| An epoch-making masterpiece written by It o—On stochastic diferential equa-tions [60] was published in1951. From then on, an important concept in the develop-ment history of mathematics and physics—stochastic diferential equation (SDE) wasborn. The models builded by the SDEs provide more realistic description for physicalsystems since incorporating random elements. Thus stochastic diferential equationshave been widely applied in many areas of life, such as fnance, population dynamics,genetic regulation, biological waste treatment, radio communication, design of verylarge scale integrated circuit and so on. Since most SDEs cann't be solved explicitly,people pay more and more attention to the numerical methods for SDEs. While stabil-ity is an important standard for measuring numerical methods, however always only beconsidered in the sense of mean square and neglected in other senses. The aim of thisthesis is to investigate the stability properties of θ-Milstein method and Runge-Kuttamethod, where stability is meant in the exponential asymptotic sense.In this thesis, we discuss linear stability and nonlinear stability properties of nu-merical methods for solving SDEs. In section2of chapter3, we consider a scalar linearIto stochastic diferential equation with multi-dimensional noise as test equation, andshow that explicit Milstein method correctly reproduces almost sure and small-momentexponential stability for sufciently small timesteps on this SDE. We also show thatwhen SDE obeys proper conditions, the semi-implicit Milstein method for this SDEis almost sure and small-moment exponential stable for sufciently small stepsizes. Insection3of chapter3, we choose linear Stratonovich SDE as test equation and prove that the p-th moment exponential stability is equivalent to the p-th moment asymp-totically stability for Runge-Kutta method. In chapter4, we show that for sufcientlysmall stepsizes, θ-Milstein method on nonlinear SDE is almost sure and small-momentexponential stable when the SDE obeys proper conditons. Now let us introduce themain results of this thesis.Linear stability analysisNow we consider the following scalar linear test SDE with multiplicative noisedriven by an m-dimensional standard Wiener process W (t)=(W1(t),···, Wm(t)), herecoefcients λ, μ1,···, μmare real numbers, and the initial value y0is non-random. Nu-merical methods in this work correspond to an equidistant discretization with stepsizeh. Furthermore, tn=nh, n∈{1,2,···} denotes the n-th step point. Applying numer-ical simulations to (1) produces approximations Yn=Y (tn)≈y(tn) with Y0=y(0).Applying θ-Milstein method to test equation (1) produceswhere θ∈[0,1] is a fxed parameter. θ=0,(2) reduces to explicit Milstein method,otherwise, it is semi-implicit Milstein method.Recalling the concept of A-stability in ordinary diferential equations, that is,problem stable implying numerical method stable for all stepsizes. Here we focus ona more fundamental property of the form—problem stable implies numerical methodstable for sufciently small stepsizes.We know the sample Lyapunov exponent of the solution to the SDE (1) is for any p>0. Then it is easy to get the following classical theorem.Theorem1. The zero solution of the SDE (1) is a.s. exponentially stable if and only if while it is pth moment exponentially stable if and only ifApplying θ-Milstein method on the test equation (1) produces numerical scheme (2). When θ=0(2) reduces to that is explicit Milstein method, now let us analyze its stability.Theorem2. If λ-1/2(?)μr2<0in (1), then there is a h*∈(0,1) such that for any h <h*, the Milstein approximation (3) has the property that Let p G (0,2]. Ifλ+(?)μr20,then there is a h*∈(0,1) such that for any h <h*, the Milstein approximation (3) has the property thatNext, we give the following theorem on stability of semi-implicit Milstein method.Theorem3. then there is a h*∈(0,1) such that for any h <h*, the semi-implicit Milstein approximation (2) has the property that Letp∈(0,2] then there is ah*∈(0,1) such that for any h<h*, the semi-implicit Milstein approximation (2) has the property that Then let's analyze the stability of Runge-Kutta method. Consider Stratonovich SDE where f(y(t)) is drift coefficient, g(y(t)) is diffusion coefficient, and W(t) is standard Wiener process. We are concerned with the following scalar linear test equation for stability analysis where a,b∈R. Applying Runge-Kutta method with order1.5constructed by Burrage to SDE (7) produces where J1=∫tntn+1odW, J10=∫tntn+1∫tnt odW(s)dt, A, B(1) and B(2) are s×s matrices α,γ(1) and γ(2) are row vectors∈Rs,I is unit matrix,(?) denotes Kronecker product, while Runge-Kutta method (9) for test equation (8) has the form Thus where Theorem4. Runge-Kutta method (9) on test equation (8) is p-th moment exponential stable if and only if E,(|Rn|p)<1, and thep-th moment exponential stability is equivalent to p-th moment asymptotic stability.Nonlinear stability analysisThroughout this thesis, we denote by (Ω,F,{Ft}t≥0,P) a complete probability space with a fiitration {Ft}t≥0satisfying the usual conditions, that is increasing and right continuous with F0containing all P-null sets. Let|·|be the Euclidean norm in Rd as well as Frobenius norm in Rd×m, and <x,y> be the inner product of x, y in Rd. Moreover, we use a∨b to denote max(a, b).We consider the following d-dimensional autonomous nonlinear Ito SDE where W(t)=(W/1(t),…,Wm(t))T is a m-dimensional Brownian motion. Similar to the deterministic cace, an SDE can always be written in autonomous form by the addition of another component representing time. Furthermore, we assume f: Rd→Rd and g: Rd→Rd×m are smooth enough so that the SDE (10) has a unique global solution y(t) on [0,∞).We begin with the linear growth assumption where K1>0is a constant. This implies The following result is about stability properties of the trivial solution to SDE (10).Theorem5. Let m=1and (11) hold. If then the solution of (10) obeys and given any ε∈(0, λ) there exists a p*E (0,1) such that for all0<p<p*We describe the Milstein scheme and the semi-implicit Milstein scheme as they applied to scalar noise SDE that is (10) with m=1. In the following expressions, fn,gn stand for f(Yn), g(Yn), respectively.1. d=1,m=1.(i) Explicit Milstein scheme(ii) Semi-implicit Milstein scheme Here θ∈(0,1] is a fixed parameter. The choice θ=1/2gives the trapezoidal Milstein Scheme and θ=1gives the backward Milstein Scheme2. d>1,m=1.(i)Explicit Milstein scheme where Jn=J(Yn) and J(y):Rd→Rd×d is the Jacobian matrix of g(y) with respect to y. The kth component of the Milstein scheme has the form(ii) Semi-implicit Milstein scheme the kth component of the semi-implicit Milstein scheme is given by Now let's analyze the almost sure and moment exponential stability of explicit Milstein Scheme as applied to scalar-noise stochastic differential equation. We assume that f(y) obeys a global linear bound of the form that is exactly a part of (11). Actually, Milstein method (13) can be obtained by adding to the Euler-Maruyama method the term1/2gng'n{(ΔWn)2-h}. Thus for investigating the stability property we should impose some hypotheses on g(y) and g'(y) for some K2>0. Accordingly, the inequality holds.Theorem6. Let (12),(17) and (18) hold. Then for any ε∈(0,λ) there is a constant h*∈(0,1) such that for any0<h<h*the Milstein scheme (13) satisfies Further, for any ε∈(0,λ) and any sufficiently small p>0, there is a constant h*∈(0,1) such that for any0<h<h*, the Milstein scheme (13) satisfiesFor the Milstein scheme (15), we assume that there is a constant K3>0such that where1≤i≤d and1≤j≤d. Then the following inequalities hold by letting gi(y)(i=1,..., d) differentiable and g(0)=0. Theorem7. Let (12),(17) and (19) hold. We also assume gi(y)(i=1,…,d) differ-entiable and g(0)=0, then for any ε∈(0,λ) there is a constant h*∈(0,1) such that for any0<h<h*the Milstein scheme (15) satisfies Further, for any ε∈(0,λ) and any sufficiently small p>0, there is a constant h*E (0,1) such that for any0<h<h*the Milstein scheme (15) satisfiesNext we study stability properties of the semi-implicit Milstein scheme for scalar-noise SDE.Theorem8. Let (17) and(18) hold. Given θ∈(0,1], assume that θK1+1/2Ï<0, where Then for any ε∈(0,|θK1+1/2Ï|) there is a pair of constants p∈(0,1) and h*∈(0,1) such that for any h<h*, the semi-implicit Milstein method (14) has the properties that andEspecially, when θ=1,(14) is backward Milstein method When analyzing stability of backward Milstein method, we can use a weaker condition instead of linear growth assumption, that is one-sides Lipschitz condition where μ∈R is a constant. Theorem9. Let (18) and(21) hold, and also assume f(0)=0. Assume that μ+1/2Ï<0, where Then for any e G (0,|μ+1/2Ï|) there is a pair of constants p∈(0,1) and h*∈(0,1) such that for any h <h*, the backward Milstein method (20) has the properties that andThe following theorem is about the semi-implicit Milstein method (16).Theorem10. Let (17) and (19) hold. We also assume gi{y)(i=1,…,d) differentiable and g(0)=0. Given θ∈(0,1], assume that θK1+1/2Ï<0, where Then for arbitrary ε∈(0,|θK1+1/2Ï|) there exist a pair of constants p∈(0,1) and h*∈(0,1) such that for any h<h*, the semi-implicit Milstein method (16) has the properties that andWe extend theorem9to the case d>1. At this time, backward Milstein method isTheorem11. Let (19) and (21) hold. We also assume gi(y)(i=1…,d) differen-tiable,f(0)=0and f(0)=0. Given θ∈(0,1], assume that μ+1/2Ï<0; where Then for arbitrary ε∈(0,μ+1/2Ï) there exist a pair of constants p∈(0,1) and h*∈(0,1) such that for any h<h*, the semi-implicit Milstein method (22) has the properties that andAn important special case of the multidimensional SDE is that of diagonal noise, where d=m, that is for each y∈Rd and j, k=1,…,m with j≠k. It is easy to see the kth component of the Milstein scheme for diagonal noise SDE has the formThe following theorem is theorem7.1in [58].Theorem12. Assume that for each integer i≥1there is a constant ki>0such that where g(y)=(g1(y)…,gm(y)). While there is a k>0such that If then the solution of (10) obeys and given any ε∈(0,λ) there exists a p*∈(0,1) such that for all0<p<p* Then, we give the following theorems on Milstein methods for SDE.Theorem13. Let (17) and (23) hold. We also assume gk,k (k=1,…, d) differentiable where L>0is a constant. Then for any ε∈(0,λ) there exits a constant h*E (0,1) such that for any0<h<h*the Milstein approximation satisfies Further, for anyε∈(0,λ) and any sufficient small p>0, there is a constant h*∈(0,1) such that for any0<h<h*the Milstein approximation satisfiesTheorem14. Let (17) and (24) hold. We also assume gk,k (k=1,…, d) differentiable and θK1+1/2Ï<0, where Then for any ε∈(0,|θ9K1+1/2Ï|) there is a pair of constants p E (0,1) and h*E (0,1) such that for any h<h*the semi-implicit Milstein method has the properties that and... |
