Modeling of applied problems by stochastic systems and their analysis using the moment equations

Diblík et al. Advances in Difference Equations 2013, 2013:152 http://www.advancesindifferenceequations.com/content/2013/1/152 R E S E A R C H Open Access Modeling of applied problems by stochastic systems and their analysis using the moment equations Josef Diblík1, Irada Dzhalladova2, Mária Michalková3and Miroslava R˚užiˇ cková3* *Correspondence: [email protected] 3University of Žilina, Žilina, Slovakia Full list of author information is available at the end of the article Abstract The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples. MSC: 34K50; 60H10; 60H30; 65C30 Keywords: stochastic systems; Markov process; moment equations; solvability; stability 1 Introduction Most of the notable achievements in theoretical economics in the last fifty years were related to finances. The first Nobel Memorial Prize in Economic Sciences was awarded in  jointly to Frisch and Tinbergen ‘for having developed and applied dynamic models for the analysis of economic processes’. Many of their works were devoted to the development of mathematical methods to the analysis of economic processes, including mathematical modeling of financial processes []. The second Nobel Memorial Prize in Economic Sciences was awarded in  to Samuelson ‘for the scientific work through which he has developed static and dynamic economic theory and actively contributed to raising the level of analysis in economic science’. In his works he studied the role of expectations in the theory of finance. The growing importance of finance theory in economics is linked to two trends: still wider use of mathematics in the modeling of economic processes, and using the results of theoretical economics in practice. The both trends have a close relationship to finance. Mathematical modeling assumes the exact determination of the parameters - they usually are expressed in the finances; application of the theory in practice assumes description of the cash flows and the risk of using models. Significant development of the theory of finance, which includes the theory of corporate finance and the theory of investment, occurred in the twentieth century. Until then, the theory of finance was developed as a theory of state finance, in the twentieth century became the theory of capital markets. The amount of significant works on the theory of financewerewrittenintheyearsto.Bachelier,thefounderofthemoderntheory of finance, has merit that the theory of finance received a mathematical basics. He anticipated many of the ideas of the twentieth century in his works: the relationship between ©2013 Diblík et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 2 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 random and diffusion processes, Markov processes, the theory of Brownian motion and much more than today lies not only in the investment theory. One of the first models of theoffersloanfundswasbuiltintheearlytwentiethcenturybyFisher.Equationstobalance between savings and investments (known as IS-LM model, and Mandella-Fleming model) are the basis of the modern macroeconomics. Its authors Hicks and Mundell are Nobel Prize winners. Mundell, in addition, created the theory of optimum currency areas, which allows to call him the father of the euro. In the theory of financial investment, there is no concept that would be such widely verified and so little credible as ‘efficient markets’. The so-called efficient market hypothesis performs a primary function - to justify the use of probabilistic calculation in the analysis of capital markets. But if markets are ‘nonlinear stochastic dynamical systems’, the use of standard statistical analysis can lead to erroneous results, especially if they are based on the model of random walks. One of the methods that permit to examine the stability of stochastic systems is a traditional method of Lyapunov functions, which was developed, for example, in the works by Barbashin [], Hasminski [], Valeev [], Zubov []andothers. Investigating the mean stability or mean square stability of solutions of differential equations with random coefficients depending on Markov process is a current problem. The theory of Markov processes was studied in the works by Chung [], Davis [], Dynkin [, ], Kolmogorov [], Lèvy [], Skorohkod [] and others. The use of the theory of Markov processes to the study of various economic processes can be found in the works by Elliot, Kopp [], Malliaris, Brock [] and Williams []. Dynamic systems considered in the present paper belong to the class of the so-called systems with random states. The works by Artem’ev [], Katz, Krasovskii []andothers are dedicated to such systems. We offer a new approach to simulation by creating algorithms for the construction of moment equations and their quantification. The origin of the theory of moment equations and their use in the examination of the stability can be found in the works by Valeev [] and his scientific school (e.g.,[]). In the present paper we derive the functional equations for particular density functions and the moment equations for the system which are used in the investigation of solvability and mean square stability. There is shown the application of the results to solve various problems of practice. 2 Statement of the problem Let (,F,P) be a probability space (see, for example, []). On the probability space, we consider the initial value problem formulated for the stochastic system dx(t) dt =At,ξ(t)x(t)+Bt,ξ(t),() x() = ϕ(ω), () where Ais an m×mmatrix with random elements, Bis an m-dimensional column vector function whose elements are random variables, ϕ:→Rm,ϕ∈C(), ξ(t)isarandom Markov process with a finite number of states θk,k=,,...,q, the probabilities of which are pk(t)=Pξ(t)=θk,k=,,...,q,() Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 3 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 and satisfy the system of linear differential equations dpk(t) dt = q  s= πks(t)ps(t)() with the transition matrix (πks(t))q k,s=. Definition  The m-dimensional random vector function x(t), the components of which are random variables is called a solution of the initial value problem (), ()ifx(t)satisfies () and initial condition () in the meaning of strong solution (defined in []) of the initial Cauchy problem. Our task is to obtain a reliable and simple method for investigating the stability of solutions of this class of systems. To solve this task, we present below the method of moment equations. On a series of examples, we demonstrate that the method is effective and useful. Definition  Let x∈Rmbe a continuous random variable depending on a random Markov process ξ(t)withqpossible states θk,k=,,...,q.Thematrices E(t)= q  k= E(k)(t), D(t)= q  k= D(k)(t), where E(k)(t)=Em xfk(t,x)dx,D(k)(t)=Em xx∗fk(t,x)dx,k=,,...,q, are called moments of the first or second order of the random variable xrespectively. The values E(k)(t)andD(k)(t), k=,,...,q, are called particular moments of the first or second order respectively. The Emin Definition denotes an m-dimensional Euclidian space, functions fk(t,x), k= ,,...,qare the particular density functions of the random variable x. Remark  The moments of the random variable xin a scalar case, x∈R,aredefined for any s=,,...,andarecalledmomentsofthe sth order. The particular moments are defined by the formula E(k) s(t)=∞ –∞ xsfk(t,x)dx,s=,,...,k=,,...,q. Several different stability statements are possible. We here recall mean square stability definition, which is based on that given in []. Definition  The trivial solution of the associated homogenous system to system ()is said to be mean square stable on the interval [,∞)ifforeachε>thereexistsδ>such that any solution x(t) of the associated system, corresponding to the initial data x,exists for all t≥ and the mathematical expectation Ex(t)<εwhenever t≥andx<δ. Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 4 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 3 Moment equations for the linear differential equations Before the initial value problem (), () formulated in the previous section will be investigated, a simpler problem will be studied. First we derive the moment equations in the scalar case of system (), that is, if instead of the system there is an equation. In the first part of this section, the linear homogenous differential equation, the coefficient of which depends on a random Markov process, with two states only is considered. In the second part, the moment equations are derived for nonhomogenous linear differential equations with qpossible states of a random process, on which the coefficients depend. 3.1 Homogenous linear differential equations On the probability space (,F,P), we consider initial value problem (), ()whereinstead of system () there is a stochastic linear homogenous differential equation of the first order of the form dx(t) dt =aξ(t)x(t), () where ais a scalar function of a random variable. We suppose that the function adepends on the random Markov process ξ(t), which has only two states θ,θwith probabilities pk(t)=Pξ(t)=θk,k=,, that satisfy the system of linear differential equations dp(t) dt =–λp(t)+νp(t), dp(t) dt =λp(t)–νp(t), λ≥,ν≥. () In the following, we use the denotations a=a(θ), a=a(θ). Theorem  Moment equations of any order s =,,,...for equation () are of the form dE() s(t) dt =saE() s(t)–λE() s(t)+νE() s(t), dE() s(t) dt =saE() s(t)+λE() s(t)–νE() s(t). () Proof We divide the time line [, ∞) into intervals of length h.Nextwereplacethe considered system of differential equations () by an approximated system of difference equations. If we denote tn=nh,h>,n= ,,..., and approximate dx(tn+)/dt with (x(tn+)–x(tn))/h, then the approximated system to system ()canbewrittenintheform x(tn+)=+haξ(tn)x(tn), n=,,... Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 5 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 or the approximated system to system ()isoftheform p(tn+)=(–hλ)p(tn)+hνp(tn), p(tn+)=hλp(tn)+(–hν)p(tn). () In accordance with the formula for total probability, we obtain relationships for the particular density functions fk(tn,x), k= ,, which satisfy the following system of functional equations: f(tn+,x)= –hλ +ha ftn,x +ha+hν +ha ftn,x +ha,() f(tn+,x)= hλ +ha ftn,x +ha+–hν +ha ftn,x +ha.() Rename ‘tn’to‘t’ and suppose that the particular density functions can be expressed in powers of parameter hby the Taylor formula. Let functions in ()berepresentedas f(tn+,x)=f(t+h,x)=f(t,x)+∂f(t,x) ∂th+Oh, –hλ +ha ftn,x +ha =–h(λ+a)+Ohft,x–hxa+Oh =–h(λ+a)+Ohf(t,x)–∂f(t,x) ∂xhxa+Oh =f(t,x)–hλf(t,x)–haf(t,x)–hax∂f(t,x) ∂x+Oh, hν +ha ftn,x +ha=hνf(t,x)+Oh, where Ois Landau order symbol. Now, using the obtained expressions and comparing the left-hand side to the right-hand side of () and assuming h→, we get ∂f(t,x) ∂t=–a ∂ ∂xxf(t,x)–λf(t,x)+νf(t,x). () Similarly, decomposition of the particular density functions in () gives the second equation ∂f(t,x) ∂t=–a ∂ ∂xxf(t,x)+λf(t,x)–νf(t,x). () Finally, multiplying equations (), ()byxs,s=,,,... andintegratingthembyparts from –∞to ∞, in accordance with Definition , a system of linear differential equations with constant coefficients () can be obtained.  Let us note that moment equations () can be derived in a different way. If system () of difference equations for probabilities is known, then the particular moments of the sth Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 6 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 order satisfy the following relations: E() s(tn+)=(–hλ)( + ha)sE() s(tn)+hν( + ha)sE() s(tn), E() s(tn+)=hλ( + ha)sE() s(tn)+(–hν)( + ha)sE() s(tn). () Particular moments contained in the first equation of () can be expressed in powers of parameter hby the Taylor formula: E() s(tn+)=E() s(t+h)=E() s(t)+dE() s(t) dt h+Oh, ( – hλ)( + ha)sE() s(tn)=+hsa–hλ+OhE() s(t), hν( + ha)sE() s(tn)=hνE() s(t)+Oh. If we put the obtained expressions into the first equation of (), then under assumption h→, we get the first equation of system (). In the same way, using the second equation of (), the second equation of system () can be constructed. Example  Let us establish conditions for s-mean stability of linear differential equation (). The characteristic equation for the system of moment equations ()iswrittenasfollows:  z–sa+λ–ν –λz–sa+ν =z+zλ+ν–s(a+a)+saa–sνa–sλa=. Therefore, the conditions of asymptotic stability of solutions of moment equations (), in accordance with the Hurwitz criterion, are of the following form (assume s=,thecase s=  is considered below): a+a<λ+ν s,aa>νa+λa s. Let us use the denotations γ≡λ+ν s,a≡νa+λa λ+ν=ap +ap , where p k=limt→+∞pk(t), k=,andais mean value of coefficients a,a. It allows us to derive a simpler form of the above conditions: a+a<γ,aa>aγ. The domains of stability for moments of various order are determined by their boundariesasitisshowninFigure. Any domain of stability includes the third quadrant where the values of coefficients a,aare negative, i.e.,a<,a<. Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 7 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 Figure 1 Domains of stability for equation (5). Using moment equations, it is also possible to determine the domain of stability for the deterministic equation dx(t) dt =ax(t), where ais independent of a random variable ξ(t). This case corresponds to the moment equations of the zeroth order, i.e.,ifs=. 3.2 Nonhomogenous linear differential equation We have derived the system of moment equations for a linear homogenous equation with random coefficient under assumptions that the random variable can only be in two states. It was a simple enough case that allowed us to understand the process of deriving the system of moment equations. Now we establish a system of moment equations in the same way for linear the nonhomogeneous differential equation dx(t) dt =at,ξ(t)x(t)+bt,ξ(t),() where ξ(t) is the Markov process which has qpossible states θ,θ,...,θq,withprobabilities pk(t)=P{ξ(t)=θk},k=,,...,q. We suppose that the probabilities satisfy the system of linear differential equations dpk(t) dt = q  s= πks(t)ps(t), () where the transition matrix (πks(t))q k,s= satisfies the following relationships: q  k= πks(t)≡, πks(t)⎧ ⎨ ⎩ ≥, k=s, ≤, k=s. Since the coefficients of studied system ()dependont,wecandenote ak(t)=a(t,θk), bk(t)=b(t,θk), k=,,...,q. Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 8 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 Theorem  Moment equations of any order s =,,...for equation ()are of the form dE(k) s(t) dt =sak(t)E(k) s(t)+sbk(t)E(k) s–(t)+ q  r= πkr(t)E(r) s(t), k=,,...,q.() Proof By dividing the time line into intervals of length h, we obtain the approximated system x(tn+)=+hatn,ξ(tn)x(tn)+hbtn,ξ(tn) to the considered system ()and pk(tn+)=pk(tn)+h q  s= πks(tn)ps(tn), k=,,...,q to system (). Particular probability density functions fk(tn,x) satisfy, in this case, the system of difference equations fk(tn+,x)=  +hak(tn)fktn,x–hbk(tn) +hak(tn) +h q  s= πks(tn) +hak(tn)fktn,x–hbk(tn) +hak(tn).() SimilarlyasintheproofofTheorem, we assume that the particular density functions can be represented in powers of parameter hbytheTaylorformula,andbythesameway as in the proof of Theorem ,weget ∂fk(t,x) ∂t=– ∂ ∂xak(t)x+bk(t)fk(t,x)+ q  s= πks(t)fs(t,x), k=,,...,q. The system of moment equations () can be derived from the last system for particular probability density functions by using the same modifications as in the proof of Theorem .  4 Moment equations for the linear differential system Now we come back to the initial problem (), ()thatwehaveformulatedinSection. We also suppose that the matrix Aand vector Bdepend on a random Markov process ξ(t) with qpossible states, the probabilities of which () satisfy the system of linear differential equations (). Moreover, we use the denotations Ak(t)=A(t,θk), Bk(t)=B(t,θk), k=,,...,q. Diblík et al. Advances in Difference Equations 2013, 2013:152 Page 9 of 12 http://www.advancesindifferenceequations.com/content/2013/1/152 Theorem  Moment equations of the first and second order respectively for system () are of the form dE(k)(t) dt =Ak(t)E(k)(t)+Bk(t)pk(t)+ q  j= πkj(t)E(j)(t), () dD(k)(t) dt =Ak(t)D(k)(t)+D(k)(t)A* k(t)+Bk(t)E(k)(t)* +E(k)(t)B* k(t)+ q  j= πkj(t)D(j)(t), k=,,...,q.() Proof The philosophy of the proof is the same as in the proof of Theorem ,onlythe calculations are more complicated, because now we work with the matrix case. In a similar way, by dividing the time line into intervals of length h, for the particular density functions fk(t,x), k=,,...,q,wegetthesystemofequations fk(tn+,x)=fk(tn,Yk)υk+h q  j= πkj(tn)fj(tn,Yj)υj, k=,,...,q,() where Yj=I+hAj(tn)–x–hBj(tn),υj=detI+hAj(tn)–,j=,,...,q, Iis the unit matrix. Assume that the particular density functions can be expressed in powers of parameter hby the Taylor formula. If we put tn=t, then decompositions of the functions on the left-hand side and on the right-hand side in ()areequalto fk(tn+,x)=fk(t+h,x)=fk(t,x)+∂fk(t,x) ∂th+Oh, υj=detI–hAj(tn)+Oh=–hTrAk(t)+Oh, Yj=I–hAj(t)+Ohx–hBj(t)=x–hAj(t)x+Bj(t)+Oh, fk(t,Yk)=fkt,x–hAk(t)x+Bk(t)+Oh =fk(t,x)–hgrad fk(t,x)Ak(t)x+Bk(t)+Oh, k,j=,,...,q, where grad f(t,x)=∂f(t,x) ∂x ,∂f(t,x) ∂x ,...,∂f(t,x) ∂xm, Tr(A) is the trace of the matrix A.