Construction Of Equivalent Stochastic Differential .
Stochastic Analysis and Applications, 26: 274–297, 2008Copyright Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362990701857129Construction of Equivalent StochasticDifferential Equation ModelsEdward J. Allen,1 Linda J. S. Allen,1Armando Arciniega,2 and Priscilla E. Greenwood31Department of Mathematics and Statistics, Texas Tech University,Lubbock, Texas, USA2Department of Mathematics, University of Texas at San Antonio,San Antonio, Texas, USA3Department of Mathematics, Arizona State University,Tempe, Arizona, USAAbstract: It is shown how different but equivalent Itô stochastic differentialequation models of random dynamical systems can be constructed. Advantagesand disadvantages of the different models are described. Stochastic differentialequation models are derived for problems in chemistry, textile engineering,and epidemiology. Computational comparisons are made between the differentstochastic models.Keywords: Chemical reactions; Cotton fiber breakage;Mathematical model; Stochastic differential equation.Epidemiology;Mathematics Subject Classification: AMS(MOS) 60H10; 65C30; 92D25; 92E20.Received April 6, 2007; Accepted June 6, 2007E.J.A. and L.J.S.A. acknowledge partial support from the Texas AdvancedResearch Program Grants 003644-0224-1999 and 003644-0001-2006, the NSFGrant DMS-0201105, and the Fogarty International Center # R01TW00698602 under the NIH NSF Ecology of Infectious Diseases initiative. P.E.G.acknowledges the Sloan Foundation, NSF Grant DMS-0441114 and NSAGrant H98230-05-1-0097 for support.Address correspondence to Edward J. Allen, Department of Mathematicsand Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA; E-mail:edward.allen@ttu.edu
Equivalent SDE Models2751. INTRODUCTIONOften, in modeling a random dynamical problem, a system of Itôstochastic differential equations is developed and studied. There appearto be three procedures for developing stochastic differential equation(SDE) models for applications in population biology, physics, chemistry,engineering, and mathematical finance. In the first modeling procedure, adiscrete stochastic model is developed by studying changes in the systemcomponents over a small time interval (e.g., [1–12]). This approach isa natural extension of the procedure used for many years in modelingdeterministic dynamical processes in physics and engineering, wherechanges in the system are studied over a small time interval and adifferential equation is obtained as the time interval approaches zero.Similarities between the forward Kolmogorov equations satisfied bythe probability distributions of discrete- and continuous-time stochasticmodels let us infer that an Itô SDE model is close to the discretestochastic model. In this procedure, the number of Wiener processes inthe resulting SDE model never exceeds the number of components inthe system. In the second procedure, the dynamical system is carefullystudied to determine all of the different independent random changesthat occur in the system. Appropriate terms are determined for thesechanges in developing a discrete-time stochastic model which is thenapproximated by a system of stochastic differential equations (e.g.,[13–17]). As the total number of different random changes may exceedthe number of components in the system, a stochastic differentialequation model is obtained where the number of Wiener processesmay exceed the number of equations. This procedure yields systemsof stochastic differential equations that are generally easy to solvenumerically. A third procedure is direct formulation of a systemof stochastic differential equations and is the most commonly usedprocedure in constructing SDE models. For a given random dynamicalsystem, specific functional forms are assumed for the elements of the driftvector and diffusion matrix. Frequently, for mathematical simplicity,these elements are assumed to be linear functions of the componentprocesses (e.g., [18, 19]). These three procedures have been used inmodeling many dynamical processes that experience random influences.In this investigation, only the first two procedures are discussed.In the next section, two stochastic differential equation systems arestudied which are produced by the first and second modeling procedures.The two systems of stochastic differential equations are structurallydifferent yet have identical probability distributions. In addition, byidentifying relations between Wiener trajectories, it is shown that asample path solution of one system is also a sample path solutionof the other system. As the stochastic models can be interchanged,conceptual or computational advantages possessed by either model can
276Allen et al.be employed in any particular problem. In Section 3, it is shown howthe two stochastic models are derived from first principles, that is,from the possible changes that may occur in the system. In Section 4,three examples from chemistry, textile engineering, and epidemiology arediscussed, illustrating the derivations of the stochastic models and somecomputational comparisons between them.2. EQUIVALENT SDE SYSTEMSLetf 0 T d d G 0 T d d m andB 0 T d d d t X1 t In addition, let X t X1 t X2 t Xd t T X TT t X2 t Xd t W t W1 t W2 t Wm t and W T W1 t W2 t Wd t where Wi t , i 1 m and Wj t , j 1 d are independent Wiener processes and m d. Considered inthis article are the two Itô SDE systems: t dX t f t X t dt G t X t dW(2.1) t f t X t dt B t X t dW t dX(2.2)andMatrices G and B are related through the d d matrix V , whereV t z G t z GT t z and B t z V 1/2 t z for z d . It is assumedthat f , G, and B satisfy certain continuity and boundedness conditions [20, 21] so that (2.1) and (2.2) have pathwise unique solutions. Since W are not defined on the same probability space, neither are X andand WX . However, it is shown that solutions to (2.1) and (2.2) have the sameprobability distribution. In addition, one can define a measure-preservingmap between the probability spaces in such a way that the correspondingsample paths X and X are identical.Notice that the d d symmetric positive semidefinite matrix V hasentries vi j t X m l 1 j l t X gi l t X g(2.3)
Equivalent SDE Models277for i j 1 d and d d symmetric positive semidefinite matrix Bhas entries that satisfy vi j t X d j l t X bi l t X b(2.4)l 1for i j 1 d. In component form, systems (2.1) and (2.2) can beexpressed asXi t Xi 0 t fi s X s ds 0 t m0 j 1 gi j s X s dWj s (2.5)for i 1 2 d, where fi is the ith entry of f and gi j is the i j entryof the d m matrix G and t t d s ds s dWj s (2.6)fi s Xbi j s XXi t Xi 0 00 j 1for i 1 d and bi j is the i j entry of the d d matrix B.It is now shown that solutions to (2.1) and (2.2) possess the sameprobability distributions; they are equivalent in distribution. To see this,consider the forward Kolmogorov equation or Fokker-Planck equationfor the probability density function p t x associated with the stochasticdifferential system (2.1), dd m 1 2 p t x p t x gi l t x gj l t x t2 i 1 j 1 xi xjl 1 d p t x fi t x xii 1(2.7) In particular, if z1 z2 d and z1 z2 , then z2 d z2 d 1 z2 1 z2 ···p t x dx1 dx2 dxd P z1 X t z1 dz1 d 1z1 1As the elements of V satisfyvi j t x m l 1gi l t x gj l t x d bi l t x bj l t x l 1systems (2.1) and (2.2) have the same forward Kolmogorov equation. t are identical.Hence, the probability density functions for X t and XIn addition to solutions of (2.1) and (2.2) having the sameprobability distribution, it is useful conceptually and for sample pathapproximation to be aware that a sample path solution of one equation
278Allen et al.is also a sample path solution of the second equation defined on anaugmented probability space. It will be shown that stochastic differentialequations (2.1) and (2.2) possess the property that a sample path solutionof one equation is also a sample path solution of the second equation,and the correspondence is measure preserving. More specifically, given t with sample path solution X t a Wiener trajectory Wto (2.1), there t with the sample path solution X t exists a Wiener trajectory W t with sampleX t to (2.2). Conversely, given a Wiener trajectory W t to (2.2), there exists a Wiener trajectory W t withpath solution X t to (2.1).the sample path solution X t X t for 0 t T is givenAssume now that a Wiener trajectory W and the sample path solution to (2.1) is X t .It is now shown that t such that (2.2) has the samethere exists a Wiener trajectory W t X t sample path solution as (2.1), i.e., Xfor 0 t T . Tosee this, an argument involving the singular value decomposition of G t G t X t is employed. Consider, therefore, the singular valuedecomposition of G t P t D t Q t for 0 t T , where P t andQ t are orthogonal matrices of sizes d d and m m, respectively,and D t is a d m matrix with r d positive diagonal entries. (Itis assumed that the rank of G t is r for 0 t T . This assumptioncan be generalized so that the rank of G t is a piecewise constantfunction of t on a partition of 0 T .) It follows that V t G t G t T P t D t DT t P T t B t 2 , where B t P t D t DT t 1/2 P T t . The t of d independent Wiener processes is now defined asvector W t t s s t P s D s D s T 1/2 D s Q s dWP s dWW00 s is a vector of length d with the first r entriesfor 0 t T , where Wequal to 0 and the next d r entries independent Wiener processes,and where D t DT t 1/2 is the d d pseudoinverse of D t DT t 1/2 .(Ifis a d m matrix with nonzero entries i i for i 1 2 rwith r d m, then is a m d matrix with nonzero entries1/ i i for i 1 2 r. See, e.g., [22] or [23] for more informationabout the singular value decomposition and pseudoinverses.) Notice that t W t T tId , where Id is the d d identity matrix verifyingE W t is a vector of d independent Wiener processes. The diffusionthat W t replaced by X t term on the right side of (2.2) with Xsatisfies t B t X t dW 1/2 t P t dW t B t P t D t D t TD t Q t dW 1/2 t P t D t DT t 1/2 P T t P t D t D t TD t Q t dW t P t dW t G t X t dW
Equivalent SDE Models279 t ; X t Hence, dX t f t X t dt B t X t dWis the sample pathsolution of (2.2). t for 0 t T is givenConversely, assume a Wiener trajectory W t . It is now shown thatand the sample path solution to (2.2) is X t such that (2.1) has the same samplethere exists a Wiener trajectory W t for 0 t T . In this case,path solution as (2.2), that is, X t X t G t the singular value decomposition of G has the form G t XP t D t Q t for 0 t T , where P t and Q t are orthogonal matricesof sizes d d and m m, respectively, and D t is a d m matrix t of m independentwith r d positive diagonal entries. The vector WWiener processes is now defined as t t t s s WQT s D s D s D s T 1/2 P T s dWQT s dW00 s is a vector of length m with the firstfor 0 t T where Wr entries equal to 0 and the next m r entries independent Wienerprocesses, and where D t is the m d pseudoinverse of D t . Notice t W t T tIm , where Im is the m m identity matrix. Thethat E W t satisfiesdiffusion term in (2.1) with X t replaced by X t dW t G t X T 1/2 T t QT t dW t G t Q t D t D t D t TP t dW 1/2 T t P t D t Q t QT t D t D t D t TP t dW t QT t dW t dW t B t X t f t X t dt G t X t dW t ; X t is the sampleThus, dXpath solution of (2.1).In effect, a sample path solution of system (2.1) with m d Wienerprocesses is also a sample path solution of stochastic system (2.2) withd Wiener processes, where the d d matrix B satisfies B2 GGT . Thatthe correspondence is measure-preserving follows from the two ways ofwriting (2.7). In summary, the following result has been proved.Theorem 2.1. Solutions to SDE systems (2.1) and (2.2) possess the sameprobability distribution. In addition, a sample path solution of one equationis a sample path solution of the second equation.3. EQUIVALENT SDE MODELING PROCEDURESIn this section, it is shown how to formulate a stochastic differentialequation (SDE) model from a random dynamical system consisting
280Allen et al.of d components, where m d distinct independent random changesmay occur to the components of the system during a small interval oftime. Two modeling procedures are described for formulating an SDEmodel as discussed in the previous sections. In the first procedure, them changes are collectively considered and means and covariances aredetermined. The first approach produces an SDE system with d Wienerprocesses. In the second procedure, each change is considered separately.The second approach produces an SDE system with m Wiener processes.In both procedures, the number of equations in the SDE model equalsthe number of components, d. In addition, the two SDE models areequivalent in that solutions to both models have the same probabilitydistribution and a sample path solution of one SDE model is also asample path solution of the other SDE model.Consider a stochastic modeling problem that involves d componentprocesses S1 S2 Sd , S S1 S2 Sd T . Suppose that there are atotal of m d possible changes that can occur to at least one of thevariables Si in a small time interval t. Suppose, in addition, that the t for j probabilities of these changes can be defined as pj t pj t S 1 2 m, where the jth change alters the ith component by the amountj i for i 1 2 d. Letm pj t S t fi t S t j i(3.1)j 1for i 1 2 d. Notice that (3.1) can be used to define a deterministicmodel consisting of a system of ordinary differential equations (ODEs): f t S t dS t dt (3.2)where f f1 f2 fd T . For t small, the ODE system (3.2) can beapproximated using Euler’s method by the formula Sn 1 i Sn i fi tn S n t (3.3)where tn n t and Sn i Si tn for i 1 d and n 0 1 .Assuming that t is a small but fixed time interval, an accuratediscrete-time stochastic model can be formulated by considering therandom changes at each time step.Let r j represent a random change of the jth kind, where to order O t 2 , r j is defined as follows: j 1 j 2 j d T with probability pj tr j 0 0 0 Twith probability 1 pj t 2For t small, rj i has approximate mean j i pj t and variance j ipj t. An accurate yet simple stochastic model for Sn 1 , given the vector
Equivalent SDE Models281S n , isS n 1 S n m r j(3.4)j 1for n 0 1 . In component form, (3.4) becomesSn 1 i Sn i m r j i(3.5)j 1for i 1 d and n 0 1 In the first modeling procedure, it can be shown [1, 5] thatif the changes are small and t is small, then the probabilitydistribution associated with the discrete-time stochastic system (3.4) canbe approximated by the solution to the forward Kolmogorov equation ddd p t x fi t x 2 1 p t x p t x vi j t x t xi2 i 1 j 1 xi xji 1(3.6) j T andwhere vi j is the i j th entry of d d matrix V mj 1 pj j f mj 1 pj j . (See, e.g., [1, 4–6] for more information about thisprocedure.) The probability distribution p t x1 x2 xd that solves(3.6) is identical to the distribution of solutions corresponding to theSDE system f t S t t dS t dt B t S t dW(3.7) S 0 S 0 t is a vector of d independentwhere the d d matrix B V 1/2 and WWiener processes. This first procedure gives the minimal number d ofGaussian processes that can be used to describe this process.The discrete stochastic model (3.4) is closely related to the SDEmodel (3.7). Specifically, the probability distribution of solutions to (3.4)is approximately the same as the probability distribution of solutions to(3.7). In addition, it is useful to notice that the drift vector and diffusionmatrix, f and B of the SDE model are equal to the expected changedivided by t and the square root of the covariance matrix of the changedivided by t, respectively. Specifically, letting j j 1 j 2 j n T ,then the expected change in S and the covariance in the change areE mm T T S pj j t f t and E S S pj j jt V tj 1j 1(3.8)where B V 1/2 .
282Allen et al.In the second modeling procedure, the m random changes in (3.4)are approximated using m independent normal random variables, j N 0 1 , j 1 2 m. The normal approximation may be justifiedby arguments involving the Central Limit Theorem or by normalapproximations to Poisson random variables. Equation (3.4) for smallbut fixed t is therefore approximated bym Sn 1 i Sn i fi tn S n t 1/2j i pj t 1/2j(3.9)j 1for n 0 1 , where fi is defined in (3.1). (See, e.g., [14–17] formore information about this procedure.) Notice the similarity betweenthe deterministic equation (3.3) and the stochastic equation (3.9). Thediscrete stochastic model (3.9) is an Euler-Maruyama approximation andconverges strongly (in the mean-square sense [20]) as t 0 to the SDEsystem f t S t t dS t dt G t S t dW(3.10) S 0 S 0 where the i j entry in the matrix G is gi j j i pj1/2 for i 1 2 d, t is a vector of m independent Wiener processes.j 1 2 m, and WThus, the SDE system (3.10) is closely related to the discrete model (3.4).Notice that the SDE system (3.10) has m Wiener processes and the d dmatrix V GGT has entriesMM gi j gl j pj V i l GGT i l j 1ji jl vi l(3.11)j 1for i l 1 d. Furthermore, the entries of G are easy to write downgiven the probabilities of the different changes based on the discrete-timeMarkov chain (3.4).Notice that the d m matrix G satisfies V GGT and the SDEsystem (3.10) can be replaced by the system (3.7) by the argument inthe previous section. Indeed, the forward Kolmogorov equations areidentical for both systems (3.7) and (3.10) and a sample path solution ofone system is a sample path solution of the other system. Finally, noticethat system (3.7) is generally more complicated than (3.10), as the d dmatrix B is the square root of V even though G is d m. Consequently,system (3.10) is generally easier to solve computationally. However, ifthe number of changes, m, is much greater than the number of systemcomponents, d, then equation (3.10) loses much of its computationaladvantages.It is interesting to note that there are other SDE systems equivalentto (3.10) that can be generated from the probabilities of the changes.
Equivalent SDE Models283For example, if the diffusion matrix G is replaced by G in (3.10), thisalternate system’s solutions will have the same probability distribution.In general, there are other diffusion matrices H with the property HH T V GGT that can replace G in (3.11). This is due to the fact that for amultivariate Gaussian process there are many ways in which the processcan be written.It is important to understand that stochastic differential equationmodels (3.7) and (3.10) approximate the actual randomly varying systemas time and system variables are continuous in the SDE models whereasdiscrete changes may be occurring at discrete times in the actualrandomly varyi
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