Stochastic Controledited Part 2 doc

Stochastic Control32 ,w),G() 1k1 t1k1    kktkkttt tttxtttxfxx kkkk )(,( k where 1k t w is a standard normal variable. The dimension of the phase space of the stochastic problem of concern here is four, since the state vector   .),,,(,,, 4 4321 RUvrxxxxx T r T t   The size of the mean state vector is four and the number of entries in the variance matrix of the state is sixteen. Since the state vector is a real-valued vector stochastic process, the condition jiij PP  holds. The total number of distinct entries in the variance matrix w’d be ten. The initial conditions are chosen as . and variablestate for the ,0)0(Prad/TU, 1.1)0(,AU/TU 01.0)0(rad, 1)0(,AU1)0( yx vr xyr       The initial conditions considered here are in canonical system of units. Astronomers adopt a normalized system of units, i.e. ‘canonical units’, for the simplification purposes. In canonical units, the physical quantities are expressed in terms of Time Unit (TU) and Astronomical Unit (AU). The diffusion parameters 2 3 (TU)0121.0   r  and 2 3 4 )( 102.2 TU AU     are chosen for numerical simulations. Here we consider a set of deterministic initial conditions, which implies that the initial variance matrix w’d be zero. Note that random initial conditions lead to the non-zero initial variance matrix. The system is deterministic at 0 tt  and becomes stochastic at 0 tt  because of the stochastic perturbation. This makes the contribution to the variance evolution coming from the ‘system non-linearity coupled with ‘initial variance terms’ will be zero at . 1 tt  The contribution to the variance evolution at 1 tt  comes from the perturbation term ),)(( txGG t T only. For , 1 tt  the contribution to the variance evolution comes from the system non-linearity as well as the perturbation term. This assumption allows to study the effect of random perturbations explicitly on the dynamical system. The values of diffusion parameters are selected so that the contribution to the force coming from the random part is smaller than the force coming from the deterministic part. It has been chosen for simulational convenience only. Fig. 1. Fig. 2. Fig. 3. Time ( TU ) The Itô calculus for a noisy dynamical system 33 ,w),G() 1k1 t1k1    kktkkttt tttxtttxfxx kkkk )(,( k where 1k t w is a standard normal variable. The dimension of the phase space of the stochastic problem of concern here is four, since the state vector   .),,,(,,, 4 4321 RUvrxxxxx T r T t   The size of the mean state vector is four and the number of entries in the variance matrix of the state is sixteen. Since the state vector is a real-valued vector stochastic process, the condition jiij PP  holds. The total number of distinct entries in the variance matrix w’d be ten. The initial conditions are chosen as . and variablestate for the ,0)0(Prad/TU, 1.1)0(,AU/TU 01.0)0(rad, 1)0(,AU1)0( yx vr xyr       The initial conditions considered here are in canonical system of units. Astronomers adopt a normalized system of units, i.e. ‘canonical units’, for the simplification purposes. In canonical units, the physical quantities are expressed in terms of Time Unit (TU) and Astronomical Unit (AU). The diffusion parameters 2 3 (TU)0121.0   r  and 2 3 4 )( 102.2 TU AU     are chosen for numerical simulations. Here we consider a set of deterministic initial conditions, which implies that the initial variance matrix w’d be zero. Note that random initial conditions lead to the non-zero initial variance matrix. The system is deterministic at 0 tt  and becomes stochastic at 0 tt  because of the stochastic perturbation. This makes the contribution to the variance evolution coming from the ‘system non-linearity coupled with ‘initial variance terms’ will be zero at . 1 tt  The contribution to the variance evolution at 1 tt  comes from the perturbation term ),)(( txGG t T only. For , 1 tt  the contribution to the variance evolution comes from the system non-linearity as well as the perturbation term. This assumption allows to study the effect of random perturbations explicitly on the dynamical system. The values of diffusion parameters are selected so that the contribution to the force coming from the random part is smaller than the force coming from the deterministic part. It has been chosen for simulational convenience only. Fig. 1. Fig. 2. Fig. 3. Time ( TU ) Stochastic Control34 Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Here, we analyse the stochastic problem involving the numerical simulation of approximate conditional moment evolutions. The approximate conditional moment evolutions, i.e. conditional mean and variance evolutions, were derived in the previous section using the second-order and bilinear approximations. The variance evolutions using the second-order approximation result reduced variances of the state variables rather than the bilinear, see figures (5), (6), (7), and (8). These illustrate that the second-order approximation of the mean evolution produces less random fluctuations in the mean trajectory, which are attributed to the second-order partials of the system non-linearity ),,( txf t i.e. . ),( ~~ 2 1 2 , qp t q qp p xx txf xx      The expectation of the partial terms leads to The Itô calculus for a noisy dynamical system 35 Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Here, we analyse the stochastic problem involving the numerical simulation of approximate conditional moment evolutions. The approximate conditional moment evolutions, i.e. conditional mean and variance evolutions, were derived in the previous section using the second-order and bilinear approximations. The variance evolutions using the second-order approximation result reduced variances of the state variables rather than the bilinear, see figures (5), (6), (7), and (8). These illustrate that the second-order approximation of the mean evolution produces less random fluctuations in the mean trajectory, which are attributed to the second-order partials of the system non-linearity ),,( txf t i.e. . ),( ~~ 2 1 2 , qp t q qp p xx txf xx      The expectation of the partial terms leads to Stochastic Control36 . ),( 2 1 2 , qp t pq qp xx txf P      The correction term qp t pq qp xx txf P      ),( 2 1 2 , involves the variance term . pq P The evolution of the variance term ij P encompasses the contributions from the preceding variances, partials of the system non-linearity, the diffusion coefficient ),()( txGG tij T  as well as the second-order partial term ),()( 2 1 , txGGP tij T qp pq   . Significantly, the variance terms are also accounted for in the mean trajectory. This explains the second-order approximation leads to the perturbed mean trajectory. This section discusses very briefly about the numerical testing for the mean and variance evolutions derived in the previous section. A greater detail is given in the Author’s Royal Society contribution. This chapter is intended to demonstrate the usefulness of the Itô theory for stochastic problems in dynamical systems by taking up an appealing case in satellite mechanics. 4. Conclusion In this chapter, the Author has derived the conditional moment evolutions for the motion of an orbiting satellite in dust environment, i.e. a noisy dynamical system. The noisy dynamical system was modeled in the form of multi-dimensional stochastic differential equation. Subsequently, the Itô calculus for ‘the Brownian motion process as well as the dynamical system driven by the Brownian motion’ was utilized to study the stochastic problem of concern here. Furthermore, the Itô theory was utilized to analyse the resulting stochastic differential equation qualitatively. The Markovian stochastic differential system can be analysed using the Kolmogorov-Fokker-Planck Equation (KFPE) as well. The KFPE- based analysis involves the definition of conditional expectation, the adjoint property of the Fokker-Planck operator as well as integration by part formula. On the other hand, the Itô differential rule involves relatively fewer steps, i.e. Taylor series expansion, the Brownian motion differential rule. It is believed that the approach of this chapter will be useful for analysing stochastic problems arising from physics, mathematical finance, mathematical control theory, and technology. Appendix 1 The qualitative analysis of the non-linear autonomous system can be accomplished by taking the Lie derivative of the scalar function  , where ,: RU   U is the phase space of the non-linear autonomous system and .)( Rx t   The function  is said to be the first integral if the Lie derivative  v L vanishes (Arnold 1995). The problem of analysing the non-linear stochastic differential system qualitatively becomes quite difficult, since it involves multi-dimensional diffusion equation formalism. The Itô differential rule (Liptser and Shirayayev 1977, Sage and Melsa 1971) allows us to obtain the stochastic evolution of the function  . Equation (6) of this chapter can be re-written as 2 2 )( ),()( 2 1 ),( )()( i t tii T i ti i i tt x x txGGtxf x x dt xd         ji t ji tij T xx x txGG      )( ),()( 2  .),( )( 1,1    BtxG x x ti rni i t       Consider the function )()( tt xEx   , where E (.) is the energy function. Thus the stochastic evolution of the energy function (Sharma and Parthasarathy 2007) can be stated as 2 2 )( ),()( 2 1 )()( i t tii T i i i i tt x xE txGGx x xE dt xdE         . )( ),()( 2 ji t ji tij T xx xE txGG      The above evolution for the stochastic differential system of this chapter assumes the following structure: , ),,,( ),()( 2 1 ),,,(),,,(),,,(),,,(),,,( 2 2 , i r tii T ji r r r rrrr x vrE txGG vrE v v vrEvrE r r vrE dt vrdE                                    where )()( 2 1 ),,,( 222 rVrvvrE rr   . A simple calculation will show that . ) 2 ())(())(( ),,,( 222 2 2 2'2'2           r r r v rvrVrvrVr dt vrdE r r r rr r Thus the derivative of the energy function for the stochastic system of concern here will not vanish leading to the non-conservative nature of the energy function. Appendix 2 The Fokker-Planck equation has received attention in literature and found applications for developing the prediction algorithm for the Itô stochastic differential system. Detailed The Itô calculus for a noisy dynamical system 37 . ),( 2 1 2 , qp t pq qp xx txf P      The correction term qp t pq qp xx txf P      ),( 2 1 2 , involves the variance term . pq P The evolution of the variance term ij P encompasses the contributions from the preceding variances, partials of the system non-linearity, the diffusion coefficient ),()( txGG tij T  as well as the second-order partial term ),()( 2 1 , txGGP tij T qp pq   . Significantly, the variance terms are also accounted for in the mean trajectory. This explains the second-order approximation leads to the perturbed mean trajectory. This section discusses very briefly about the numerical testing for the mean and variance evolutions derived in the previous section. A greater detail is given in the Author’s Royal Society contribution. This chapter is intended to demonstrate the usefulness of the Itô theory for stochastic problems in dynamical systems by taking up an appealing case in satellite mechanics. 4. Conclusion In this chapter, the Author has derived the conditional moment evolutions for the motion of an orbiting satellite in dust environment, i.e. a noisy dynamical system. The noisy dynamical system was modeled in the form of multi-dimensional stochastic differential equation. Subsequently, the Itô calculus for ‘the Brownian motion process as well as the dynamical system driven by the Brownian motion’ was utilized to study the stochastic problem of concern here. Furthermore, the Itô theory was utilized to analyse the resulting stochastic differential equation qualitatively. The Markovian stochastic differential system can be analysed using the Kolmogorov-Fokker-Planck Equation (KFPE) as well. The KFPE- based analysis involves the definition of conditional expectation, the adjoint property of the Fokker-Planck operator as well as integration by part formula. On the other hand, the Itô differential rule involves relatively fewer steps, i.e. Taylor series expansion, the Brownian motion differential rule. It is believed that the approach of this chapter will be useful for analysing stochastic problems arising from physics, mathematical finance, mathematical control theory, and technology. Appendix 1 The qualitative analysis of the non-linear autonomous system can be accomplished by taking the Lie derivative of the scalar function  , where ,: RU   U is the phase space of the non-linear autonomous system and .)( Rx t   The function  is said to be the first integral if the Lie derivative  v L vanishes (Arnold 1995). The problem of analysing the non-linear stochastic differential system qualitatively becomes quite difficult, since it involves multi-dimensional diffusion equation formalism. The Itô differential rule (Liptser and Shirayayev 1977, Sage and Melsa 1971) allows us to obtain the stochastic evolution of the function  . Equation (6) of this chapter can be re-written as 2 2 )( ),()( 2 1 ),( )()( i t tii T i ti i i tt x x txGGtxf x x dt xd         ji t ji tij T xx x txGG      )( ),()( 2  .),( )( 1,1    BtxG x x ti rni i t       Consider the function )()( tt xEx   , where E (.) is the energy function. Thus the stochastic evolution of the energy function (Sharma and Parthasarathy 2007) can be stated as 2 2 )( ),()( 2 1 )()( i t tii T i i i i tt x xE txGGx x xE dt xdE         . )( ),()( 2 ji t ji tij T xx xE txGG      The above evolution for the stochastic differential system of this chapter assumes the following structure: , ),,,( ),()( 2 1 ),,,(),,,(),,,(),,,(),,,( 2 2 , i r tii T ji r r r rrrr x vrE txGG vrE v v vrEvrE r r vrE dt vrdE                                    where )()( 2 1 ),,,( 222 rVrvvrE rr   . A simple calculation will show that . ) 2 ())(())(( ),,,( 222 2 2 2'2'2           r r r v rvrVrvrVr dt vrdE r r r rr r Thus the derivative of the energy function for the stochastic system of concern here will not vanish leading to the non-conservative nature of the energy function. Appendix 2 The Fokker-Planck equation has received attention in literature and found applications for developing the prediction algorithm for the Itô stochastic differential system. Detailed Stochastic Control38 discussions on the Fokker-Planck equation, its approximate solutions and applications in sciences can be found in Risken (1984), Stratonovich (1963). The Fokker-Planck equation is also known as the Kolmogorov forward equation. The Fokker-Planck equation is a special case of the stochastic equation (kinetic equation) as well. The stochastic equation is about the evolution of the conditional probability for given initial states for non-Markov processes. The stochastic equation is an infinite series. Here, we explain how the Fokker-Planck equation becomes a special case of the stochastic equation. The conditional probability density ).()() , ,,(), ,,(), ,,( 143232121 nnnnnn xpxxpxxxxpxxxxpxxxp   In the theory of the Markov process, the above can be re-stated as ).()() ()(), ,,( 1322121 nnnn xpxxpxxpxxpxxxp   Thus, ),(),() ,(),(), ,,( 1,32,21,21 13221 nnnttttttn xqxxqxxqxxqxxxp nn    where ),( 1, 1 iitt xxq ii   is the transition probability density, ni   1 and . 1 ii tt   The transition probability density is the inverse Fourier transform of the conditional characteristic function, i.e. . 2 1 ),( )(),( 1, 11 1 duEeexxq iiii ii xxiuxxiu iitt        (11) For deriving the stochastic equation, we consider the conditional probability density ),( 21 xxp where ).()(),( 22121 xpxxpxxp  After integrating over the variable 2 x , the above equation leads to .)(),()( 2221,1 21 dxxpxxqxp tt   (12) Equation (12) in combination with equation (11) leads to .)()( 2 1 )( 22 )()( 1 2121 dudxxpEeexp xxiuxxiu     (13) The conditional characteristic function is the conditional moment generating function and the n th order derivative of the conditional characteristic function )( 21 xxiu Ee  evaluated at the 0u gives the n th order conditional moment. This can be demonstrated by using the definition of the generating function of mathematical science, i.e. n n n uxux )(),( 0     , where ),( ux  can be regarded as the generating function of the sequence )}({ x n  . As a result of this, the characteristic function n n n xxiu xx n iu Ee )( ! )( 21 0 )( 21     . After introducing the definition of the conditional characteristic function, equation (13) can be recast as dudxxpxx n iu exp n n n xxiu 22 0 21 )( 1 )())( ! )( ( 2 1 )( 21       .)()())( 2 1 ( ! 1 2 0 221 )( 21 dxxpxxdueiu n n n xxiu n        (14) The term duiue n xxiu )( 2 1 )( 21    within the second integral sign of equation (14) becomes )()( 21 1 xx x n      and leads to the probability density )( 1 xp .)()()()( ! 1 2221 0 21 1 dxxpxxxx xn n n n         15) Consider the random variables 1 t x and , 2 t x where 21 tt  . The time instants 1 t and 2 t can be taken as ., 21 tttt     For the short hand notation, introducing the notion of the stochastic process, taking xxxx   21 ,  , equation (15) can be recast as dxxpxxxx xn xp n n n )()()()( ! 1 )( 0            dxxpxkxx xn n n n )()()()( ! 1 0           , The Itô calculus for a noisy dynamical system 39 discussions on the Fokker-Planck equation, its approximate solutions and applications in sciences can be found in Risken (1984), Stratonovich (1963). The Fokker-Planck equation is also known as the Kolmogorov forward equation. The Fokker-Planck equation is a special case of the stochastic equation (kinetic equation) as well. The stochastic equation is about the evolution of the conditional probability for given initial states for non-Markov processes. The stochastic equation is an infinite series. Here, we explain how the Fokker-Planck equation becomes a special case of the stochastic equation. The conditional probability density ).()() , ,,(), ,,(), ,,( 143232121 nnnnnn xpxxpxxxxpxxxxpxxxp   In the theory of the Markov process, the above can be re-stated as ).()() ()(), ,,( 1322121 nnnn xpxxpxxpxxpxxxp   Thus, ),(),() ,(),(), ,,( 1,32,21,21 13221 nnnttttttn xqxxqxxqxxqxxxp nn    where ),( 1, 1 iitt xxq ii   is the transition probability density, ni   1 and . 1 ii tt   The transition probability density is the inverse Fourier transform of the conditional characteristic function, i.e. . 2 1 ),( )(),( 1, 11 1 duEeexxq iiii ii xxiuxxiu iitt        (11) For deriving the stochastic equation, we consider the conditional probability density ),( 21 xxp where ).()(),( 22121 xpxxpxxp  After integrating over the variable 2 x , the above equation leads to .)(),()( 2221,1 21 dxxpxxqxp tt   (12) Equation (12) in combination with equation (11) leads to .)()( 2 1 )( 22 )()( 1 2121 dudxxpEeexp xxiuxxiu     (13) The conditional characteristic function is the conditional moment generating function and the n th order derivative of the conditional characteristic function )( 21 xxiu Ee  evaluated at the 0u gives the n th order conditional moment. This can be demonstrated by using the definition of the generating function of mathematical science, i.e. n n n uxux )(),( 0     , where ),( ux  can be regarded as the generating function of the sequence )}({ x n  . As a result of this, the characteristic function n n n xxiu xx n iu Ee )( ! )( 21 0 )( 21     . After introducing the definition of the conditional characteristic function, equation (13) can be recast as dudxxpxx n iu exp n n n xxiu 22 0 21 )( 1 )())( ! )( ( 2 1 )( 21       .)()())( 2 1 ( ! 1 2 0 221 )( 21 dxxpxxdueiu n n n xxiu n        (14) The term duiue n xxiu )( 2 1 )( 21    within the second integral sign of equation (14) becomes )()( 21 1 xx x n      and leads to the probability density )( 1 xp .)()()()( ! 1 2221 0 21 1 dxxpxxxx xn n n n         15) Consider the random variables 1 t x and , 2 t x where 21 tt  . The time instants 1 t and 2 t can be taken as ., 21 tttt   For the short hand notation, introducing the notion of the stochastic process, taking xxxx  21 ,  , equation (15) can be recast as dxxpxxxx xn xp n n n )()()()( ! 1 )( 0            dxxpxkxx xn n n n )()()()( ! 1 0           , Stochastic Control40 where )( )( xk xx n n     and the time interval condition 0  leads to )()()( ! 1 )()( 1 0 xpxk xn xpxp Lt n n n            or ).()()( ! 1 )( 1 xpxk xn xp n n n       (16) The above equation describes the evolution of conditional probability density for given initial states for the non-Markovian process. The Fokker-Plank equation is a stochastic equation with ixk i  2,0)( . Suppose the scalar stochastic differential equation of the form ,),(),( tttt dBtxgdttxfdx  using the definition of the coefficient )(xk n of the stochastic equation (16), i.e. ),( )( xk xx n n     0  , we have ),,()( 1 txfxk  ),,()( 2 2 txgxk  and the higher-order coefficients of the stochastic equation will vanish as a consequence of the Itô differential rule. Thus, the Fokker-Planck equation ).( ),( 2 1 )(),()( 2 22 xp x txg xptxf x xp        Acknowledgement I express my gratefulness to Professor Harish Parthasarathy, a Scholar and Author, for introducing me to the subject and explaining cryptic mathematics of stochastic calculus. 5. References Arnold, V. I. (1995). Ordinary Differential Equations, The MIT Press, Cambridge and Massachusetts. Dacunha-Castelle, D. & Florens-Zmirou, D. (1986). Estimations of the coefficients of a diffusion from discrete observations, Stochastics, 19, 263-284. Jazwinski, A. H. (1970). 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[...]... in the simulation;   25 000 17500 26 25 26 .25 15.75 11. 025  17500 24 500 55 12 73.50 55. 125 46.305     26 25 55 12. 5 1653.75 27 .5 625 24 .80 625 0 24 .310 125     26 .25 73.5 27 .5 625 0.55 125 0 0.5788 12 0.64 827 0     15.75 55. 125 24 .80 625 0.5788 12 0.694575 0.875164  11. 025 46.305 24 .310 125 0.64 827 0 0.875164 1 .22 523 3 The spreading behaviour of a cloud of contaminants The characteristics of a spreading... in the stochastic differential equations Paths of 5 particles driven by CN for 52 days 4 2. 5 x 10 x 10 2 1.5 1.5 1 y[grid−index m] 1 y[grid−index m] Paths of 5 particles driven by BM for 52 days 4 2. 5 2 57 0.5 0 −0.5 0.5 0 −0.5 −1 −1 −1.5 −1.5 2 2 2. 5 2. 5 2 −1.5 −1 −0.5 0 0.5 x[grid−index n] 1 1.5 2 2. 5 2. 5 2. 5 4 2 −1.5 −1 −0.5 0 0.5 x[grid−index n] x 10 (a) x 10 2 2.5 4 x 10 Paths of a particle... x 10 3 4 x 10 (b) a dispersion after of 5000 particles after 51 days by CN 4 2. 5 2 a dispersion after of 5000 particles after 51 days by BM 4 2. 5 2 1.5 x 10 2 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 2 2. 5 2. 5 2 2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 4 2. 5 2. 5 2 −1.5 −1 −0.5 0 x 10 (c) 0.5 1 1.5 2 2.5 4 x 10 (d) Fig 4 Dispersion of a cloud of 5000 particles released in the idealised whirl pool... and driven by BM for 52 days 4 2. 5 2 x 10 2 1.5 1.5 1 y[grid−index m] 1 y[grid−index m] 1.5 (b) Paths of a particle labeled 1 and driven by CN for 52 days 4 2. 5 1 0.5 0 −0.5 0.5 0 −0.5 −1 −1 −1.5 −1.5 2 2. 5 2. 5 2 2 −1.5 −1 −0.5 0 0.5 x[grid−index n] 1 1.5 2 2.5 4 2. 5 2. 5 2 −1.5 −1 −0.5 0 0.5 x[grid−index n] x 10 (c) 1 1.5 2 2.5 4 x 10 (d) Fig 5 Tracking of a single marked particle in the whirl... al (20 09) In Section 3.4 it is suggested that for t TL a turbulent mixing coefficient similar to constant dispersion coefficient D such that D = can be defined 2 2 TL ∏ 2 i 2 56 Stochastic Control 3 4 x 10 a cloud of 5000 particles at initial time 3 1 a cloud of 5000 particles at initial time 2 1 y[grid−index m] y[grid−index m] 2 4 x 10 0 −1 2 0 −1 2 −3 −3 2 −1 0 1 x[grid−index n] 2 3 4 −3 −3 2 −1... in a sense that for t TL the variance of a cloud of particles shortly after deployment is then given by the following equation: Var[ Xt ] = 1 2 2 σ α1 TL t2 2 (19) 2 With the constant dispersion coefficient D = 1 2 2 TL , the variance of the cloud of particles, 1 2 therefore initially grows with the square of time: Var[ X (t)] = D 2 t TL (20 ) 52 Stochastic Control 3.4 The general long term behaviour... follows that, E [uτ us ] = Var[ Xt ] = 2 −|τ −s| 1 2 α1 TL e TL 2 t 0 t 0 −(τ −s) 1 2 α1 TL e TL dτds 2 (17) The integration of equation (17) can easily be yielded by separately considering the regions τ < s and τ > s, and it can be shown that t3 t2 − 3 ··· 2 2TL 6TL = 3 2 2 TL 1 = Var[ Xt ] 2 2 TL t2 σ 2 2 t3 1 1 − +··· 2 6 (18) Since the short time analysis, eqn (18) are of interest in this section... and variance: −t −2t Var[ X (t)] 2T T 2 = 2 2 TL 1 − L (1 − e TL ) + L (1 − e TL ) 1 t t 2t Thus, a position of a particle observed over a long time span as modelled by the coloured noise process u1 (t) behaves much like the one driven by Brownian motion with variance parameter Application of coloured noise as a driving force in the stochastic differential equations 51 2 2 2 2 TL Hence, the dispersion... the particle along the y direction proceeds completely analogously Let us now compute the variance of the general equations for position given by eqn. (23 ) Var[ X (t)] n = 2 ( TL )2 ∏i=1 2 i t 0 [1 − ]2 dk (24 ) For σ > 0, αi > 0, and TL > 0, the process again behaves like a Brownian process with n 2 variance parameters TL 2 ∏i=1 2 as t → ∞ Thus the appropriate diffusion coefficient from i 2 T 2. .. variance variance variance variance 1 .2 1 0.005 of of of of u1(t) u2(t) u3(t) u (t) 4 0.8 0.6 0 0.4 −0.005 −0.01 0 0 .2 20 40 time 60 80 100 ( a) α1 = 2 = α3 = α4 = 1 100 20 40 time 60 80 100 ( b) α1 = 2 = α3 = α4 = 1 u1(t) u2(t) u3(t) u4(t) u5(t) u (t) 80 auto−covariance 0 0 60 40 6 20 0 20 −40 0 20 40 time 60 80 100 ( c) α1 = 2 = α3 = α4 = α5 = α6 = 1 Fig 3 (a)Shows that the mean goes to zero, . simulation;         25 000 17500 26 25 26 .25 15.75 11. 025 17500 24 500 55 12 73.50 55. 125 46.305 26 25 55 12. 5 1653.75 27 .5 625 24 .80 625 0 24 .310 125 26 .25 73.5 27 .5 625 0.55 125 0 0.5788 12 0.64 827 0 15.75 55. 125 24 .80 625 . simulation;         25 000 17500 26 25 26 .25 15.75 11. 025 17500 24 500 55 12 73.50 55. 125 46.305 26 25 55 12. 5 1653.75 27 .5 625 24 .80 625 0 24 .310 125 26 .25 73.5 27 .5 625 0.55 125 0 0.5788 12 0.64 827 0 15.75 55. 125 24 .80 625 . , ),,,( ),()( 2 1 ),,,(),,,(),,,(),,,(),,,( 2 2 , i r tii T ji r r r rrrr x vrE txGG vrE v v vrEvrE r r vrE dt vrdE                                    where )()( 2 1 ),,,( 22 2 rVrvvrE rr   . A simple calculation will show that . ) 2 ())(())(( ),,,( 22 2 2 2 2& apos ;2& apos ;2           r r r v rvrVrvrVr dt vrdE r r r rr r