBM with drift
$\mu\in\mathbb{R}$, skewness parameters $\beta_{1},...,\beta_{n}\in[-1,1]$ and
barriers $x_{1}\in\mathbb{R}$, $x_{j+1}:=\frac{j}{n}+x_{1},\text{ }\forall
j\in\\{1,...,n-1\\}$.

The skewness parameter $\beta$ of the limit $1$-SBM is given by,

$\beta:=\frac{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})}{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})}.$
(3.3)

If $n$ is even,

$\beta=\frac{\displaystyle\sum^{n}_{j=1}\beta_{j}+\sum_{j_{1}<j_{2}<j_{3}}\beta_{j_{1}}\beta_{j_{2}}\beta_{j_{3}}+...+\sum_{j_{1}<...<j_{n-1}}\beta_{j_{1}}...\beta_{j_{n-1}}}{\displaystyle
1+\sum_{j_{1}<j_{2}}\beta_{j_{1}}\beta_{j_{2}}+\sum_{j_{1}<...<j_{4}}\beta_{j_{1}}\beta_{j_{2}}\beta_{j_{3}}\beta_{j_{4}}+...+\beta_{1}\beta_{2}...\beta_{j_{n}}}.$
(3.4)

If $n$ is odd,

$\beta=\frac{\displaystyle\sum^{n}_{j=1}\beta_{j}+\sum_{j_{1}<j_{2}<j_{3}}\beta_{j_{1}}\beta_{j_{2}}\beta_{j_{3}}+...+\beta_{j_{1}}...\beta_{j_{n}}}{\displaystyle
1+\sum_{j_{1}<j_{2}}\beta_{j_{1}}\beta_{j_{2}}+\sum_{j_{1}<...<j_{4}}\beta_{j_{1}}\beta_{j_{2}}\beta_{j_{3}}\beta_{j_{4}}+...+\sum_{j_{1}<...<j_{n-1}}\beta_{1}\beta_{2}...\beta_{j_{n-1}}}.$
(3.5)

###### Proof.

Refer to [21] and [15]. ∎

The M-SBM framework also only considers one half-plane at a time, so that the
transition density (or distribution) of the upper plane is assumed to be the
same for the lower half plane, which is not always the case (except for
BGCSP). We show below that whilst BGCSPs are a special case of M-SBMs, they
have some unique properties that make them of particular interest among the
larger class.

###  3.3. Constructing BGC Stochastic Processes

We can compliment Mazzonetto by condensing all possible local barrier
combinations to the following four possible global barrier combinations that
comprise a lower barrier $\alpha_{j}$ and an upper barrier $\alpha_{k}$, as
shown in Figure 7.

(a). (b).

(c). (d).

Figure 7. Generalised Barrier Combination Arguments

The diagrammatic summary of possible cases represented in Figure 7 is formally
stated as Lemma 3.7 which is then used in Theorem 3.8. This Theorem formally
expresses that the barriers of an M-SBM merge to a 1-SBM, to which the process
converges.

######  Lemma 3.7.

If any $|\alpha_{j}|<1$ for some $|\alpha_{k}|=1$, or similarly for any
$|\alpha_{k}|<1$, then the barrier $\pm 1$ will dominate the barrier $\neq\pm
1$, almost surely, as shown in Figure 7. Furthermore, if there are more than
two fully reflective barriers, they will merge and effectively operate as one
of the four possible combinations of Figure 7.

###### Proof.

We first assume that there exists only one reflective barrier $|\alpha_{j}|=1$
and $n$ semipermeable barriers $|\alpha_{j}|<1$. We then consider the two
SDEs;

$\begin{array}[]{rcl}X_{t}&=&\mu_{1}\,dt+\sigma_{1}\,dW_{t}+\underbrace{\alpha_{1}\,dL^{\alpha_{1}}_{t}}_{|\alpha_{i}|=1}+\underbrace{\alpha_{2}\,dL^{\alpha_{2}}_{t}+...+\alpha_{n}\,dL^{\alpha_{n}}_{t}}_{|\alpha_{i}|<1}\\\
Y_{t}&=&\mu_{2}\,dt+\sigma_{2}\,dW_{t}+\underbrace{\alpha_{1}\,dL^{\alpha_{1}}_{t}}_{|\alpha_{i}|=1}\end{array},$
(3.6)

where $Y_{t}$ is an unconstrained Itô diffusion and $X_{t}$ is a constrained
Itô diffusion according to the above barrier constraints. Let
$\delta=\sum^{n}_{i=1}\alpha_{i}$, so $\delta<1$ or $\delta\geqslant 1$.

If $\delta<1$, then $\alpha_{i}$ will dominate $\delta$ as it will vanish
(i.e. $\delta\rightarrow 0$) such that,

$\displaystyle\sup_{t\rightarrow\infty}\Big{\\{}|X_{t}|-|Y_{t}|\Big{\\}}=0.$

If $\delta\geqslant 1$, then $\delta$ will dominate $\alpha_{1}$ and merge
(i.e. $\alpha_{1}\rightarrow\delta$) such that,

$\displaystyle\sup_{t\rightarrow\infty}\Big{\\{}|X_{t}|-|Y_{t}|\Big{\\}}=X_{t}.$

Next, assume that there exist two fully reflective barriers $|\alpha_{j}|=1$,
$|\alpha_{k}|=1$ and $n$ semipermeable barriers $|\alpha_{i}|<1$. (3.6) now
equates to,

$\begin{array}[]{rcl}X_{t}&=&\mu_{1}\,dt+\sigma_{1}\,dW_{t}+\underbrace{\alpha_{1}\,dL^{\alpha_{1}}_{t}}_{|\alpha_{i}|=1}+\underbrace{\alpha_{2}\,dL^{\alpha_{2}}_{t}+...+\alpha_{n}\,dL^{\alpha_{n}}_{t}}_{|\alpha_{j}|=1,\text{
}|\alpha_{k}|=1}\\\
&&+\underbrace{\alpha_{1}\,dL^{\alpha_{1}}_{t}}_{|\alpha_{i}|=1}+\underbrace{\alpha_{2}\,dL^{\alpha_{2}}_{t}+...+\alpha_{n}\,dL^{\alpha_{n}}_{t}}_{\delta=|\alpha_{i}|<1}\\\
Y_{t}&=&\mu_{2}\,dt+\sigma_{2}\,dW_{t}+\underbrace{\alpha_{1}\,dL^{\alpha_{1}}_{t}}_{\delta=|\alpha_{i}|<1}\end{array}.$
(3.7)

If $\delta<1$, then $\alpha_{j}$ and (or) $\alpha_{k}$ will dominate $\delta$
and as it will vanish (i.e. $\delta\rightarrow 0$) and if $\delta\geqslant 1$,
then $\delta$ will dominate $\alpha_{j}$ and (or) $\alpha_{k}$ hence merge to
$\alpha_{j}$ and $\alpha_{k}$, such that
$\sup_{t\rightarrow\infty}\Big{\\{}|X_{t}|-|Y_{t}|\Big{\\}}=0$ and
$\sup_{t\rightarrow\infty}\Big{\\{}|X_{t}|-|Y_{t}|\Big{\\}}=X_{t}$,
respectively.

Finally, if there are more than $N\geqslant 3$ fully reflective barriers
$|\alpha_{i}|=1$ and $n$ semipermeable barriers, then the new barriers will
effectively be a linear combination of any two possible combinations in Figure
7, depending on how the fully reflective barriers of $N$ are defined,
completing the proof for all scenarios. ∎

To contrast Figure 4 for two reflective constant barriers, for BGCSP we have
two hidden reflective barriers which also constrain the interior between the
boundaries, as shown in Figure 8, where (a) shows the multiple barriers in
$\mathbb{R}^{2}$, and (b) shows how the multiple barriers are projected from
$\mathbb{R}^{3}$.

(a). Barrier in $\bm{\mathbb{R}^{2}}$ \- Contour plot (b). Barrier in
$\bm{\mathbb{R}^{3}}$\- Contour & surface plot

Figure 8. Diffusion Between Two BGC Reflective Barriers

Leveraging the work of Ramirez, [26], we partition $X$ into countably infinite
intervals $I_{k}=(x_{k},x_{k+1})$, $\forall k\in\mathbb{R}$ forming the
sequence $\\{I_{-\infty},...,I_{-1},I_{1},...,I_{\infty}\\}$ such that the
standard conditions are met; $I_{k}\bigcap I_{k+1}=\varnothing\text{ }\text{
}\forall k\in\mathbb{R}$, $\varnothing\notin X$ and
$\bigcup^{\infty}_{k=-\infty}I_{k}=X$. We wish to shrink the size of each
interval $|I_{k}|=|x_{k+1}-x_{k}|$ to zero as we apply more and more
intervals, where $\lim_{k\rightarrow\infty}|x_{k+1}-x_{k}|\rightarrow 0$ and
$\int^{\infty}_{-\infty}I_{k}\,dk<\infty$. This is because we wish to
constrain the Itô diffusion by the BGC function $\Psi(X_{t},t)\in\mathbb{R}$.

In terms of BGC stochastic processes, we effectively have a
$(\beta_{-n},...,\beta_{-1},\beta_{1},...,\beta_{n})$-SBM and will express it
as,

$\displaystyle{\begin{cases}\begin{array}[]{ccl}\displaystyle
dX_{t}&=&\mu\,dt+\sigma\,dW_{t}+\underbrace{\underbrace{\sum^{-1}_{j=-n}\beta_{j}\,dL^{x_{j}}_{t}}_{<0}+\underbrace{\sum^{n}_{j=1}\beta_{j}\,dL^{x_{j}}_{t}}_{>0}}_{\displaystyle\Psi(X_{t},t)}\\\
\displaystyle X_{0}&=&0\\\
\displaystyle\mathcal{E}&=&\big{\\{}\mu,\sigma,(\beta_{-n},...,\beta_{-1},\beta_{1},...,\beta_{n})\big{\\}}\in\mathbb{R}\\\
\displaystyle
L^{x_{j}}_{t}&=&\int^{t}_{0}\mathds{1}_{\\{X_{s}=z_{j}\\}}\,dL^{x_{j}}_{s},\text{
}j\in\\{-n,...,-1,1,...,n\\}\end{array},\end{cases}}$ (3.8)

as illustrated in Figure 9.

Figure 9. Constructing BGC Stochastic Processes from M-SBM Framework

######  Theorem 3.8.

(Skewness Parameter of BGC Stochastic Processes). Let us denote by
$(X^{(n)}_{t})_{t}$ the $(\beta_{-n},...,\beta_{n})$-SBM with drift $\mu$ and
barriers $x_{-n},...,x_{n}$, and denote by $(Y_{t})_{t}$ the 2-SBM (i.e.
$\beta_{1},\beta_{2}$-SMB) with drift $\mu$, diffusion $\sigma$ and barrier
$x_{1}$. Let us assume $X^{(n)}_{0}=Y_{0}$, then $X^{(n)}$ converges to $Y$ in
the following sense,

$\lim_{n\to\infty}\left\\{\mathbb{E}\Big{[}\sup_{s\in[0,t]}|X^{(n)}_{s}-Y_{s}|\Big{]}\right\\}=0,\text{
}\forall t\in[0,T].$

The same holds in the case of $n>2$ barriers merging. In this case
$(X^{(n)}_{t})_{t}$ is the $(\beta_{-n},...,\beta_{n})$-SBM with drift
$\mu\in\mathbb{R}$, diffusion $\sigma\in\mathbb{R}$, skewness parameters
$\beta_{-n},...,\beta_{n}\in[-1,1]$ and barrier $x_{1}\in\mathbb{R}$,
$x_{j+1}:=\frac{j}{n}+x_{1},\text{ }\forall j\in\\{1,...,n-1\\}$. Then
$\beta=0$.

###### Proof.

In contrast to the skewness parameter of the limit $1$-SBM in (3.3), the
corresponding skewness parameter of the limit $2$-SBM for BGCSP is given by
(3.9),

$\begin{array}[]{rcl}\beta&:=&\underbrace{\frac{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})}{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})}}_{\bm{>0}}+\underbrace{\frac{\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{-n}_{j=-1}(1-\beta_{j})}{\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})}}_{\bm{<0}}\\\
&=&\leavevmode\resizebox{422.77661pt}{}{$\frac{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})\right]\left[\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})\right]+\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\right]\left[\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{-n}_{j=-1}(1-\beta_{j})\right]}{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\right]\left[\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})\right]}$}\\\
&=&\leavevmode\resizebox{422.77661pt}{}{$\frac{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})\right]}{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})\right]}$}\\\
&&+\leavevmode\resizebox{422.77661pt}{}{$\frac{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})\right]}{\left[\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{n}_{j=1}(1+\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1-\beta_{j})\right]}.$}\end{array}$
(3.9)

Noting that due to the symmetry of BGCSP about the origin,

$\displaystyle\prod^{-n}_{j=-1}(1-\beta_{j})=\prod^{n}_{j=1}(1+\beta_{j}),\quad\displaystyle\prod^{-n}_{j=-1}(1+\beta_{j})=\prod^{n}_{j=1}(1-\beta_{j}),$

which allows the $\prod^{n}_{j=1}(1+\beta_{j})$ terms to factor out in (3.9)
giving,

$\begin{array}[]{rcl}\beta&=&\leavevmode\resizebox{422.77661pt}{}{$\frac{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\left[\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})-\left.\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\left[\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})+\left.\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}$}\\\
&&+\leavevmode\resizebox{422.77661pt}{}{$\frac{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\left[\prod^{-n}_{j=-1}(1+\beta_{j})-\prod^{-n}_{j=-1}(1-\beta_{j})-\prod^{n}_{j=1}(1-\beta_{j})+\left.\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}{\displaystyle\prod^{n}_{j=1}(1+\beta_{j})\left[\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})+\left.\prod^{n}_{j=1}(1-\beta_{j})\prod^{-n}_{j=-1}(1+\beta_{j})\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}$}\end{array}$

which expands to,

$\begin{array}[]{rcl}\beta&=&\leavevmode\resizebox{422.77661pt}{}{$\frac{\displaystyle\left[\prod^{-n}_{j=-1}(1-\beta_{j})-\left.\left(\prod^{n}_{j=1}(1-\beta_{j})\right)^{2}\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}{\displaystyle\left[\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})+\left.\left(\prod^{n}_{j=1}(1-\beta_{j})\right)^{2}\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}$}\\\
&&+\leavevmode\resizebox{422.77661pt}{}{$\frac{\displaystyle\left[-\prod^{-n}_{j=-1}(1-\beta_{j})+\left.\left(\prod^{n}_{j=1}(1-\beta_{j})\right)^{2}\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}{\displaystyle\left[\prod^{-n}_{j=-1}(1+\beta_{j})+\prod^{-n}_{j=-1}(1-\beta_{j})+\prod^{n}_{j=1}(1-\beta_{j})+\left.\left(\prod^{n}_{j=1}(1-\beta_{j})\right)^{2}\middle/\prod^{n}_{j=1}(1+\beta_{j})\right.\right]}.$}\end{array}$
(3.10)

It is clear that the numerator equates to 0 and so $\beta=0$, completing the
proof. ∎

######  Remark 3.9.

Due to the bi-directionality of BGC stochastic processes, then $n$ in (3.4) is
always even, so $\beta=0$. From Portenko, [25], if $|x_{i}|\leqslant 1$, then
$|\sum^{n}_{i=1}\alpha_{i}|\geqslant 1$ is of special interest. With BGCSP,
$\alpha_{-n}+\alpha_{n}=0$,
$\alpha_{-n+1}+\alpha_{n-1}=0$,…,$\alpha_{-1}+\alpha_{1}=0$ due to their
symmetry about the origin, hence $|\sum^{n}_{i=1}\alpha_{i}|=0$ as well.

######  Theorem 3.10.

(Cylindrical BGCSPs are 2-SBMs). For a complete filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}\\}_{t\geqslant 0},\mathbb{P})$ and a BGC
function $\Psi(y):\mathbb{R}\rightarrow\mathbb{R}$, $\forall y\in\mathbb{R}$,
then the corresponding BGC Itô diffusion is defined as follows,

$dY_{t}=f(Y_{t},t)\,dt+g(Y_{t},t)\,dW_{t}-\underbrace{\operatorname{sgn}[Y_{t}]\Psi(Y_{t},t)}_{\textbf{BGC}},$
(3.11)

where $f(Y_{t},t)$ is the drift coefficient, $g(Y_{t},t)$ is the diffusion
coefficient, $\operatorname{sgn}[x]$ is the usual sign function, $f(Y_{t},t)$,
$g(Y_{t},t)$, $\Psi(Y_{t},t)$ are convex functions and the 2-SBM is defined
by,

$\displaystyle{\begin{cases}\begin{array}[]{rcl}\displaystyle
dX_{t}&=&\mu\,dt+\sigma\,dW_{t}+\underbrace{\beta_{-1}\,dL^{x_{-1}}_{t}}_{<0}+\underbrace{\beta_{1}\,dL^{x_{1}}_{t}}_{>0}\\\
\displaystyle X_{0}&=&0,\text{
}\mathcal{E}\Big{(}\mu,\sigma,(\beta_{-1},\beta_{1})\Big{)}\\\ \displaystyle
L^{x_{j}}_{t}&=&\int^{t}_{0}\mathds{1}_{\\{X_{s}=z_{j}\\}}\,dL^{x_{j}}_{s},\text{
}j\in\\{-1,1\\}\end{array},\end{cases}}$ (3.12)

then, $Y_{t}\rightarrow X_{t}$ almost surely.

###### Proof.

It is conceivable that under general non-constant $f(X_{t},t)$ and
$g(X_{t},t)$ and some generalized BGC function
$\Psi^{\prime}(f(X_{t},t),g(X_{t},t),X_{t},t)$ that $\Psi^{\prime}(x)$ could
modulate $X_{t}$ such that it is bounded above and below by a constant barrier
at $a$ and $b$, respectively, where $b=-a$. For this theorem, we are required
to prove that constant over time (i.e. cylindrical) BGC functions
$\Psi(X_{t},t)$ will converge almost surely to a 2-SBM. We know from at least
Krykun, [14] that if $|\alpha_{i}|\leqslant 1$, $i\in\\{1,...,n\\}$, there
exists a strong solution to (3.12). Since the BGC functions
$\Psi(X_{t},t)\in\mathbb{R}$ are convex, there exists some value $\kappa$ for
both a hidden lower barrier $\mathfrak{B}_{L}$ and a hidden upper barrier
$\mathfrak{B}_{L}$ that are induced by $\Psi(X_{t},t)$. For BGCSP, there is no
fully reflective barrier defined in advance as there is with M-SBM. However,
there are still two fully reflective barriers in BGCSP because the BGC term
$\Psi(X_{t},t)$ will enable the constrained Itô process $Y_{t}$ to eventually
be overtaken by the underlying unconstrained Itô process $X_{t}$ such that
$|X^{(n)}_{s}|\geqslant|Y_{s}|$ giving,

$\mathfrak{B}_{U}=\kappa\text{ for
}\lim_{n\uparrow\kappa}\left\\{\mathbb{E}\Big{(}\sup_{s\in[0,t]}\Big{|}|X^{(n)}_{s}|-|Y_{s}|\Big{|}\Big{)}\right\\}=0,$
(3.13) $\mathfrak{B}_{L}=-\kappa\text{ for
}\lim_{n\downarrow-\kappa}\left\\{\mathbb{E}\Big{(}\sup_{s\in[0,t]}\Big{|}|X^{(n)}_{s}|-|Y_{s}|\Big{|}\Big{)}\right\\}=0.$
(3.14)

For this to be true, it must be shown that $\kappa>0$ exists. We create a
small neighborhood $\mathcal{N}$ about the initial point $x_{0}$ of radius
$\epsilon\in\mathbb{R}_{+}$ such that
$\mathcal{N}(x_{0})=(x_{0}-\epsilon,x_{0}+\epsilon)$. As
$\epsilon\rightarrow+\infty$, $\mathfrak{B}_{L}$ and $\mathfrak{B}_{U}$ will
eventually lie in $\mathcal{N}(x_{0})$.

If $x_{0}>0$, then $\sup\\{\mathcal{N}(x_{0})\\}=x_{0}+\min(\epsilon)$ such
that $\mathfrak{B}_{U}=\sup\\{\mathcal{N}(x_{0})\\}=\kappa$.
If $x_{0}=0$, then $\sup\\{\mathcal{N}(x_{0})\\}=\inf\\{\mathcal{N}(x_{0})\\}$
such that $\mathfrak{B}_{L}=-\mathfrak{B}_{U}=|\mathfrak{B}_{U}|=\kappa$.
If $x_{0}<0$, then $\inf\\{\mathcal{N}(x_{0})\\}=x_{0}-\min(\epsilon)$ such
that $\mathfrak{B}_{L}=\inf\\{\mathcal{N}(x_{0})\\}=-\kappa$.

Hence $\kappa$ exists and its value is
$\kappa=f(\Psi(X_{t},t),X_{t},t,\mu,\sigma)$ for some function
$f:\mathbb{R}\rightarrow\mathbb{R}$. Having found $\kappa$, we know that the
reflectiveness at $\pm\kappa$, i.e. $|\beta_{\kappa}|=1$,
$|\beta_{-\kappa}|=1$ and before $\pm\kappa$, i.e. $|\beta_{i}|<1$,
$|\beta_{-i}|<1$. Hence,
$\beta_{-\kappa},...,\beta_{-1},\beta_{1},...,\beta_{\kappa}$ for $X_{t}$ must
be scaled for $Y_{t}$ by $\Psi(X_{t},t)$ and since $\Psi(X_{t},t)$ is strictly
convex and symmetrical about the origin, then the ordering is preserved,

$\frac{\beta_{-\kappa}}{\Psi(\kappa,t)}<...<\frac{\beta_{-1}}{\Psi(\kappa,t)}<\frac{\beta_{1}}{\Psi(\kappa,t)}<...<\frac{\beta_{\kappa}}{\Psi(\kappa,t)}.$
(3.15)

(3.15) ensures that a strong solution to BGCSP exists within a 2-SBM
framework, completing the proof. ∎

So far, our formulations of BGC functions have been expressed in the general
form $\Psi(X_{t},t)$, but we have considered BGC barriers induced by time-
independent convex surfaces which can be specified by just $\Psi(X_{t})$,
hence M-SBM is related to BGCSPs with $\Psi(X_{t})$. However, since the
barriers have been specified to be able to change not only under space
(distance) but over time as well, we demonstrate this additional complexity of
BGCSPs in Figure 10.

Figure 10. Example BGC Function $\Psi(X_{t},t)$ Constraining BGCSPs over Space
and Time, more so than in M-SBM

Having developed the M-SBM and 2-SBM frameworks for BGCSP, we can now support
this by numerical simulations in the Results and Discussion section.

##  4\. Results and Discussion

In the following simulations, the underlying unconstrained Itô diffusions have
drift $\mu=0$ and diffusion $\sigma=1$, resulting in just the Wiener process.
This is so that the subsequent impact of BGC can be easily compared. Despite
this, we still refer to these as the more general Itô diffusions because these
parameters can be modified for one’s specific requirements.

To validate the existing M-SBM research and to support our comparison of BGCSP
with M-SBM, we develop Algorithm 1 which is used to progressively introduce
additional reflective barriers. In the subsequent series of simulations, we
introduce 2, 4, 8, 16 and finally 32 semipermeable barriers, with increasing
reflectiveness (i.e. decreasing permeability) the further the Itô diffusion is
from the origin, which are simulated via Algorithm 1.

1 # Pseudocode based on R

2 INPUT:

3 $\mu=drift,\text{ }\sigma=diffusion,\text{ }i=simulation\text{ }index,\text{
}$s$=\\#\text{ }simulations=10,000,\text{ }t=time\text{ }steps=1001,\text{
}j=time\text{ }index,\text{ }Print\\_Simulations=TRUE$

4 OUTPUT:

5 $ID\\_value\leftarrow matrix(0:0,nrow=TimeSteps,ncol=Simulations)$

6 $T\\_1000\leftarrow matrix(0:0,nrow=Simulations,ncol=1)$

7 for  _(i=1:s)_ do

8 for  _(j=1:t)_ do

9 if  _(t $==$1)_ then

10 $ID\\_value[t,i]\leftarrow 0$

11 else

12 $dt=(t/TimeSteps)$

13 $ID\\_value[t,i]\leftarrow(\mu*dt+\sigma*rnorm(1))$

14 $Sum\\_ID\\_value\leftarrow sum(ID\\_value[,i])$

15

16 $\\#$ UPPER BARRIERS$================================$

17 if  _$((Sum\\_ID\\_value >0)\text{ }\&\&\text{
}(Sum\\_ID\\_value<=UpperBarrier\\_01))$_ then

18 Do nothing;

19 else if  _$((Sum\\_ID\\_value >UpperBarrier\\_01)\text{ }\&\&\text{
}(Sum\\_ID\\_value<=UpperBarrier\\_02))$_ then

20
$ID\\_value[t,i]\leftarrow(ID\\_value[t,i]-abs(ID\\_value[t,i]*ID\\_value[t,i])/100)$

21 else if  _$(Sum\\_ID\\_value >UpperBarrier\\_16)$_ then

22 $ID\\_value[t,i]\leftarrow(-abs(ID\\_value[t,i]))$

23 else

24 $ID\\_value[t,i]\leftarrow ID\\_value[t,i]$

25 end if

26

27 $\\#$ LOWER BARRIERS$================================$

28 if  _$((Sum\\_ID\\_value <0)\text{ }\&\&\text{
}(Sum\\_ID\\_value>=LowerBarrier\\_01))$_ then

29 Do nothing;

30

31 else if  _$(Sum\\_ID\\_value <LowerBarrier\\_16)$_ then

32 $ID\\_value[t,i]\leftarrow(abs(ID\\_value[t,i]))$

33 else

34 $ID\\_value[t,i]\leftarrow ID\\_value[t,i]$

35 end if

36

37 end if

38 if  _(Print_Simulations $==$TRUE)_ then

39 if  _(i $==$1)_ then

40 $plot(T,cumsum(ID_{v}alue[,i]),type="l",ylim=c(yMax,yMin))$

41 else

42 $lines(T,cumsum(ID_{v}alue[,i]),type="l",ylim=c(yMax,yMin))$

43 end if

44

45 end if

46

47 end for

48 $T\\_1000[i]<-sum(ID\\_value[,i])$

49 end for

Algorithm 1 Approximating BGC Stochastic Processes via Successive Reflective
Barriers

The simplest application of Algorithm 1 is shown for two fully reflective
barriers in Figure 11.

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 11. 10,000 Simulations of 1,000 Step 1-Dimensional Itô Diffusions With
2 Reflective Barriers

Figure 11 has 2 fully reflective barriers at $\pm 10$ generated using
Algorithm 1. This was then increased to 4 barriers (2 fully reflective and 2
semipermeable) as shown in Figure 12.

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 12. 10,000 Simulations of 1,000 Step 1-Dimensional Itô Diffusions With
4 Reflective Barriers

In Figure 12, we make the barriers at $\pm 20$ fully reflective and the
barriers at $\pm 10$ now to be semipermeable. Although it may not yet be
apparent due to the thickness of the drawn barriers, we have and will continue
to increase the thickness of the barriers to highlight the increasing
reflectiveness (and decreasing semipermeability) the further the Itô
diffusions are from the origin. The total number of barriers was doubled again
to result in 8 barriers (2 fully reflective and 6 semipermeable), as shown in
Figure 13.

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 13. 10,000 Simulations of 1,000 Step 1-Dimensional Itô Diffusions With
8 Reflective Barriers

In Figure 13(b), we start to notice a corrugation or ‘crinkling’ of the
density due to 2 fully reflective barriers and 6 semipermeable barriers. This
was doubled again to 16 barriers (2 fully reflective and 14 semipermeable), as
shown in Figure 14.

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 14. 10,000 Simulations of 1,000 Step 1-Dimensional Itô Diffusions With
16 Reflective Barriers

In Figure 14(b), we notice that the corrugation effect has become more
pronounced due to another doubling of the number of barriers. This was doubled
again to 32 barriers (2 fully reflective and 30 semipermeable), as shown in
Figure 15.

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 15. 10,000 Simulations of 1,000 Step 1-Dimensional Itô Diffusions With
32 Reflective Barriers

Finally, in Figure 15(b), we notice the most amount of the corrugation effect
due to yet another doubling of the number of semipermeable barriers. Due to
the importance of this density as a sufficient approximation of BGC densities,
we plot the density again in Figure 16 without the barriers depicted and
slightly larger.

Figure 16. Typical Density of BGC Itô Diffusions Approximated by 10,000
Simulations of 1,000 Step 1-Dimensional Itô Diffusions With 32 Reflective
Barriers

Figure 16 shows that after 32 barriers, we effectively arrive at the typical
density of BGC Itô diffusions, which have an infinite number of increasingly
reflective barriers, with just 2 fully reflective hidden barriers. If we take
the limit of this numerical approximation process, the number of such barriers
$n$ would approach infinitely many barriers, of smaller and smaller size and
hence smaller constraining contribution. These approximation barriers are thus
replaced by the main BGC function itself,
$\Psi(X_{t},t)=(\frac{X_{t}}{10})^{2}$, as shown in Figure 17, where the
algorithm for BGCSP was stated in [32].

(a). 1,000 Simulations, (b). 10,000 Simulation Density

Figure 17. 10,000 Simulations of 1,000 Step BGC 1-Dimensional Itô Diffusions

From Figure 17, we see the typical characteristics of BGCSPs; 1). a certain
amount of discretization or banding at various local times, 2). the emergence
of two hidden reflective barriers that are not known exactly in advance and
can only be estimated, 3). the density is ‘corrugated’ or ‘rough’.

The random component of the Itô diffusions, i.e. the $dW_{t}$ term is sampled
from a standard normal distribution that is then constrained by BGCSP. The
density of Figure 17(b) has no discontinuities. However, if we sample the path
increments from a discrete binary (i.e. binomial) random distribution, we
obtain a random walk that is then constrained by BGCSP. When a histogram is
derived for the corresponding simulated data rather than fitting a density
through the distribution, we obtain Figure 18, which shows the
discontinuities, also evident in [33].

(a). $\bm{t\in[0,1000]}$, (b). $\bm{t=1000}$.

Figure 18. Histogram of 10,000 Simulations of 1,000 Step BGC 1-Dimensional Itô
Diffusions Showing Discretization or Banding, for Sampling $dW_{t}$ from a
Binomial Distribution

Figure 18 shows that 1). reflection occurs at the barriers as seen by the
peaks on either side of the distribution (more prominent in (a)), and 2).
discretization or banding occurs at prominent local times that are contracted
together due to the BGC function $\Psi(X_{t},t)$. Figure 18(a) shows similar
‘corrugation’ as in the continuous case in Figure 16. Figure 18(b) shows that
at $t=T$, the BGC Itô diffusion is less likely to be at the origin and more
likely to be near the barrier(s). To further demonstrate the characteristics
of BGCSP, a detailed plot of 10,000 simulations are shown in Figures 19 and
20.

Figure 19. 200 BGC Itô Diffusions Sampled from the Standard Normal
Distribution

Figure 20. 1,000 BGC Itô Diffusions Sampled from the Standard Normal
Distribution

##  5\. Conclusions

This paper has extended the previous theoretical research on BGCSP by
comparing them to a type of multi-skew Brownian motion (M-SBM). This was
achieved both theoretically by leveraging existing research, and heuristically
by generating new simulations. Working within the M-SBM framework, we proved
one Lemma and two Theorems for BGCSPs. This research provides a richer
framework in which the semipermeable barriers are modulated in a non-constant
manner over distance $X$, allowing for a new constraining regime that is more
complex than the Ornstein-Uhlenbeck process (OUP) and yet still related to it.
BGCSPs have applications in many fields requiring the constraining of the
underlying stochastic process in a gradual manner where the two ultimate
reflective barriers are not known in advance.

## References

  * Appuhamillage and Sheldon, [2012] Appuhamillage, T. and Sheldon, D. (2012).  First passage time of skew brownian motion.  J. Appl. Probab., 49(3):685–696.
  * Atar and Budhiraja, [2015] Atar, R. and Budhiraja, A. (2015).  On the multi-dimensional skew brownian motion.  Stochastic Process. Appl., 125(5):1911–1925.
  * Ball and Roma, [1998] Ball, C. A. and Roma, A. (1998).  Detecting mean reversion within reflecting barriers: application to the european exchange rate mechanism.  Applied Mathematical Finance, 5(1):1–15.
  * Bramson et al., [2010] Bramson, M., Dai, J., and Harrison, J. (2010).  Positive recurrence of reflecting brownian motion in three dimensions.  The Annals of Applied Probability, 20(2):753–83.
  * Budhiraja and Dupuis, [2003] Budhiraja, A. and Dupuis, P. (2003).  Large deviations for the emprirical measures of reflecting brownian motion and related constrained processes in r+.  Electronic Journal of Prob., 8.
  * Dereudre et al., [2015] Dereudre, D., Mazzonetto, S., and Roelly, S. (2015).  An explicit representation of the transition densities of the skew brownian motion with drift and two semipermeable barriers.  arXiv preprint arXiv:1509.02846.
  * Dua et al., [1976] Dua, S., Khadilkar, S., and Sen, K. (1976).  A modified random walk in the presence of partially reflecting barriers.  J. Appl. Prob., 13:169–175.
  * Gairat and Shcherbakov, [2017] Gairat, A. and Shcherbakov, V. (2017).  Density of skew brownian motion and its functionals with application in finance.  Mathematical Finance, 27(4):1069–1088.
  * Gupta, [1966] Gupta, H. (1966).  Random walk in the presence of a multiple function barrier.  Journ. Math. Sci., 1:18–29.
  * Harrison and Shepp, [1981] Harrison, J. and Shepp, L. (1981).  On skew brownian motion.  Ann. Probab., 9(2):309–313.
  * Harrison, [1986] Harrison, M. (1986).  Brownian Motion and Stochastic Flow Systems.  John Wiley & Sons.
  * Hu et al., [2012] Hu, Q., Wang, Y., and Yang, X. (2012).  The hitting time density for a reflected brownian motion.  Computational Economics, 40(1):1–18.
  * Itô and McKean, [1965] Itô, K. and McKean, H. (1965).  Diffusion Processes and Their Sample Paths.  Springer, New York.
  * Krykun, [2017] Krykun, I. (2017).  Convergence of skew brownian motions with local times at several points that are contracted into a single one.  Journal of Mathematical Sciences, 221(5):671–678.
  * LeGall, [1984] LeGall, J. (1984).  One-dimensional stochastic differential equations involving the local times of the unknown process.  Springer, Berlin, Heidelberg.
  * Lehner, [1963] Lehner, G. (1963).  One-dimensional random walk with a partially reflecting barrier.  Ann. Math. Stat., 34:405–412.
  * Lejay, [2006] Lejay, A. (2006).  On the constructions of the skew brownian motion.  Probab. Surv., 3:413–466.
  * L’epingle, [2009] L’epingle, D. (2009).  Boundary behavior of a constrained brownian motion between reflecting-repellent walls.
  * Linetsky, [2005] Linetsky, V. (2005).  On the transition densities for reflected diffusions.  Advances in Applied Probability, 37(2):435–460.
  * Lions and Sznitman, [1984] Lions, P. and Sznitman, A. (1984).  Stochastic differential equations with reflecting boundary conditions.  Communications on Pure and App. Maths, 37(4):511–537.
  * Mazzonetto, [2016] Mazzonetto, S. (2016).  On the exact simulation of (skew) Brownian diffusions with discontinuous drift (Doctoral dissertation, Universität Potsdam).
  * Mazzonetto, [2019] Mazzonetto, S. (2019).  Rates of convergence to the local time of oscillating and skew brownian motions.  arXiv preprint arXiv:1912.04858.
  * Ouknine et al., [2015] Ouknine, Y., Russo, F., and Trutnau, G. (2015).  On countably skewed brownian motion with accumulation point.  Electron. J. Probab., 20(82):1–27.
  * Percus, [1985] Percus, O. (1985).  Phase transition in one-dimensional random walk with partially reflecting boundaries.  Adv. Appl. Prob., 17:594–606.
  * Portenko, [1976] Portenko, N. (1976).  Generalized diffusion processes.  Springer, Berlin.
  * Ramirez, [2011] Ramirez, J. (2011).  Multi-skewed brownian motion and diffusion in layered media.  Proceedings of the American Mathematical Society, 139(10):3739–3752.
  * Sacerdote et al., [2016] Sacerdote, L., Tamborrino, M., and Zucca, C. (2016).  First passage times of two-dimensional correlated processes: Analytical results for the wiener process and a numerical method for diffusion processes.  Journal of Computational and Applied Mathematics, 296:275–292.
  * [28] Taranto, A. and Khan, S. (2020a).  Bi-directional grid absorption barrier constrained stochastic processes with applications in finance and investment.  Risk Governance & Control: Financial Markets & Institutions, 10(3):20–33.
  * [29] Taranto, A. and Khan, S. (2020b).  Drawdown and drawup of bi-directional grid constrained stochastic processes.  Journal of Mathematics and Statistics, 16(1):182–197.
  * [30] Taranto, A. and Khan, S. (2020c).  Gambler’s ruin problem and bi-directional grid constrained trading and investment strategies.  Investment Management and Financial Innovations, 17(3):54–66.
  * [31] Taranto, A. and Khan, S. (2021a).  Application of bi-directional grid constrained stochastic processes to algorithmic trading.  Journal of Mathematics and Statistics, 17(1):22–29.
  * [32] Taranto, A. and Khan, S. (2021b).  Hidden geometry of bi-directional grid constrained stochastic processes.  arXiv preprint.
  * Taranto et al., [2020] Taranto, A., Khan, S., and Addie, R. (2020).  Iterated logarithm bounds of bi-directional grid constrained stochastic processes.  arXiv Preprint: Modern StochAstic Models & ProbleMs Of Actuarial MaTHematics (MAMMOTH) Conference, pages 1–21.
  * Weesakul, [1961] Weesakul, B. (1961).  The random walk between a reflecting and an absorbing barrier.  Ann. Math. Statist., 32:765–769.