Stability theory for numerical methods for stochastic differential ...juengel/sde/Buckwar1.pdf · Maruyama-type methods: strong order 1 2 (add. noise: 1), weak order 1. Milstein-type

Stability theory for numerical methods forstochastic differential equations, Part I

Evelyn Buckwar

JKU

Institute for Stochastics

Summer School Vienna, 2nd September 2013

Evelyn Buckwar (JKU) Stability analysis for SODEs Vienna 2013 1 / 32

Outline

1 Introduction and motivation

2 Stability of numerical methods

3 Linear stability analysis for ODEs

4 Linear stability analysis for numerical methods for ODEs


Stochastic differential equations

dX (t)=F (t,X (t))dt +∑m

r=1 Gr (t,X (t))dWr (t), t ∈ [0,T ], X (0) = X0

coefficients: (globally Lipschitz) F : [0,T ]× Rd → Rd ,G = (G1, . . . ,Gm) : [0,T ]× Rd → Rd×m;

Wiener process:W = W (t, ω) = (W1(t, ω), . . . ,Wm(t, ω)), t ∈ [0,T ], ω ∈ Ω is anm-dim. Wiener process on (Ω,F , Ftt∈[0,T ],P).


Solutions of stochastic differential equations

dX (t)=aX (t)dt+bdW (t), BX (t)=eat(1 + b∫ t0 e−asdW (s))

dX (t)=aX (t)dt+bX (t)dW (t),BX (t)=exp((a− 1

2b2)t + bW (t))


θ-Methods

dX (t)=F (t,X (t))dt +∑m

r=1 Gr (t,X (t))dWr (t), t ∈ [0,T ], X (0) = X0

Notation: tn = n · h, n = 0, 1, . . ., Itn,tn+1r = Wr (tn+1)−Wr (tn) ∼

√hN (0, 1),

Itn,tn+1r1,r2 =

∫ tn+1

tn

∫ s

tndWr1(u)dWr2(s)

I θ-Maruyama-method:

Xn+1 = Xn + h (θF (tn+1,Xn+1) + (1− θ)F (tn,Xn)) +m∑r=1

Gr (tn,Xn) I tn,tn+1r

I θ-Milstein-method:

Xn+1 = Xn + h (θF (tn+1,Xn+1) + (1− θ)F (tn,Xn)) +m∑r=1

Gr (tn,Xn) I tn,tn+1r

+m∑

r1,r2=1(Gr1)′x Gr2(tn,X (tn)) I tn,tn+1

r1,r2

We will mostly be concerned with these two methods, as they are illustrativeexamples for various issues!


Other methods exist, e.g., Stochastic LinearMulti-step Methods

I tn,tn+1r = Wr (tn+1)−Wr (tn) ∼

√hN (0, 1), I tn,tn+1

r1,r2 =∫ tn+1

tn

∫ s

tndWr1(u)dWr2(s)

and tn = n · h, n = 0, 1, . . .,

I BDF2-Maruyama-method:Xn − 4

3Xn−1 + 13Xn−2 = h 2

3F (tn,Xn)

+m∑r=1

Gr (tn−1,Xn−1) Itn−1,tnr − 1

3

m∑r=1

Gr (tn−2,Xn−2) Itn−2,tn−1r

I Milstein variant of the two-step Adams-Bashforth-method:Xn − Xn−1 = h

(32F (tn−1,Xn−1)− 1

2F (tn−2,Xn−2))

+m∑r=1

Gr (tn−1,Xn−1) Itn−1,tnr +

m∑r1,r2=1

(Gr1)′x Gr2(tn−1,X (tn−1)) Itn−1,tnr1,r2

and, of course, many Runge-Kutta methods exist....


Two Objectives - Two Modes of Convergence


Two Objectives - Two Modes of Convergence

Strong Approximations: Compute (several to many) single paths, strongconvergence criterion (mean-square convergence), p order of method:

max1≤n≤N

(E|X (tn)− Xn|2

) 12 ≤ C hp, for h→ 0 .

Weak Approximations: Compute (using many paths) the expectation of afunction Ψ of the solution, weak convergence criterion, p order of method:

max1≤n≤N

|EΨ(X (tn))− EΨ(Xn)| ≤ C hp, for h→ 0 .

approx. E(Ψ(Xn)) by M realisations 1M

M∑i=1

Ψ(X(i)n )


A Graph showing mean-square order of convergenceof numerical methods fordX (t)=( 2

1+t X (t) + (1+t)2)dt + ε (1+t)2 dW1(t), t ∈ [0, 1], X (0) = 1


What does convergence say?

Maruyama-type methods: strong order 12 (add. noise: 1), weak order 1.

Milstein-type methods: strong order 1, weak order 1.

Convergence is a ’limit property’ !It says that the numerical solution approaches the exact solution,

when the step-size goes to 0.

However in practice on a computer this will not happen, we have tochoose a step-size and will try to avoid choosing a very, very, very tinyone, to keep computing time at a reasonable level!


Disambiguation: Stability

Numerical stability, Zero-stability, Dahlquist stability, Lax stability :robustness of a numerical scheme wrt perturbations such as round-offerror, ’measured’ over finite interval for step-size to zero, necessaryfor convergence!

Lyapunov stability: characterises qualitative behaviour of equilibriawrt perturbations in the i.v., fundamental problem ’does the(convergent) numerical method have the same stability behaviour asthe continuous problem and if under which conditions on thestep-size?’, ’measured’ for ’fixed step-size’ and t going to infinity.


An illustration of a numerically unstable, thus notconverging scheme

α=−1, β=0.01

−10

0

10

20

30

40

50

60

70

80

90

100

−2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6log(h)

unstable method

log(

||err

or||_

L2)

dX (t) = αX (t)dt + βX (t)dW1(t) using the numerically unstable scheme

Xn − 3Xn−1 + 2Xn−2 = hα( 12Xn−1 − 3

2Xn−2) + β(Xn−1Itn−1,tn1 − 2Xn−2I

tn−2,tn−1

1 )


In contrast, convergent schemes and a different’problem’


Some more detailed illustrations






Linear stability analysis for ODEs

I We will be concerned with Lyapunov stability henceforth.I (Linear) stability analysis of Numerical Methods for Ordinary

Differential Equations

dx(t)= f (t, x(t))dt, t ∈ [0,T ], x(0) = x0

is well established and described in every textbook on this topic, e.g.,– Solving Ordinary Differential Equations. Part I and II. E. Hairer,(S. P. Nørsett,) and G. Wanner, Springer.– Numerical Methods for Ordinary Differential Systems: The Initial ValueProblem. J. Lambert, John Wiley & Sons, 1991.– Dynamical Systems and Numerical Analysis. A. R. Humphries andA. M. Stuart, Cambridge University Press, 1996.



I Question: given an ODE x ′(t) = f (x(t)) and a numerical method, doesthe (convergent) method share the qualitative properties of the ODE andif so, under which restrictions on the step-size?

I (Usually) first step: linear stability analysis, using the test equationx ′(t) = λx(t), λ ∈ C. This means: apply the method to the test equation,determine its stability behaviour and compare with that of the testequation.

I Based on: linearisation and centering of nonlinear ODE around anequilibrium, the resulting linear system x′(t) = Ax(t) (A the Jacobian of fevaluated at equilibrium) is then diagonalised and the system thusdecoupled, justifying the use of the scalar test equation.



What does that mean in detail?


Linear stability analysis for ODEs, the set-up (1)(Stability. Philip Holmes and Eric T. Shea-Brown (2006), Scholarpedia,

1(10):1838.)

Consider autonomous (ODEs)

dx(t)

dt= x ′(t) = f (x(t)) , (1)

where x ∈ Rd and f = (f1, . . . , fd) : Rd → Rd , we denote a solution to (1)by x(t), with initial conditions x(0).Equilibria xe (sometimes called equilibrium points, fixed points orstationary points), are special constant solutions

x(t) ≡ xe wheredxe

dt= f (xe) = 0 , (2)

which is equivalent to requiring fj(xe1 , . . . , x

ed ) = 0 for all 1 ≤ j ≤ d .


Linear stability analysis for ODEs, the set-up (2)Lyapunov stabilityxe is a stable equilibrium if for every neighbourhood U of xe there is aneighbourhood V ⊆ U of xe such that every solution x(t) starting inV (x(0) ∈ V ) remains in U for all t ≥ 0. Notice that x(t) need not approach xe .If xe is not stable, it is unstable.Asymptotic stabilityAn equilibrium xe is asymptotically stable if it is Lyapunov stable and additionallyV can be chosen so that |x(t)− xe | → 0 as t →∞ for all x(0) ∈ V . (Anequilibrium that is Lyapunov stable but not asymptotically stable is sometimescalled neutrally stable.)Exponential stability

An equilibrium xe is exponentially stable if there is a neighbourhood V of xe and

a constant a > 0 such that |x(t)− xe | < e−at as t →∞ for all x(0) ∈ V .

Exponentially stable equilibria are also asymptotically stable, and hence Lyapunov

stable.

For scalar, linear x ′(t) = λx(t) with solution x(t) = x0 eλt it is obvious that the zero

solution (starting with x0 = 0) is asymptotically stable iff λ < 0 (or R(λ) < 0 forcomplex ODEs).


Linear stability analysis for ODEs, the set-up (3)LinearisationSuppose that x = xe is an equilibrium, so that if x(0) = xe , then x(t) ≡ xe . Letx(t) = xe + p(t), where p(t) = (p1, . . . , pd) is a small perturbation |p(t)| 1.Substitute into (1) and expand f in a multi-variable, vector-valued Taylor series(assuming f is smooth enough) to obtain:

dxe

dt+

dp(t)

dt= f (xe + p(t)) = f (xe) + Df (xe)p(t) + O(|p(t)|2). (3)

Here, Df (xe) denotes the d × d Jacobian matrix of partial derivatives

(∂fi∂xj

),

evaluated at the equilibrium xe , and O(|p(t)|2) denotes terms of quadratic andhigher order in the components p1, . . . , pd . In particular, for g(p) = O(|p(t)|k)

then lim|p(t)|→0g(p(t))|p(t)|k ≤ M <∞. Thus, for small enough |p(t)|, the first order

term Df (xe) dominates. Taking into account that dxedt and f (xe) vanish and

ignoring the small term O(|p(t)|2, we get the linear system:

dp(t)

dt= Df (xe)p(t) . (4)


Linear stability analysis for ODEs, the set-up (4)The general solution p(t) of Eqn. (4) is determined by the eigenvalues and eigenvectorsof the Jacobian matrix Df (xe). Note: in studying stability we only want to knowwhether the size of solutions grows, stays constant, or shrinks as t →∞.Recall that, if λ is a real eigenvalue with eigenvector ~v , then there is a solution to thelinearised equation of the form p(t) = c~veλt . If λ = α± iβ is a complex conjugate pairwith eigenvectors ~v = ~u ± i ~w (where ~u, ~w are real) then

p1(t) = eαt(~u cosβt − ~w sinβt)

andp2(t) = eαt(~u sinβt + ~w cosβt)

are two linearly-independent solutions. In both cases the real part of λ (almost)determines stability. Since any solution of the linearised equation can be written as alinear superposition of terms of these forms (except in the case of multiple eigenvalues),we can deduce that

If all eigenvalues of Df (xe) have strictly negative real parts, then |p(t)| → 0 ast →∞ for all solutions.

If at least one eigenvalue of Df (xe) has a positive real part, then there is asolution p(t) with |p(t)| → +∞ as t →∞.

If some pairs of complex-conjugate eigenvalues have zero real parts with distinctimaginary parts, then the corresponding solutions oscillate and neither decay norgrow as t →∞.


Linear stability analysis for ODEs, the set-up (5)

Most important point:Definition: xe is a hyperbolic or non-degenerate equilibrium of x ′(t) = f (x(t)) ifall the eigenvalues of Df (xe) have non-zero real parts.

Equipped with the linear analysis sketched above, and recognising that theremainder terms ignored in passing from Eqn. (3) to (4) can be made as small aswe wish by selecting a sufficiently small neighborhood of xe , we can determinethe stability of hyperbolic equilibria of x ′(t) = f (x(t)) from their linearisation:

Proposition: If xe is an equilibrium of x ′(t) = f (x(t)) and all the eigenvalues ofthe Jacobian matrix Df (xe) have strictly negative real parts, then xe isexponentially (and hence asymptotically) stable. If at least one eigenvalue hasstrictly positive real part, then xe is unstable.

This is essentially the contents of the Hartman-Grobman Theorem in DynamicalSystems Theory, or can be shown in the context of Lyapunov Stability Theoryusing converse Lyapunov theorems.


Linearised stability for difference equations

Analogous results exist for stability of equilibria of difference equations ofthe form

xn+1 = f (xn) , (5)

but here the magnitude rather than the sign of the eigenvalues isimportant. A solution xe is called a fixed point if xe = f (xe).

Definition: xe is a hyperbolic or non-degenerate fixed point of thedifference equation xn+1 = f (xn) if no eigenvalue of Df (xe) hasmagnitude 1.

Proposition: If xe is a fixed point of the difference equation xn+1 = f (xn)and all the eigenvalues of Df (xe) have magnitudes strictly strictly lessthan 1, then xe is asymptotically stable. If at least one eigenvalue hasmagnitude greater than 1, then xe is unstable.


Linear stability analysis for numerical methods forODEs

The procedure for the linear stability analysis for numerical methods for ODEs thenconsists of the following steps:

Obtain the linearisation of the ODE (1) around an equilibrium, to get the linear

system (4), i.e.,dp(t)dt

= Df (xe)p(t) with equilibrium being the zero solution.

If possible, multiply this system with a suitable matrix to obtain a systemx′(t) = Dx(t) with a diagonal matrix D. This effectively decouples the ODEsystem into scalar, usually complex ODEs x ′(t) = λx(t), λ ∈ C, the so-called ’testequation’. This simplifies the computations considerably, but may not alwaysfeasable.

Apply your favorite numerical method to the linear system or the ’test equation’and determine the stability properties of the resulting difference equation.

Due to the just described background on ’linearised stability’ the results obtained for thelinear system or the scalar ’test equation’ via this procedure yield meaningful informationabout the behaviour of the numerical method applied to the nonlinear ’original’ ODE!


Example:

Consider the test equation x ′(t) = λx(t), λ ∈ C and its discretisation by theexplicit (forward) and implicit (backward) Euler methods (corresponds to θ = 0and θ = 1 in the θ-method xn+1 = xn + h (θf (tn+1, xn+1) + (1− θ)f (tn, xn)) forthe general ODE):Explicit Euler:

xn+1 = xn + h λxn , or xn+1 = (1 + h λ)n+1x0 ,

The zero solution is asymptotically stable, if |1 + hλ| < 1.Implicit Euler:

xn+1 = xn + h λxn+1 , or xn+1 = (1

1− h λ)n+1x0 ,

The zero solution is asymptotically stable, if | 11− h λ | < 1 or |1− h λ| > 1.


Example:

The Stability RegionDefinition: The region of (absolute) stability of a numerical method for aninitial value problem is the set of (complex) values λ h for which the zerosolution of the difference equation resulting from applying the numericalmethod to the test equation is asymptotically stable.The corresponding region of stability for the test equationx ′(t) = λx(t), λ ∈ C is the set of (complex) values λ for which the zerosolution of the test equation is asymptotically stable, i.e, the left half-planeof the complex plane.


Example:


A-stability

The concept of A-stability describes the desirable property of a numericalmethod to have (at least) the same linear stability property as the testequation itself, often it is phrased as ’the stability region of the numericalmethod includes that of the test equation’.

Obviously, the forward Euler method is not A-stable, the backward Eulermethod is A-stable.

Caution: The backward Euler method can produce ’stable solutions’ evenwhen the zero solution of the test equation is unstable, it is ’overdamping’


Further directions

The already mentioned books, e.g., Dynamical Systems and NumericalAnalysis. A. R. Humphries and A. M. Stuart, Cambridge University Press,1996, present a large body of results that go beyond Linear StabilityAnalysis to find comparisons between the dynamical behaviour of theODE and its numerical approximation. Topics include:

Investigations of non-autonomous ODEs and their numericalcounterparts.

Investigation of dissipative systems.

Convergence of Invariant Sets.

. . .


Summary

I The property of (Lyapunov) stability of a method is one of the mostimportant factors determining its performance when applied to differentclasses of problems in practice.I We have set up the framework for the concept ’linear stability analysis’

of numerical methods for ODEs.I In particular, we have provided a justification for using linear systems or

linear, scalar test equations to determine stability properties of thenumerical method.I It should be clear that a linear stability analysis in the stochastic setting

requires an analogous framework.I This is the topic of the next two lecures.


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Stability theory for numerical methods for stochastic differential ...juengel/sde/Buckwar1.pdf · Maruyama-type methods: strong order 1 2 (add. noise: 1), weak order 1. Milstein-type