Stability analysis of numerical methods for stochastic systems with additive noise

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1 Stability aalysis of umerical metods for stoctic systems wit additive oise Yosiiro SAITO Abstract Stoctic differetial equatios (SDEs) represet pysical peomea domiated by stoctic processes As for determiistic ordiary differetial equatios (ODEs), various umerical metods are proposed for SDEs We ave proposed two types of umerical stability for SDEs, amely mea-square stability ad trajectory stability However we ave cosidered te aalysis for te test equatio wit multiplicative oise oly, but ot additive I tis ote we will study umerical stability aalysis for a test system wit additive oise, ad will sow some results for te Euler-Maruyama metod 1 Itroductio Numerical stability for stoctic differetial equatios (SDEs) bee studied Tere sould be two types of test equatios, wit additive oise ad multiplicative oise We ave proposed te umerical mea-square stability (MS-stability) ad trajectory stability (T-stability) for a scalar SDE wit oe multiplicative oise [7, 6] However we ave ot discussed umerical stability for additive oise I tis ote we will study umerical stability of te Euler-Maruyama metod for a stoctic system wit additive oise Stability aalysis for SDE wit additive oise ca be see i [, 3] Cosider te d-dimesioal SDE of Ito-type give by d f ( t, )dt g( t, )dw, t, () (1) were f +1, g +1 ad W(t) is a scalar Wieer process ( t ) A Wieer process is a Gaussia process wit te property tat W, W W ( s) mit, s We sumed tat bot f ad g are sufficietly smoot so tat equatio (1) a uique solutio Te oise is called additive if g(t, x) does ot deped o x: oterwise it is called multiplicative Hereafter we sall coose te ce of additive oise, ie d f ( t, )dt g( t)dw () Let be te umerical approximatio to ( t ) ( ) wit costat step-size For te SDE (), te k t compoet of te Euler-Maruyama metod te form

2 k k k k 1 f ( t, ) g ( t ) W, (3) were W stads for te icremet of te Wieer process [1, 3] Now we cosider te followig liear stoctic test system wit oe additive oise d Ldt bdw, (4) were L is d d real matrix, b is d-dimesioal real vector ad W (t) is a scalar stadard Wieer process Te eigevalues i ( i 1,,, d ) of L are sumed distict ad satisty i ( i 1,,, d ) Te tere exists a o-sigular matrix T suc tat 1 T LT diag 1,,, d 1 We ow defie Y T ad o pre-multiplyig (4) by 1 dy Ydt T b dw 1 T, we obtai As for umerical aalysis of ordiary differetial equatios (ODEs), we coclude tat te liear stability aalysis for te system (4) is reduced to te followig scalar test equatio d dt dw, (),,, (5) Te exact solutio of (5) is We observe tat e Terefore we ave ad t e t t ( ts) s e dw ( ) t, [ ] e t [ ] (1 e ) (6) [ ] (7) t from [5] Stability of Euler-Maruyama metod First we will study umerical stability of te Euler-Maruyama metod wit respect to mea Numerical stability i mea is correspodig to te property (6) Te Euler-Maruyama metod for te test equatio (5) is

3 1 W (8) Takig te expectatio of eac side, we obtai 1 ( 1 ) (1 1 Tus if 1 1, we ave ) (9) Te we ca give te followig defiitio for umerical metods of SDEs Defiitio 1 Te umerical metod is said to be umerical stable i mea if te umerical solutio for te test equatio (5) is satisfied For te Euler-Maruyama metod, we ave te followig result Teorem 1 For te test equatio (5) te Euler-Maruyama metod is umerical stable i mea if 1 1 Note tat te stability coditio 1 1 is te same Euler metod for ODEs [4] I geeral te umerical stability coditios for te additive ce are coicidet wit tem for ODEs [] Next we will focus o te umerical solutio i mea square Like i mea, we get te followig relatio Cotiuig te iteratio, [ 1 ] 1 [ ] 1 1 ] [ ] 1 1 [ [ ] 1 1 ( 1) 1 [ ] 1 1 ( 1) ( 1) [ ] ( 1) ( 1) 1 1 [ ]

4 If te Euler-Maruyama metod is umerical stable i mea for (5), we ave te followig property of te umerical solutio wit respect to mea square, [ ] ( ) Note tat te equilibrium value i mea square sese is differet from te true value However te equilibrium value lim [ ] olds te followig property (1) Tis property ca be foud i Yua ad Mao [8] Tus we will give te followig defiitio for te ymptotic property i mea square Defiitio Te umerical metod is said to be ymptotically cosistet i mea square if te umerical solutio for te test equatio(5) satisfies lim(lim [ ]) We terefore obtai te followig result for te Euler-Maruyama metod Teorem If te Euler-Maruyama metod satisfies te umerical stable coditio i mea, ie 1 1, te Euler-Maruyama metod is ymptotically cosistet i mea square 3 Coclusios ad Future pects We studied umerical stability for a test system wit additive oise ad gave some results for te Euler-Maruyama metod We will aalyze te oter metods, eg stoctic teta, Ruge-Kutta type, multistep ad implicit metods REFERENCES [1] Gard, T C, 1988, Itroductio to Stoctic Differetial Equatios, Marcel Dekker [] Heradez, DB ad R Spigler, 199, A-stability of Ruge-Kutta metods for systems

5 wit additive oise, BIT 3, [3] Kloede, P E, ad E Plate, 199, Numerical Solutio of Stoctic Differetial Equatios, Spriger-Verlag [4] Lambert, J D, 1991, Numerical Metods for Ordiary Differetial Systems, Wiley [5] Mao,, 1997, Stoctic Differetial Equatios ad Applicatios, Horwood [6] Saito, Y ad T Mitsui, 1993, T-stability of umerical sceme for stoctic differetial equatios, WSSIAA, [7] Saito, Y ad T Mitsui, 1996, Stability aalysis of umerical scemes for stoctic differetial equatios, SIAM J Numer Aal 33, [8] Yua, C ad Mao,4, Stability i Distributio of Numerical Solutios for Stoctic Differetial Equatios, Stoc aal appl,

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat