A dynamical system with random parameters as a mathematical model of real phenomena

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