Most Probable Phase Portraits of Stochastic Differential Equations and Its Numerical Simulation

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1 Mos Probable Phase Porrais of Sochasic Differenial Equaions and Is Numerical Simulaion Bing Yang, Zhu Zeng and Ling Wang 3 School of Mahemaics and Saisics, Huazhong Universiy of Science and Technology, yang_bobby@qq.com School of Mahemaics and Saisics, Cenral China Normal Universiy, zhuzeng_syle@qq.com 3 School of Mahemaics and Saisics, Cenral China Normal Universiy, wanglingccnu@qq.com Absrac. A pracical and accessible inroducion o mos probable phase porrais is given. The reader is assumed o be familiar wih sochasic differenial equaions and Euler-Maruyama mehod in numerical simulaion. The aricle firs inroduce he mehod o obain mos probable phase porrais and hen give is numerical simulaion which is based on Euler-Maruyama mehod. All of hese are given by examples and easy o undersand. Key Words. Mos probable phase porrais, Euler-Maruyama mehod, numerical simulaion, sochasic differenial equaions, MATLAB Equaion Secion (Nex). Inroducion A phase porrai is a geomeric represenaion of he rajecories of a dynamical sysem in he phase plane. For deerminisic dynamical sysems, phase porrais provide geomeric picures of dynamical orbis, a leas for lower dimensional sysems. However, a sochasic dynamical sysem is quie differen from he deerminisic case. There have been some opions of phase porrais already. Bu hey are all limied in some ways. Hence, in his aricle we explain a new kind of phase porrais mos probable phase porrais, which is firs proposed by Prof. Duan in his recen published book [, 5.3]. In he nex secion, we inroduce you he hisory of phase porrais for SDEs, including some examples o show how o ge mos probable phase porrais. In secion 3 we explain our moivaion in his aricle is o esablish he numerical mehod of mos

2 B. Yang, Z. Zeng & L. Wang probable phase porrais. In secion 4, we show our numerical mehod hrough examples and compare i wih he real resul. Finally, we give a summary of our aricle. Equaion Secion (Nex). Hisory. Earlier mehods in phase porrais There are wo apparen opions of phase porrais, which are mean phase porrais and almos sure phase porrais... Mean phase porrais Le us consider a simple linear SDE sysem dx 3X db (.) The mean EX evolves according o he linear deerminisic sysem d EX 3 EX (.) d which is he original sysem wihou noise. In oher words, he mean phase porrai will no capure he impac of noise in his simple linear SDE sysem. The siuaion is even worse for nonlinear SDE sysem. For example, consider EX dx ( X X )d db (.3) 3 Take mean on boh side of his SDE o ge d EX 3 EX E( X ) (.4) d Thus, we do no have a closed differenial equaion for he evoluion of mean, because E( X ) ( EX ). This is a heoreical difficul for analyzing mean phase 3 3 porrais for sochasic sysem. The same difficuly arises for mean-square phase porrais and higher-momen phase porrais... Almos sure phase porrais Anoher possible opion is o plo sample soluion orbis for an SDE sysem, mimicking deerminisic phase porrais. If we plo represenaive sample orbis in he sae space, we will see i could hardly offer useful informaion for undersanding dynamics. Moreover, each sample orbi is a possible oucome of a realisic orbi X of he sysem. Bu which sample orbi is mos possible or maximal likely? This is deermined by he maximizers of he probabiliy densiy funcion p x,of X, a every ime.. New mehod in phase porrais -- Mos probable phase porrais In he las secion, we discussed wo sors of phase porrais. Mean phase porrais and

3 Mos probable phase porrais 3 almos sure phase porrais. Mean phase porrais has difficulies for nonlinear SDE sysems and higher-momen phase porrais. Almos sure phase porrais shows us a very complicaed picure. I is difficul o find useful informaion. In his secion, we inroduce you a deerminisic geomeric ool mos probable phase porrais, which is firs proposed by Professor J. Duan, see [, 5.3]. The mos probable phase porrais provide geomeric picures of mos probable or maximal likely orbis of sochasic dynamical sysems. I is based on Fokker-Planck equaions. n For an SDE sysem in R dx b( X )d ( X )db, X (.5) The Fokker-Planck equaion for he probabiliy densiy funcion p( x, ) of X is p( x, ) A* p( x, ) (.6) Wih iniial condiion p( x, ) ( x ). Recall ha he Fokker-Planck operaor is Where xx i j T A* p Tr( H( p )) ( bp ) (.7) H ( ) is he Hessian marix, Tr evaluaes he race of a marix, and T H( p ) denoes he marix muliplicaion of H,, T p. We ake wo simple example o show you how o ge mos probable phase porrais. Example.: Consider a scalar SDE wih addiive noise where, are real consans. dx X d db, X x (.8) We solve he SDE firs. We ry for a soluion of he form X( ) X ( )X ( ) (.9) where and dx X dx X ( ) dx C( )d D( )db X ( ) x The funcion C(), D() o be chosen. Then (.) (.)

4 4 B. Yang, Z. Zeng & L. Wang dx X dx X dx d Xd C( )X d D( )X db Xd db (.) C =, D ( X ) will work. Thus (.) reads For his, X ( ) x dx ( X ) db (.3) Now, we begin o solue X. We se Y log X, hen using Io' ˆ s fomula, we have Thus dy dx d X X d (.4) Y c, X C e (.5) And we know ha X ( ), i s easy o ge C. So we ge According o (.3), we can ge We conclude ha X e (.6) s s X e db x (.7) ( s ) X( ) X ( )X ( ) xe e db s (.8) This is a Gaussian process. Once we have is mean and variance, we can hen have is probabiliy densiy funcion p x,. Is mean is and he variance is e x X (.9)

5 Mos probable phase porrais 5 Var X X e x ( s ) e db s ( s ) e ds e So he probabiliy densiy funcion for he soluion X is (.) p x, e e x exp x e x exp e x ( e ) ( e ) (.) By seing x p or jus observe p x,, we obain he maximizer a ime : for every x m x e x (.). Thus, he mos probable dynamical sysem is x m x (.3) m This is he same as he corresponding deerminisic dynamical sysem x x, in he sysem wih addiive noise. The phase porrai for he corresponding deerminisic sysem and he mos probable phase porrai are in Figure. x Figure. Top: Phase porrai for he corresponding deerminisic sysem (noise is absen) in Example.. Boom: Mos probable phase porrai for Example.. Example.: Consider a scalar SDE wih muliplicaive noise. dx X d X db, X x (.4)

6 6 B. Yang, Z. Zeng & L. Wang where, are real consans. We solve he SDE firs. Le Y =logx (.5) and so dx X d dy d(logx ) by Ioˆ ' s fomula X X ( ) d db Consequenly X ( )B (.6) x e (.7) Then le us find ou is probabiliy disribuion funcion F(x,) and hen is probabiliy densiy funcion p(x,). The sae is an equilibrium sae. When he iniial sae x is posiive (negaive), he soluion is also posiive (negaive). The disribuion funcion is calculaed as For F( x, )= ( X x) ( x e x) x (l og ( ) ( )) x ( ) B x ( e e ) x x ( B (log( ) ( ) ) ) x e ( )B d x, he disribuion funcion F is non-zero only if x and x (.8) have he same sign (i.e., x x is posiive). The probabiliy densiy funcion is hen x (log( ) ( ) ) x p(x, )= xf(x, ) exp( ) (.9) x Seing, we obained x p m 3 ( ) x x e. (.3)

7 Mos probable phase porrais 7 I can be checked ha his is indeed he maximizer of p(x,)a ime. Thus, he mos probable dynamical sysem is Equaion Secion (Nex) 3. Moivaion x m 3 ( ) xm (.3) As we see in he firs secion, he mehods o ge he mos probable phase porrai can be commen as he following four seps. Sep: figure ou he soluion of he sochasic differenial equaions; Sep: figure ou he probabiliy densiy funcion Sep3: figure ou he mos probable soluion by seing Sep4: plo he mos probable phase porrai. p(x,) of he soluion; For many simple sochasic differenial equaions, we can ge he mos probable phase porrais easily by hese four seps. However, for some more complicaed SDEs, solving he analyical soluion is no easy a all and hus we canno ge he MPPP easily. Therefore, i is significan o find numerical ways o ge he MPPP. This passage will show you our numerical mehod laer. Equaion Secion (Nex) x p 4. Numerical mehod o ge mos probable phase porrais According o he four seps lised in he former secion, we will follow he hree seps o ge he mos probable phase porrai by MATLAB simulaion: Sep: plo many discreized pahs of he soluion o he sochasic differenial equaions; Sep: ge he mos probable poins a every discree values. Sep3: plo he discreized mos probable phase porrais. Then we will follow he hree seps o inroduce you our mehod. And we will ake he SDE in Example. as example. 4. Numerical simulaion of SDEs -- Euler-Maruyama mehod A scalar, auonomous SDE can be wrien in inegral form as X x f ( X,s ) ds g( X,s ) db (4.) s s s ; MPPP: abbreviaion of Mos Probable Phase Porrais.

8 8 B. Yang, Z. Zeng & L. Wang %% EM Euler-Maruyama mehod on linear SDE % % SDE is dx = alpha*x*d + bea*db, X() = Xzero, % where alpha =, bea= and Xzero =. % % Discreized Brownian pah over [,] has d = ^(-7). % Euler-Maruyama uses imesep d. %% I: iniial daas T = ; % righ poin of he ime inerval Xzero = ; % iniial value of x M = ^5; % number of pahs N = ^7*T; % number of seps in every pah randn('sae',m*n) d = T/N; % discreized ime sep db = sqr(d)*randn(m,n); % Brownian incremens X = ones(m,n); % preallocae for efficiency %% II: figure ou he value of X in every discreized ime of every pah for i=:m Xemp = Xzero; for j = :N Xemp = Xemp + Xemp*d + db(i,j); X(i,j) = Xemp; %% III: plo he pahs for i=:m plo([:t/n:t],[xzero,x(i,:)]),hold on xlabel('','fonsize',) ylabel('x','fonsize',6,'roaion',,'horizonalalignmen','righ' ) Lising : M-file Xpahs.m where f and g are scalar funcions and he iniial condiion is I is usual o rewrie (4.) in differenial equaion form as X x. dx f ( X, ) d g( X, ) db, X x (4.) To apply a numerical mehod o (4.) over [, T], we firs discreize he inerval. T Le for some posiive ineger N, and he pariion is N N T

9 Mos probable phase porrais 9 Our numerical approximaion o (EM) mehod akes he form X( ) j will be denoed j j j j j j j j X j. The Euler Maruyama X X f ( X, ) g( X, )( B( ) B( )), j,,,n (4.3) In his aricle, we will use discreized Brownian pahs in [, 57] o generae he incremens needed B( j ) B( j ) in (4.3). We will apply he EM mehod o he linear SDE in Example. dx X d db, X x (4.4) and we se,x. You can see he code in Lising. Run he code, we can ge 5 discreized pahs of X, jus as Figure 4. shows. Figure 4.: EM approximaion of he soluion for (4.4), from Xpahs.m 4. Numerical simulaion of MPPP Now we have ge he many simulaion pahs of SDE. The nex sep is o figure ou he mos probable poins a every discree values. We use he funcion ksdensiy in MATLAB o achieve our goal. For every discreized ime, here are M poins in M pahs of X and we use funcion ksdensiy in MATLAB o ge he kernel smoohing densiy esimaion of X. Then we can easily find he mos probable poins in every ime, which is corresponding o he highes poins of kernel smoohing densiy esimaion a every ime. From Example., we can see ha he rue MPPP is m rue MPPP in blue line o see if he simulaion resul maches well. x e x. And we plo he

10 B. Yang, Z. Zeng & L. Wang %% Ploing MPPP % % SDE is dx = alpha*x*d + bea*db, X() = Xzero, % where alpha =, bea= and Xzero =. % % Discreized Brownian pah over [,] has d = ^(-7). % Euler-Maruyama uses imesep d. %% I: iniial daa T = ; % righ poin of he ime inerval Xzero = ; % iniial value of x M = ^5; % number of pahs N = ^8*T; % number of seps in every pah randn('sae',m*n) d = T/N; % discreized ime sep db = sqr(d)*randn(m,n); % Brownian incremens X = ones(m,n); % preallocae for efficiency %% II: figure ou he value of X in every discreized ime of every pah for i=:m Xemp = Xzero; for j = :N Xemp = Xemp + Xemp*d + db(i,j); X(i,j) = Xemp; %% III: figure ou he mos probable poins a every discree values. MPPP = zeros(,n); % preallocae for efficiency for =:N Xem = X(:,); [h,xi] = ksdensiy(xem); [hmax,i] = max(h); MPPP() = xi(i); %% IV: plo he real mos probable phase porrais = [:T/N:T]; MPPP_rue = exp(); plo(,mppp_rue,'b-'),hold on MPPP_err = abs(mppp(n)-mppp_rue(n))/mppp_rue(n) %% V: plo he discreized mos probable phase porrais. plo([:t/n:t],[xzero,mppp],'r-') xlabel('','fonsize',) ylabel('mppp','fonsize',6) Lising : M-file MPPP.m

11 Mos probable phase porrais Running he code in Lising, we can ge Figure 4.. Figure 4.: True MPPP(blue line) and simulaed MPPP(red line) for (4.4), from MPPP.m When we se T o be, 4, 6, 8, we can ge Figure 4.3 T= T=4 T=6 T=8 Figure 4.3: MPPPs when T =, 4, 6, 8 respecively

12 B. Yang, Z. Zeng & L. Wang %% III: figure ou he value of X in every discreized ime of every pah for i=:m Xemp = Xzero; for j = :N Xemp = Xemp + f(xemp,)*d + g(xemp,)db(i,j); X(i,j) = Xemp; Lising 3: M-file MPPP.m From Figure 4.3, we can see ha our mehod works very well. Furhermore, we have o noe ha he numerical mehod is broadly applicable for all SDEs. For one-dimension SDEs,, dx f X d g X db, X x (4.5) We jus need o correspondingly change he second par of he MATLAB code in Lising o be Lising 3. For wo-dimension SDEs, his mehod could also work, ake (4.6) for example. dx Y d B dy X d B We can ge is numerical simulaion resul by he code in Lising 4. Running he code, we can ge he MPPP of X and Y, see in Figure 4.4. (4.6) Figure 4.4: simulaed MPPP of X and Y for (4.6), from MPPP_.m

13 Mos probable phase porrais 3 %% Ploing MPPP of -dimension SDE % SDE is dx = Y*d + db % dy = X*d + db, X()=Xzero, Y()=Yzero % Discreized Brownian pah over [,] has d = ^(-7). % Euler-Maruyama uses imesep d. %% I: iniial daas T = ; % righ poin of he ime inerval Xzero = ; Yzero = ; % iniial value of x M = ^5; % number of pahs N = ^7*T; % number of seps in every pah randn('sae',*m*n) d = T/N; % discreized ime sep db = sqr(d)*randn(m,n); db = sqr(d)*randn(m,n);% Brownian incremens X = ones(m,n); Y = ones(m,n); % preallocae for efficiency %% II: figure ou he value of X in every discreized ime of every pah for i=:m Xemp = Xzero; Yemp = Yzero; for j = :N Xemp = Xemp - Yemp*d - db(i,j); Yemp = Yemp + Xemp*d + db(i,j); X(i,j) = Xemp; Y(i,j) = Yemp; %% III: figure ou he mos probable poins a every discree values. MPPPx = zeros(,n) ; MPPPy = zeros(,n); % preallocae for efficiency for =:N Xem = X(:,); Yem = Y(:,); [hx,xi] = ksdensiy(xem); [hy,yi] = ksdensiy(yem); [hxmax,ix] = max(hx); [hymax,iy] = max(hy); MPPPx() = xi(ix); MPPPy() = yi(iy); %% IV: plo he discreized mos probable phase porrais. plo([:t/n:t],[xzero,mpppx],'r*',[:t/n:t],[yzero,mpppy],'g*') xlabel('','fonsize',6) ylabel('mppp','fonsize',6) %plo([xzero,mpppx],[yzero,mpppy],'r*') %xlabel('x','fonsize',6) %ylabel('y','fonsize',6,'roaion',,'horizonalalignmen','righ ') Lising 4: M-file MPPP_.m

14 4 B. Yang, Z. Zeng & L. Wang And he phase porrais in X-Y phase plain is Figure 4.5: simulaed MPPP in X-Y plain for (4.6), from MPPP_.m Jus as we noed in one-dimension SDEs, he mehod is also applicable for he general cases dx f ( X, Y, ) d g ( X, Y, ) db dy f ( X, Y, ) d g ( X, Y, ) db (4.7) For hree-dimension and more higher dimension SDEs, obviously he mehod can also work. Equaion Secion (Nex) 5. Summary Recalling all of he above, we have esablished a beer way o ge phase porrais of SDEs. For simple SDEs, i s easy o ge is analyical soluion by he four seps in secion 3. For more complicaed SDEs, we can use he numerical simulaion mehod in secion 4 and he numerical mehod is broadly applicable in all SDEs. Though i seems ha we have solved he quesion o ge MPPP, here are sill many problems o be solved laer. The mehod above is broadly applicable for all SDEs, bu for some SDEs, i canno fi he real resul very well or he MPPP is very unsable because of he sochasic facor. Thus i sill have spaces for furher improvemen. Moreover, analyses he convergence of he mehods is anoher problem o be solved, which may also be helpful for furher improvemen of he mehod.

15 Mos probable phase porrais 5 References: [] J. Duan, An inroducion o sochasic dynamics, Cambridge Universiy Press, 4 [] D. J. Higham, An algorihmic inroducion o numerical simulaion of sochasic differenial equaions, SIAM Review 43(),