symme y
S
S
A icle
A Dynamical Sys em wi h Random Pa ame e s as a
Ma hema ical Model o Real Phenomena
Jose Diblík 1,* , I ada Dzhallado a 2and Mi osla a R˚užiˇcko á 3
1Facul y o Ci il Enginee ing, B no Uni e si y o Technology, Ve eˇ í 331/95, 602 00 B no, Czech Republic
2Depa men o Compu e Ma hema ics and In o ma ion Secu i y, Na ional Uni e si y o Economics,
03068 Kyi , Pe emogy 54/1, Uk aine; [email p o ec ed]
3Facul y o Ma hema ics, Uni e si y o Białys ok, K. Ciołkowskiego 1M, 15-245 Białys ok, Poland;
mi osla a. [email p o ec ed] o [email p o ec ed]
*Co espondence: [email p o ec ed].cz
Recei ed: 30 Sep embe 2019; Accep ed: 28 Oc obe 2019; Published: 30 Oc obe 2019


Abs ac :
In many cases, i is di icul o ind a solu ion o a sys em o di e ence equa ions wi h
andom s uc u e in a closed o m. Thus, a andom p ocess, which is he solu ion o such a sys em,
can be desc ibed in ano he way, o example, by i s momen s. In his pape , we conside sys ems o
linea di e ence equa ions whose coe icien s depend on a andom Ma ko o semi-Ma ko chain
wi h jumps. The momen equa ions a e de i ed o such a sys em when he andom s uc u e is
de e mined by a Ma ko chain wi h jumps. As an example, h ee p ocesses: Th ea s o secu i y in
cybe space, adioca bon da ing, and s abili y o he o eign cu ency exchange ma ke a e modelled
by sys ems o di e ence equa ions wi h andom pa ame e s ha depend on a semi-Ma ko o
Ma ko p ocess. The momen equa ions a e used o ob ain he condi ions unde which he p ocesses
a e s able.
Keywo ds:
Ma ko and semi-Ma ko chain; andom ans o ma ion o solu ions;
L2
-s abili y; jumps
o solu ions; momen equa ions
1. In oduc ion
Many eal p ocesses in bo h esea ch and p ac ice can be well modelled by dynamical sys ems
wi h andom pa ame e s. Dynamical sys ems o his kind a e widely used o modelling in biology
and medicine [1], in sociology o socioeconomics [2,3], and in inance [4–8]. They can also be applied
o modelling secu i y and isk managemen in cybe space [9,10] and in many o he a eas.
I is known ha , in gene al, di e ence equa ions wi h a andom s uc u e canno be sol ed in
a closed o m excep o se e al classes o such equa ions. In hese cases o equa ions and sys ems,
s a is ical cha ac e is ics o he solu ion a e sough . Thus, inding a solu ion o an equa ion wi h
a andom s uc u e in a b oade sense means inding he s a is ical cha ac e is ics o he solu ion.
The e a e se e al me hods o ob aining such a solu ion. By one o hem he p oblem o de e mining
he p obabili y densi y unc ion o a solu ion ans o ms in o one o ind a solu ion o he Kolmogo o
o wa d (Fokke –Plank) equa ion, which is a pa ial di e en ial equa ion ha desc ibes he ime
e olu ion o he pd [
11
]. Ano he op ion is o ind an app oxima e solu ion o s ochas ic di e en ial
equa ions ha can be ob ained di ec ly using nume ical me hods [
12
,
13
]. A ex book by Kloeden and
Pla en [
14
] desc ibes many such algo i hms. A s a is ical app oach o bio idelic modelling is used
in [15,16].
The me hod we p opose is one o momen equa ions. I ies o ind he momen s o he solu ion
wi hou using he densi y unc ion. I cons uc s a de e minis ic sys em o momen equa ions o
a gi en dynamical sys em wi h andom s uc u e ha can be sol ed using well-known me hods,
Symme y 2019,11, 1338; doi:10.3390/sym11111338 www.mdpi.com/jou nal/symme y
Symme y 2019,11, 1338 2 o 14
wi h he solu ions o such a sys em o momen equa ions beha ing like momen s o solu ions o he
sys em wi h andom s uc u e. The o igin o his me hod can be ound in he wo ks by Valee and his
scien i ic school [
17
]. The de i a ion o momen equa ions o some classes o dynamical sys ems wi h
andom coe icien s and hei use o sol ing ce ain eal p oblems can be ound in ou p e ious wo ks.
The p oblem o na iga ion o a a ge , o example, is conside ed in [18] using he momen equa ions
o a non-homogenous linea sys em o equa ions wi h andom s uc u e, which was de i ed in [
19
].
The momen equa ions o sys ems o linea and nonlinea di e en ial and di e ence equa ions wi h a
andom s uc u e de e mined by a Ma ko o semi-Ma ko p ocess a e also de i ed in [20–22].
In [
9
], momen equa ions a e ob ained o sys ems o di e ence equa ions i he andom s uc u e
is de e mined by a semi-Ma ko chain wi h jumps. In he p esen pape , we deal wi h he de i a ion
o momen equa ions o sys ems o di e ence equa ions wi h andom coe icien s i hese coe icien s
depend on a Ma ko chain wi h jumps. To de i e momen equa ions, we use he esul s o [9].
We will s udy sys ems in a p obabili y space
(Ω
,
F
,
P)
, whe e
Ω
is he sample space,
F
is he se
o all possible e en s, and
P
is some p obabili y measu e on
Ω
. Le a sequence
ξ={ξi}∞
i=1
o andom
a iables
ξi:Ω→S
,
i=
0,1,2,
. . .
be a andom Ma ko o semi-Ma ko chain. In ou conside a ions,
Sis he s a e space o all andom a iables o which he e exis s a squa ed i s -o de momen .
In such a p obabili y space, we conside a non-s a iona y sys em o linea di e ence equa ions
Xk+1=A(k,ξk)Xk,k=0, 1,2, . . . , (1)
X0=ϕ, (2)
whe e he s a e unc ion
Xk
is an
m
-dimensional column ec o - unc ion wi h he ini ial s a e
X0=ϕ
,
A
is an
m×m
ma ix whose elemen s depend on he Ma ko o semi-Ma ko chain
ξ
wi h jumps a
poin s kj,j=0,1,2, . . ..
The s a e
m
-dimensional column ec o - unc ion
Xkk=
0,1,2,
. . .
is called a solu ion o ini ial
Cauchy p oblem
(1)
,
(2)
wi hin he meaning o a s ong solu ion i i sa is ies
(1)
wi h ini ial condi ion
(2)
,
see [23].
The es o he pape is o ganized as ollows. In Sec ion 2, some p elimina y ema ks and auxilia y
esul s a e in oduced. The main esul s conce ning momen equa ions o sys em
(1)
wi h Ma ko
swi ching a e p o ed in Sec ion 3. In Sec ion 4, selec ed h ee p ocesses a e modelled by di e ence
equa ions wi h andom s uc u e depending on a Ma ko o semi-Ma ko chain. The e we use he
momen equa ions o sys em
(1)
ob ained in Sec ion 3 o de e mine he s abili y domain o p ocesses
men ioned abo e. The las Sec ion 5sugges s di ec ions o u he esea ch in his a ea.
2. P elimina y Rema ks
Assume ha he andom Ma ko o semi-Ma ko chain
ξ
can be in
n
possible s a es
θ1
,
θ2
,
. . .
,
θn
.
Thus, o any ealiza ion
θs
,
s=
1,
. . .
,
n
, he ini ial Cauchy p oblem
(1)
,
(2)
de e mines an ini ial
Cauchy p oblem o non-s a iona y sys ems o linea equa ions
Xk+1,s=As(k)Xk,s,k=0, 1, . . . , s=1, . . . , n, (3)
X0=ϕ, (4)
whe e A(k,ξk=θs) = As(k),k=0,1, . . . , s=1, . . . , n, i ξk=θs,s=1, . . . , n.
Le
Ns(k)
,
s=
1,
. . .
,
n
, be
m×m
undamen al ma ices o solu ions o ini ial Cauchy
p oblem
(3)
,
(4)
such ha
Ns(
0
) = I
,
s=
1,
. . .
,
n
whe e
I
is he
m×m
iden i y ma ix. Then,
he solu ions o (1), (2) can be w i en in he o m
Xk,s=Ns(k)X0,s,k=0, 1, . . . , s=1, . . . , n.
Symme y 2019,11, 1338 3 o 14
Now, he jumps o solu ions a e associa ed wi h he momen s o jumps,
kj
,
j=
0,1,2,
. . .
, and,
o any m×m egula cons an ma ix Cls,l,s=1, . . . , n, hey ha e he ollowing o m
Xk=Ns(k−kj−1)Xkj−1,kj−1≤k≤kj,j=1, 2, . . . ,
Xkj=Cls Ns(k−kj−1)Xkj−1, de Cls 6=0, l,s=1, . . . , n,
Xk=Nl(k−kj)Xkj,kj≤k≤kj+1.
(5)
In he cons uc ion o momen equa ions, i is app op ia e o use he p obabili y densi y unc ion
o he andom a iable. Thus, we ex end his concep o he case o a disc e e andom a iable
X
using
he in eg able Di ac del a unc ion, as he unc ion
(x) = ∑
s
psδ(x−xs),
which means ha he disc e e andom a iable Xcan assume alue xswi h p obabili y
ps=P(X=xs),s=1, 2, . . . , ∑
s
ps=1.
Gi en his ex ension, we can de e mine he momen s o a disc e e andom a iable in he same
way as o a con inuous one.
De ini ion 1.
Le
Xk∈Rm
be a andom a iable depending on a andom chain
ξ
wi h
n
possible s a es
θs
,
s=1, 2, . . . , n. The ec o unc ion
E(1){Xk}=
n
∑
s=1
E(1)
s{Xk}
whe e
E(1)
s{Xk}=Z
Rm
x s(k,x)dx,s=1, . . . , n, (6)
is called he i s -o de momen o he andom a iable
Xk
. The alues
E(1)
s{Xk)}s=
1,
. . .
,
n
, a e called he
i s -o de pa icula momen s co esponding o he s a es o he andom a iable
Xk
wi h pa icula densi y
unc ions s(k,x), k =0, 1, . . ..
The ma ices
E(2){Xk}=
n
∑
s=1
E(2)
s{Xk}
whe e
E(2)
s{Xk}=Z
Rm
x x∗ s(k,x)dx,s=1, . . . , n, (7)
a e called he second-o de momen s o he andom a iable
Xk
. The alues
E(2)
s{Xk}
,
s=
1,
. . .
,
n
, a e called
he second-o de pa icula momen s.
De ini ion 2.
The i ial solu ion o sys em
(1)
is said o be
L2
-s able, i , o any solu ion
Xk
,
k=
0,1,
. . .
o
sys em (1), he se ies
∞
∑
k=0E(1)kXkk2con e ges.
Rema k 1.
I is easy o see ha he i ial solu ion o sys em
(1)
is
L2
-s able i and only i he ma ix se ies
∞
∑
k=0E(2)Xk, o
∞
∑
k=0E(1){XkXT
k}is con e gen .
Symme y 2019,11, 1338 4 o 14
2.1. Ma ko Chain ξis Ma ko ian
Assume ha he andom chain
ξ
is Ma ko ian ha can be in
n
possible s a es
θ1
,
θ2
,
. . .
,
θn
wi h p obabili ies
ps(k) = Pξk=θs,k=0, 1,2, . . . , s=1, 2, . . . , n
sa is ying he sys em o di e ence equa ions
ps(k+1) =
n
∑
l=1
πsl pl(k),k=0,1,2, . . . , s=1, 2, . . . , n
whe e
πsl(k+
1
) = Pξk+1=θlξk=θs
,
k=
0,1,2,
. . .
,
s
,
l=
1,2,
. . .
,
n
a e ansi ion p obabili ies
om one s a e o ano he .
2.2. Ma ko Chain ξis Semi-Ma ko ian
Assume ha a andom chain
ξ
is semi-Ma ko ian wi h
n
possible s a es
θ1
,
θ2
,
. . .
,
θn
.
The ansi ion in ensi ies
qls(k)l
,
s=
1,
. . .
,
n
om s a e
θs
o
θl
a ime
k
sa is y he ollowing condi ions
qls(k)⩾0,
∞
∑
k=1
qls(k) = πls,l,s=1, . . . , n, (8)
qs(k) =
n
∑
l=1
qls(k),s=1, . . . , n,
∞
∑
k=1
qs(k) = 1.
When o mula ing ou esul s, we use he concep o ma ix ope a o s.
De ini ion 3.
In he p obabili y space
(Ω
,
F
,
P)
, le wo andom a iables
X:Ω→Rm
and
Y:Ω→Rm
be
de ined wi h p obabili y densi y unc ions 1(x)and 2(y) espec i ely. Then, he ope a o
L: 1(x)→ 2(y)o 2(y) = L 1(x)
is said o be s ochas ic.
De ine he ope a o
ψ(k):=diagψ1(k), . . . , ψn(k),ψs(k)≡
∞
∑
i=k+1
qs(i),s=1, . . . , n. (9)
In [
9
], he momen equa ions o sys em (1) wi h semi-Ma ko swi ching a e de i ed. A simila
esul o a sys em (1) in which he semi-Ma ko chain is ans o med in o a Ma ko chain will be
de i ed using he esul s ob ained in [9].
Theo em 1
([
9
])
.
Le
Xk
,
k=
0,1,2,
. . .
,be solu ions o sys em
(
1
)
wi h semi-Ma ko swi ching and
jumps
(
5
)
. Then, he ec o
E(1)Xk
o he i s -o de momen s is de e mined by he sys em o equa ions o
pa icula momen s o he i s o de E(1)
sXk,s=1, 2, . . . , n, k =1, 2, . . .,
E(1)
sXk=ψ(k)Ns(k)E(1)
sX0+
k
∑
kj=k1
ψs(k−kj)Ns(k−kj)Vs(kj), (10)
Vs(k) =
n
∑
i=1
qsi(k)Csi Ni(k)E(1)
iX0+
k−k1
∑
kj=k1
n
∑
i=1
qsi(k−kj)Csi Ni(k−kj)Vi(kj). (11)
Symme y 2019,11, 1338 5 o 14
The ma ix
E(2)Xk
o he second-o de momen s is de e mined by he sys em o pa icula second-o de
momen s E(2)
sXk,s=1, 2, . . . , n, k =1, 2, . . .,
E(2)
sXk=ψs(k)Ns(k)E(2)
sX0NT
s(k)
+
k
∑
kj=k1
ψs(k−kj)Ns(k−kj)Ws(kj)NT
s(k−kj), (12)
Ws(k) =
n
∑
i=1
qsi(k)Csi Ni(k)E(2)
iX0NT
i(k)CT
si
+
k−k1
∑
kj=k1
n
∑
i=1
qsi(k−kj)Csi Ni(k−kj)Wi(kj)NT
i(k−kj)CT
si. (13)
Theo em 2 ([9]).Le he sums
Is=
∞
∑
k=0
ψs(k)Ns(k)NT
s(k),s=1, . . . , n
con e ge and
Is>
0,
s=
1,
. . .
,
n
. Then, o he i ial solu ion o sys em
(1)
wi h jumps o solu ions
(5)
o be
L2-s able, i is necessa y and su icien ha he ollowing equi alen condi ions hold:
(1)
he e exis s a solu ion
Bs=E(2)
sX0+Ws>
0 o sys em o ma ix Equa ions
(12)
and
(13)
unde he
condi ion E(2)
sX0>0,
(2) he successi e app oxima ions
B(j+1)
s=E(2)
sX0+
n
∑
l=1
∞
∑
k=1
qsl(k)Csl Nl(k)B(j)
lNT
l(k)CT
sl, (14)
B(0)
s=0, s=1, . . . , n,j=0, 1,2, . . . .
a e con e gen .
3. Momen Equa ions o Di e ence Sys ems wi h Random Jumps
Fi s we conside sys em (1) wi h semi-Ma ko swi ching and piecewise cons an coe icien s
Xk+1=A(ξk)Xk,k=0, 1,2, . . . . (15)
I we deno e
As=A(ξk=θs),
hen, in each o he ealiza ions o he semi-Ma ko chain ξ, sys em (15)can be w i en as
Xk+1=AsXk,k=0, 1,2, . . . , s=1, 2, . . . , n
wi h he undamen al ma ices o solu ions being in he o m
Ns(k) = Ak
s,k=0, 1,2, . . . , s=1, 2, . . . , n. (16)
Theo em 3.
The ec o
E(1)Xk
o he i s -o de momen s o sys em
(15)
is de e mined by he sys em o
equa ions o pa icula momen s o he i s o de E(1)
sXk,s=1, 2, . . . , n, k =1, 2, . . .,
E(1)
sXk=ψs(k)Ak
sE(1)
sX0+
k
∑
j=1
ψs(k−j)Ak−j
sVs(j), (17)

Symme y 2019,11, 1338 6 o 14
Vs(k) =
n
∑
i=1
qsi(k)Csi Ak
iE(1)
iX0+
k−1
∑
j=1
n
∑
i=1
qsi(k−j)Csi Ak−j
iVi(j). (18)
The ma ix
E(2)Xk
o he second-o de momen s is de e mined by he sys em o pa icula second-o de
momen s E(2)
sXk,s=1, 2, . . . , n, k =1, 2, . . .,
E(2)
sXk=ψs(k)Ak
sE(2)
sX0AT
sk+
k
∑
j=1
ψs(k−j)Ak−j
sWs(j)AT
sk−j, (19)
Ws(k) =
n
∑
i=1
qsi(k)Csi Ak
iE(2)
iX0AT
ikCT
si +
k−1
∑
j=1
n
∑
i=1
qsi(k−j)Csi Ak−j
iWi(j)AT
ik−jCT
si. (20)
P oo .
The conclusion o he heo em ollows di ec ly om Theo em 1wi h
(16)
applied, and he ac
ha kj=0, 1,2, . . . o j=0, 1,2, . . ..
Nex , we conside sys em (1) wi h Ma ko swi ching and piecewise cons an coe icien s
Xk+1=AsXk,k=0, 1,2, . . . (21)
whe e As=A(ξk=θs)in each o he ealiza ions o he Ma ko chain ξ.
Condi ions
(8)
and
(9)
imply ha he semi-Ma ko chain is ans o med in o a Ma ko chain
unde he assump ion:
ψs(k) = πk
ss,s=1, 2, . . . , n,k=1, 2, . . . ,
qjs(k) = (0, i j=s,
πjs πk−1
ss , i j6=s,j,s=1, 2, . . . , n,k=1, 2, . . . . (22)
Theo em 4.
The i s -o de momen ec o
E(1)Xk
and he second-o de momen ma ix
E(2)Xk
o
sys em (21)a e de e mined by he sys em o equa ions o he i s -o de pa icula momen s
E(1)
sXk+1=
n
∑
i=1
πsiCsi AiE(1)
iXk,s=1, 2, . . . , n,k=1, 2, . . . , (23)
and he second-o de pa icula momen s
E(2)
sXk+1=
n
∑
i=1
πsiCsi AiE(2)
iXkAT
iCT
si,s=1, 2, . . . , n,k=1,2, . . . , (24)
espec i ely.
P oo . Using (22), sys em (17), (18) akes he o m
E(1)
sXk=πk
ss Ak
sE(1)
sX0+
k
∑
j=1
πk−j
ss Ak−j
sVs(j), (25)
Vs(k) =
n
∑
i=1
i6=s
πsiCsi πk−j
ii Ak
iE(1)
iX0+
k−1
∑
j=1
πk−j−1
ii Ak−j−1
sVi(j)!. (26)
Symme y 2019,11, 1338 7 o 14
Ma ching he igh -hand sides o sys ems (25) and (26), we ob ain he equa ions
Vs(k) =
n
∑
i=1
i6=s
πsiCsi AiE(1)
iXk−1,s=1, 2, . . . , n. (27)
Replacing kwi h k+1 in (25), we ob ain he equa ions
E(1)
sXk+1=πk+1
ss Ak+1
sE(1)
sX0+
k
∑
j=1
πk+1−j
ss Ak+1−j
sVs(j),
o , wi h espec o (27), we ob ain
πss AsE(1)
sX0+Vs(k+1) = πss AsE(1)
sX0+
n
∑
i=1
i6=s
πsiCsi AiE(1)
iXk,
which can be w i en in he o m (23). Simila ly, i is possible o p o e (24).
Rema k 2.
I sys em
(1)
has no jumps, hen he momen Equa ions
(23)
and
(24)
coincide wi h hose ob ained
in [6].
4. Model P oblems
In his sec ion, we will p esen h ee model p oblems: Th ea s o secu i y in cybe space,
adioca bon da ing, and s abili y o o eign cu ency exchange ma ke . These p ocesses a e modelled
by di e ence equa ions wi h andom pa ame e s ha depend on a semi-Ma ko o Ma ko p ocess.
P ocess s abili y is in es iga ed using he momen equa ions de i ed abo e.
4.1. Th ea s o Secu i y in Cybe space Modelled by a Sys em wi h Semi-Ma ko Pa ame e s
The e a o a i icial in elligence is coming and he cybe space is becoming a place o nume ous
con lic s ha di e in o m and me hod, in ensi y and deg ee o h ea s. This o ces us o de elop new
solu ions and models ha allow us o an icipa e he h ea in ad ance.
The e a e a ious c i e ia o in o ma ion sys em secu i y isks classi ica ion ha p o ide an
o e iew o mos o he h ea models. The cu en endency is o desc ibe he di e si y o si ua ions o
exposu e o limi ed in o ma ion on a ious h ea s, conside ing he desc ip ion o he g ea es possible
numbe o ac o s in luencing he sa e y o in o ma ion. This is usually a classi ica ion a chi ec u e ha
guides o ganiza ions o implemen in o ma ion secu i y s a egies.
Ou app oach o his p oblem is di e en . We c ea e a dynamical sys em o di e ence
equa ions wi h coe icien s depending on a Ma ko o semi-Ma ko chain as a ma hema ical model.
The eme gence o eal h ea s o cybe a acks wi h ce ain p obabili ies implemen s he ansi ion o
he sys em om one s a e o ano he . Thus, he ansi ion om one s a e o ano he is a ec ed by he
simples s eams o e en s wi h he co esponding in ensi y o de ec ion o elimina ion. The andom
pa ame e s o h ea s ob ained om he s a is ics o hei occu ence and elimina ion can se e as he
inpu pa ame e s. Ou goal is o de e mine he s abili y domain o an in o ma ion sys em, ha is, i he
sys em is eady o ope a e in condi ions o i s secu i y.
Suppose ha he wo k o an in o ma ion sys em is desc ibed by sys em
(15)
wi h jumps
o solu ions
xk+1=c xk,k=0,1,2,. . .
and he semi-Ma ko chain in (15) can be in h ee possible s a es:
θ1—
he sys em ope a es in a h ea - ee en i onmen , o h ea s appea wi h ansi ion in ensi ies
q1s,s=1,2,3, bu do no ep esen damage wi h p obabili y q1;
Symme y 2019,11, 1338 8 o 14
θ2—
he sys em ope a es in an en i onmen whe e h ea s occu wi h ansi ion in ensi ies
q2s
,
s=
1,2,3, bu he p og amme is eady o e lec i wi h p obabili y q2;
θ3—
he sys em ope a es in an en i onmen whe e h ea s occu wi h ansi ion in ensi ies
q2s
,
s=
1,2,3, and he p og amme is no eady o e lec i wi h p obabili y q3;
We wan o de e mine he condi ions unde which he compu e sys em can wo k wi hou leakage
o in o ma ion as a esul o he h ea ac ion. Deno e
a(ξ=θs)≡as,s=1,2,3,
and suppose ha he ansi ion in ensi ies a e gi en as
q12(1) = a,q12(2) = b,q12(3) = 1−(a+b),
q13(1) = e,q13(2) = d,q13(3) = 1−(e+d),
q23(1) = k,q23(2) = l,q23(3) = 1−(k+l)
and equal o ze o in o he cases. Then, sys em (14) akes he o m
b1=E(2)
1X0+c2a a2
2+ba4
2+ (1−(a+b))a6
2b2
+c2ea2
3+da4
3+ (1−(e+d)a6
3)b3,
b2=E(2)
2X0+c2(a a2
1+ba4
1+ (1−(a+b))a6
1)b1
+c2(ka2
3+la4
3+ (1−(k+l)a6
3))b3,
b3=E(2)
3X0+c2(e a2
1+da4
1+ (1−(e+d))a6
2)b2.
Thus, condi ions o L2-s abili y can be exp essed in he o m
c4a2
2a2
1a+ba2
2+ (1−a−b)a4
2a+ba2
1+ (1−a−b)a4
1<1,
c6a2
1a2
2a2
3a+ba2
1+ (1−a−b)a4
1k+la2
2+ (1−k−l)a4
2
×e+da2
3+ (1−e−d)a4
3
+c6a2
1a2
2a2
3e+da2
1+ (1−e−d)a4
1a+ba2
2+ (1−a−b)a4
2
×k+la2
3+ (1−k−l)a4
3
+c4a2
1a2
3e+da2
1+ (1−e−d)a4
1e+da2
3+ (1−e−d)a4
3
+c4a2
2a2
3k+la2
2+ (1−k−l)a4
2k+la2
3+ (1−k−l)a4
3
+c4a2
1a2
2a+ba2
1+ (1−a−b)a4
2(a+ba2
1+ (1−a−b)a4
1)<1.
(28)
The ac ual bounda ies o he in o ma ion sys em s abili y a ea can be de e mined in a speci ic
case, see Figu e 1.
Symme y 2019,11, 1338 9 o 14
Figu e 1.
The ac ual bounda ies o he o eign exchange ma ke s abili y a ea as well as he in o ma ion
sys em s abili y a ea i a=b=d=e=k=l=1
3and c=1.
4.2. S abili y o Fo eign Cu ency Exchange Ma ke
The la ges ma ke in he wo ld is cu en ly he o eign exchange ma ke , which de e mines he
exchange a es o each cu ency. The p ima y unc ion o he o eign exchange ma ke is he ans e
o pu chasing powe om one cu ency o ano he . The demand o a coun y’s cu ency depends
on he coun y’s balance o paymen s. In pa icula , cen al banks a e esponsible o main aining
in la ion in he in e es o sus ainable economic g ow h and, a he same ime, con ibu ing o he
o e all s abili y o he inancial sys em. A big ac o a ec ing he exchange a es is he in e es a e paid
by a coun y’s cen al bank, he money supply c ea ed by he coun y’s cen al bank and a coun y’s
economic g ow h and inancial s abili y. Comme cial banks ope a e in he o eign exchange ma ke ,
buy and sell cu encies o hei clien s.
A o eign exchange isk a ises when a bank holds asse s o liabili ies in a o eign cu ency,
which causes exchange a e luc ua ions and a ec s he bank’s p o i and capi al. The ac ual a es may
di e signi ican ly om he end i d as ic changes in he coun y’s economy occu ha may lead o a
cu ency c isis.
The bank’s ma ke ac i i y can be desc ibed by sys em
(15)
. Le he semi-Ma ko chain in
(15)
be
in h ee possible s a es:
θ1— he e is a cu ency c isis, aξk=a1,
θ2— he e is a s able o eign cu ency exchange ma ke , aξk=a2,
θ3— he e is a ma ke wi h cu ency es ic ions, aξk=a3.
I we deno e ansi ion in ensi ies as men ioned abo e in he p e ious p oblem, he condi ions
o he s abili y o he o eign exchange ma ke will be iden ical o hose ob ained he e, i.e.,
(28)
. In a
special case, i
a=b=d=e=k=l=1
3, (29)
we ge he condi ion o s abili y in he o m