BIT Numerical Mathematics(2006)46:261–282
DOI:10.1007/s10543-006-0060-5
Published online:2June2006–c Springer2006
ASYMPTOTIC MEAN-SQUARE STABILITY OF
TWO-STEP METHODS FOR STOCHASTIC
ORDINARY DIFFERENTIAL EQUATIONS
E.BUCKW AR1,R.HORV´ATH-BOKOR2and R.WINKLER1,
1Department of Mathematics,Humboldt-Universit¨a t zu Berlin,Unter den Linden6 ail:{buckwar,winkler}@mathematik.hu-berlin.de
2Department of Math.and Computer Science,University of Veszpr´e m,Egyetem utca10 8201Veszpr´e ail:hrozsa@almos.vein.hu
Abstract.
We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square n of the exact solution.As in deterministic numerical analysis we u a linear time-invariant test equation and perform a linear stability analysis.Standard approaches ud either to analy deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes.In order to obtain sufficient conditions for asymp-totic mean-square stability of stochastic linear two-step-Maruyama methods we con-struct and apply Lyapunov-type functionals.In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth-and Adams–Moulton-methods,the Milne–Simpson method and the BDF method.
AMS subject classification(2000):60H35,65C30,65L06,65L20.
Key words:stochastic linear two-step-Maruyama methods,mean-square asymptotic stability,linear stability analysis,Lyapunov functionals.
1Introduction.
Our aim is to study when a numerical approximation generated by a stochastic linear two step scheme shares asymptotic properties of the exact solution of an SDE of the form
(1.1)
d X(t)=f(t,X(t))d t+G(t,X(t))d W(t),t∈J,X(t0)=X0, wher
histe J=[t0,∞),f:J×R n→R n,G:J×R n→R n×m.Later we consider also complex-valued functions f,G,X.The driving process W is an m-dimensional Received December2004.Accepted in revid form March2006.Communicated by Anders Szepessy.
This rearch was supported by the DFG grant234499and the DFG Rearch Center Mathematics for Key Technologies in Berlin.
262 E.BUCKW AR ET AL.
Wiener process on the given probability space (Ω,F ,P )with a filtration (F t )t ∈J .The initial value X 0is a given F t 0-measurable random variable (it can be de-terministic data,of cour),independent of the Wiener process and with finite cond moments.We assume that there exists a path-wi unique strong solution X ={X (t ),t ∈J }of Equation (1.1)and we indicate the dependence of this solution upon the initial data by writing X (t )≡X (t ;t 0,X 0).The numerical methods to be considered are generally drift-implicit linear two-step Maruyama methods with constant step-size h which for (1.1)take the form
α2X i +1+α1X i +α0X i −1=h [β2f (t i +1,X i +1)+β1f (t i ,X i )+β0f (t i −1,X i −1)]
(1.2)
+γ1G (t i ,X i )∆W i +γ0G (t i −1,X i −1)∆W i −1,
for i =1,2,...,where t i =i ·h,i =0,1,...,and ∆W i =W (t i +1)−W (t i ).For normalization we t α2=1.We require given initial values X 0,X 1∈L 2(Ω,R n )that are F t 0-and F t 1-measurable,respectively.We emphasize that an explicit discretization is ud for the diffusion term.For β2=0,the stochastic multi-step scheme (1.2)is explicit,otherwi it is drift-implicit.See also [3,4,7,8,11,15,16,19,20].
Given a reference solution X (t ;t 0,X 0)of (1.1),the concept of asymptotic mean-square stability in the n of Lyapunov concerns the question whether or not solutions X D 0(t )=X (t ;t 0,X 0+D 0)of (1.1)exist and approach the refer-ence solution when t tends to infinity.The distance between X (t )and X D 0(t )is measured in the mean-square in L 2(Ω),and the terms D 0∈L 2(Ω)are small perturbations of the initial value X 0.
Already in the deterministic ca it is a difficult problem to answer the ques-tion when numerical approximations share asymptotic stability properties of the exact solution in general .Including stoch
astic components into the problem does not simplify the analysis.In deterministic numerical analysis,the first step in this direction is a linear stability analysis ,where one applies the method of interest to a linear test equation and discuss the asymptotic behaviour of the resulting recurrence equation ([9,10,12]).Well-known notions like A -stability of numerical methods refer to the stability analysis of linear test equations.In this article we would like to contribute to the linear stability analysis of stochas-tic numerical methods and thus we choo as a test equation the linear scalar complex stochastic differential equation
d X (t )=λX (t )d t +µX (t )d W (t ),t ≥0,X (0)=X 0,λ,µ,X 0∈C ,(1.3)with th
e complex geometric Brownian motion as exact solution.In complex arithmetic we denote by ¯ηthe complex conjugate o
f a complex scalar η∈C .The method (1.2)applied to (1.3)takes the followin
g form:
X i +1+α1X i +α0X i −1=h [β2λX i +1+β1λX i +β0λX i −1]
(1.4)+γ1µX i ∆W i +γ0µX i −1∆W i −1,i ≥1,
where α2=1.Subquently we will assume that the parameters in (1.4)are chon such that the resulting scheme is mean-square convergent (e [8]).Thenmambo
MEAN-SQUARE STABILITY OF TWO-STEP METHODS FOR SODEs263 the coefficients of the stochastic terms have to satisfyγ1=α2=1andγ0=α1+α2.We will in particular discuss stochastic versions of the explicit and implicit Adams ,the Adams–Bashforth and the Adams–Moulton method,respectively,the Milne–Simpson method and the BDF method. Section2contains definitions of the notions of stability discusd in this arti-cle.In Section3we will discuss the difficulties that ari when applying standard approaches for performing a linear stability analysis either for stochastic one-step methods or for deterministic multi-step schemes to the ca of stochastic multi-step methods.In Section4we give a Lyapunov-type theorem ensuring asymptotic mean-square stability of the zero solution of a class of stochastic dif-ference equations under sufficient conditions on the parameters.The method of construction of Lyapunov functionals[17]is briefly sketched.Then we construct appropriate Lyapunov functionals in four different ways and thus obtain four ts of sufficient conditions on the parameters.In Section5results for stochastic linear two-step methods,in particular the explicit and implicit Adams meth-ods,the Milne–Simpson method and the BDF method are prented.Section6 summarizes ourfindings and points out open problems.
2Basic notions.
We will be concerned with mean-square stability of the solution of Equation (1.1),with respect to perturbations D0in the initial data X0.We here recall various definitions,which are bad on tho given in[13].
Definition2.1.The reference solution X of the SDE(1.1)is termed
solution X D0(t)exists for all t≥t0and
E|X D0(t)−X(t)|2< ,
whenever t≥t0and E|D0|2<δ;
2.asymptotically mean-square stable,if it is mean-square stable and if there
exists aδ≥0such that whenever E|D0|2<δ
E|X D0(t)−X(t)|2→0for t→∞.
一对一雅思外教It is well known for which parametersλ,µ∈C the solutions
X(t)=X0e(λ−12|µ|2)t+µW(t)
impala(2.1)
of the linear test equation(1.3)approach zero in the mean-square n.The following result can be in[1,pp.189–190],[13,14,22,23].Its proof us the fact,that E eµW(t)−12|µ|2t=1.osaka
Theorem2.1.The zero solution of(1.3)is asymptotically mean-square stable if
Re(λ)<−1
2
|µ|2.
(2.2)ironcurtain
264 E.BUCKW AR ET AL.
三年级下册英语试卷
Now we formulate analogous definitions for the discrete recurrence equation (1.2)with solution {X i }∞i =0.We denote by {X i }∞i =0={X i (X 0,X 1)}∞i =0the ref-erence solution and by X D 0,D 1i ∞i =0= X i (X 0+D 0,X 1+D 1)∞i =0 a solution of (1.2)where the initial values have been perturbed.
Definition 2.2.The reference solution {X i }∞i =0of (1.2)is said to be
if for each ε>0,there exists a value δ>0such that,whenever E |D 0|2+|D 1|2 <δ,
E X D 0,D 1i −X i 2<ε,i =1,2,...;
2.asymptotically mean-square stable if it is mean-square stable and if there exists a value δ>0such that,whenever E |D 0|2+|D 1|2 <δ,
E X D 0,D 1i −X i 2→0as i →∞.
Recall that applying a convergent stochastic two-step method (1.2)to our test equation (1.3)results in the stochastic difference equation (1.4)with γ1=1,γ0=1+α1.For simplification in our subquent analysis we rewrite (1.4)as X i +1=a X i +c X i −1+b X i ξi +d X i −1ξi −1,(2.3)
a =−α1+λh β11−λh β2,c =−α0+λh β01−λh β2,(2.4)
b =µh 121−λh β2,d =µh 12(1+α1)1−λh β2,(2.5)
where ξi −1=h −12∆W i −1,and ξi =h −12∆W i are N (0,1)Gaussian random
variables,independent of each other.Obviously this recurrence equation admits the zero solution {X i }∞i =0={0}∞i =0,which will be the reference solution in the subquent stability analysis.
We would like to add a remark concerning the choice of the linear test equation (1.3).The scalar linear test equation (1.3)is less significant for SDEs than the corresponding scalar linear test equation is for ODEs.Generally,it is not possible to decouple systems of linear equations of the form
d X (t )=A X (t )d t +m
r =1B r X (t )d W r (t ),
where X is a vector-valued stochastic process,to scalar equations,since the eigenspaces of the matrices A,B 1,...,B m may differ.The results for the scalar test equation are thus only significant for linear systems where the matrices A,B 1,...,B m are simultaneously diagonalizable with constant tran
sformations.We refer to [21]for a linear stability analysis of one-step methods applied to systems of linear equations.As a first step in the area of linear stability analysis for stochastic multi-step schemes we restrict our attention to scalar linear test equations (e Section 6).
MEAN-SQUARE STABILITY OF TWO-STEP METHODS FOR SODEs265 3Review of standard approaches for a linear stability analysis. Several approaches for an investigation in linear stability analysis exist,either for stochastic one-step methods or for deterministic multi-step methods.How-ever,it turns out that standard methods can not easily be extended to the ca of stochastic multi-step methods.We here describe the difficulties arising with standard approaches.
3.1The approach for stochastic linear one-step schemes.
In[14]the author considers the stochasticθ-method and investigates its asymp-totic mean-square stability properties.The method,applied to the test equation (1.3)has the form
lantivyX i+1=X i+θhλX i+1+(1−θ)hλX i+√
hµX iξi,
whereθ∈[0,1]is afixed parameter.Rewritten as a one-step recurrence it reads
X i+1=( a+ bξi)X i,
(3.1)
where a=1+(1−θ)λh
1−θλh
, b=
µh12
1−θλh
.
Squaring both sides of Equation(3.1)and taking the expectation,one obtains an exact one-step recurrence for E|X i|,
E|X i+1|2=(| a|2+| b|2)E|X i|2,
(3.2)
玉米的功效与作用
漂亮的英文单词
which allows a direct derivation of conditions for asymptotic mean-square sta-bility of the zero solution.For comparison we now apply this approach to the stochastic multi-step method.Squaring both sides of(2.3)yields
|X i+1|2=|a+bξi|2|X i|2+|c+dξi−1|2|X i−1|2
+2 {(a+bξi)X i(c+dξi−1)X i−1},
and the last term is not so easily resolved.Either one resorts to inequalities, such as2AB≤A2+B2,or one follows the recurrence further down.The latter approach provides forλ,µ∈R an exact recurrence of the form
E|X i+1|2=(|a|2+|b|2)E|X i|2+(|c|2+|d|2)E|X i−1|2
+2a(ac+bd)i−2
j=0
c j E|X i−j|2+2a E(c+dξ0)X0X1.
In any ca one does not immediately obtain conditions for asymptotic mean-square stability of the zero solution as in the ca of the one-step recurrence (3.2).
