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Lyapunov stability theory for ODEsStability of SDEs
Stability of Stochastic Differential EquationsPart 1: Introduction
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
In 1892, A.M. Lyapunov introduced the concept of stabilityof a dynamic system.Roughly speaking, the stability means insensitivity of thestate of the system to small changes in the initial state orthe parameters of the system.For a stable system, the trajectories which are “close" toeach other at a specific instant should therefore remainclose to each other at all subsequent instants.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
In 1892, A.M. Lyapunov introduced the concept of stabilityof a dynamic system.Roughly speaking, the stability means insensitivity of thestate of the system to small changes in the initial state orthe parameters of the system.For a stable system, the trajectories which are “close" toeach other at a specific instant should therefore remainclose to each other at all subsequent instants.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
In 1892, A.M. Lyapunov introduced the concept of stabilityof a dynamic system.Roughly speaking, the stability means insensitivity of thestate of the system to small changes in the initial state orthe parameters of the system.For a stable system, the trajectories which are “close" toeach other at a specific instant should therefore remainclose to each other at all subsequent instants.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Consider a d-dimensional ordinary differential equation (ODE)
dx(t)dt
= f (x(t), t) on t ≥ 0,
where f = (f1, · · · , fd)T : Rd × R+ → Rd . Assume that for everyinitial value x(0) = x0 ∈ Rd , there exists a unique globalsolution which is denoted by x(t ; x0). Assume furthermore that
f (0, t) = 0 for all t ≥ 0.
So the ODE has the solution x(t) ≡ 0 corresponding to theinitial value x(0) = 0. This solution is called the trivial solutionor equilibrium position.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
DefinitionThe trivial solution is said to be stable if, for every ε > 0, thereexists a δ = δ(ε) > 0 such that
|x(t ; x0)| < ε for all t ≥ 0.
whenever |x0| < δ. Otherwise, it is said to be unstable.
The trivial solution is said to be asymptotically stable if it isstable and if there moreover exists a δ0 > 0 such that
limt→∞
x(t ; x0) = 0
whenever |x0| < δ0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
If the ODE can be solved explicitly, it would be rather easyto determine whether the trivial solution is stable or not.However, the ODE can only be solved explicitly in somespecial cases.Fortunately, Lyapunov in 1892 developed a method fordetermining stability without solving the equation. Thismethod is now known as the method of Lyapunov functionsor the Lyapunov method.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
If the ODE can be solved explicitly, it would be rather easyto determine whether the trivial solution is stable or not.However, the ODE can only be solved explicitly in somespecial cases.Fortunately, Lyapunov in 1892 developed a method fordetermining stability without solving the equation. Thismethod is now known as the method of Lyapunov functionsor the Lyapunov method.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Notation
Let K denote the family of all continuous nondecreasingfunctions µ : R+ → R+ such that µ(0) = 0 and µ(r) > 0 if r > 0.
Let K∞ denote the family of all functions µ ∈ K such thatlimr→∞ µ(r) =∞.
For h > 0, let Sh = x ∈ Rd : |x | < h.
Let C1,1(Sh × R+; R+) denote the family of all continuousfunctions V (x , t) from Sh×R+ to R+ with continuous first partialderivatives with respect to every component of x and to t .
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Basic ideas of the Lyapunov method
Let x(t) be a solution of the ODE and V ∈ C1,1(Sh × R+; R+).Then v(t) = V (x(t), t) represents a function of t with thederivative
dv(t)dt
= V (x(t), t),
where V (x , t) = Vt(x , t) + (Vx1(x , t), · · · ,Vxd (x , t))f (x , t).
If dv(t)/dt ≤ 0, then v(t) will not increase so the “distance”of x(t) from the equilibrium point measured by V (x(t), t)does not increase.If dv(t)/dt < 0, then v(t) will decrease to zero so thedistance will decrease to zero, that is x(t)→ 0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Basic ideas of the Lyapunov method
Let x(t) be a solution of the ODE and V ∈ C1,1(Sh × R+; R+).Then v(t) = V (x(t), t) represents a function of t with thederivative
dv(t)dt
= V (x(t), t),
where V (x , t) = Vt(x , t) + (Vx1(x , t), · · · ,Vxd (x , t))f (x , t).
If dv(t)/dt ≤ 0, then v(t) will not increase so the “distance”of x(t) from the equilibrium point measured by V (x(t), t)does not increase.If dv(t)/dt < 0, then v(t) will decrease to zero so thedistance will decrease to zero, that is x(t)→ 0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Theorem
Assume that there exist V ∈ C1,1(Sh ×R+; R+) and µ ∈ K suchthat
V (0, t) = 0, µ(|x |) ≤ V (x , t)
andV (x , t) ≤ 0
for all (x , t) ∈ Sh × R+. Then the trivial solution of the ODE isstable.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
Concept of stabilityThe Lyapunov method
Theorem
Assume that there exist V ∈ C1,1(Sh × R+; R+) andµ1, µ2, µ3 ∈ K such that
µ1(|x |) ≤ V (x , t) ≤ µ2(|x |)
andV (x , t) ≤ −µ3(|x |)
for all (x , t) ∈ Sh × R+. Then the trivial solution of the ODE isasymptotically stable.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
Consider a d-dimensional stochastic differential equation (SDE)
dx(t) = f (x(t), t)dt + g(x(t), t)dB(t) on t ≥ 0,
where f : Rd × R+ → Rd and g : Rd × R+ → Rd×m, andB(t) = (B1(t), · · · ,Bm(t))T is an m-dimensional Brownianmotion.As a standing hypothesis in this course, we assume that both fand g obey the local Lipschitz condition and the linear growthcondition.Hence, for any given initial value x(0) = x0 ∈ Rd , the SDE hasa unique global solution denoted by x(t ; x0). Assumefurthermore that
f (0, t) = 0 and g(0, t) = 0 for all t ≥ 0.
Hence the SDE admits the trivial solution x(t ; 0) ≡ 0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
When we try to carry over the principles of the Lyapunovstability theory to to SDEs, we face the following problems:
What is a suitable definition of stochastic stability?With what should the derivative dv(t)/dt or V (x , t) bereplaced?What conditions should a stochastic Lyapunov functionsatisfy?
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
When we try to carry over the principles of the Lyapunovstability theory to to SDEs, we face the following problems:
What is a suitable definition of stochastic stability?With what should the derivative dv(t)/dt or V (x , t) bereplaced?What conditions should a stochastic Lyapunov functionsatisfy?
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
When we try to carry over the principles of the Lyapunovstability theory to to SDEs, we face the following problems:
What is a suitable definition of stochastic stability?With what should the derivative dv(t)/dt or V (x , t) bereplaced?What conditions should a stochastic Lyapunov functionsatisfy?
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
It turns out that there are various different types of stochasticstability. In this course, we will only concentrate on
stability in probability;pth moment exponential stability;almost sure exponential stability.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
It turns out that there are various different types of stochasticstability. In this course, we will only concentrate on
stability in probability;pth moment exponential stability;almost sure exponential stability.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
It turns out that there are various different types of stochasticstability. In this course, we will only concentrate on
stability in probability;pth moment exponential stability;almost sure exponential stability.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
DefinitionThe trivial solution of the SDE is said to be stochastically stableor stable in probability if for every pair of ε ∈ (0,1) and r > 0,there exists a δ = δ(ε, r) > 0 such that
P|x(t ; x0)| < r for all t ≥ 0 ≥ 1− ε
whenever |x0| < δ. Otherwise, it is said to be stochasticallyunstable.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
DefinitionThe trivial solution is said to be stochastically asymptoticallystable if it is stochastically stable and, moreover, for everyε ∈ (0,1), there exists a δ0 = δ0(ε) > 0 such that
P limt→∞
x(t ; x0) = 0 ≥ 1− ε
whenever |x0| < δ0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
DefinitionThe trivial solution is said to be stochastically asymptoticallystable in the large if it is stochastically stable and, moreover, forall x0 ∈ Rd
P limt→∞
x(t ; x0) = 0 = 1.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
DefinitionThe trivial solution is said to be almost surely exponentiallystable if for all x0 ∈ Rd
lim supt→∞
1t
log(|x(t ; x0)|) < 0 a.s.
It is said to be pth moment exponentially stable if for all x0 ∈ Rd
lim supt→∞
1t
log(E |x(t ; x0)|p) < 0.
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
Outline
1 Lyapunov stability theory for ODEsConcept of stabilityThe Lyapunov method
2 Stability of SDEsSDEsDefinition of stochastic stabilityDiffusion operator
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
To figure out with what the derivative dv(t)/dt or V (x , t) shouldbe replaced, we naturally consider the Itô differential of theprocess V (x(t), t), where x(t) is a solution of the SDE and V isa Lyapunov function.
According to the Itô formula, we of course requireV ∈ C2,1(Sh × R+; R+), which denotes the family of allnonnegative functions V (x , t) defined on Sh × R+ such thatthey are continuously twice differentiable in x and once in t .
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
By the Itô formula, we have
dV (x(t), t) = LV (x(t), t)dt + Vx(x(t), t)g(x(t), t)dB(t),
where
LV (x , t) = Vt(x , t)+Vx(x , t)f (x , t)+12
trace[gT (x , t)Vxx(x , t)g(x , t)
],
in which Vx = (Vx1 , · · · ,Vxd ) and Vxx = (Vxi xj )d×d .
Xuerong Mao FRSE Stability of SDE
Lyapunov stability theory for ODEsStability of SDEs
SDEsDefinition of stochastic stabilityDiffusion operator
We shall see that V (x , t) will be replaced by the diffusionoperator LV (x , t) in the study of stochastic stability. Forexample, the inequality V (x , t) ≤ 0 will sometimes be replacedby LV (x , t) ≤ 0 to get the stochastic stability. However, it is notnecessary to require LV (x , t) ≤ 0 to get other stabilities e.g.almost sure exponential stability.
The study of stochastic stability is therefore much richer thanthe classical stability of ODEs. Let us begin to explore thiswonderful world of stochastic stability.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stability of Stochastic Differential EquationsPart 2: Stability in Probability
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
In this part, we shall see how the classical Lyapunov method isdeveloped to study stochastic stability in such a similar way thatthe results in this part are natural generalizations of theLyapunov stability theory for ODEs. Of course, such resultsmay not be surprising but we will see some surprising results inthe next part.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Theorem
Assume that there exist V ∈ C2,1(Sh ×R+; R+) and µ ∈ K suchthat
V (0, t) = 0, µ(|x |) ≤ V (x , t)
andLV (x , t) ≤ 0
for all (x , t) ∈ Sh × R+. Then the trivial solution of the SDE isstochastically stable.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Proof. Let ε ∈ (0,1) and r ∈ (0,h) be arbitrary. Clearly, we canfind a δ = δ(ε, r) ∈ (0, r) such that
1ε
supx∈Sδ
V (x ,0) ≤ µ(r).
Now fix any x0 ∈ Sδ and write x(t ; x0) = x(t) simply. Define
τ = inft ≥ 0 : x(t) 6∈ Sr.
(Throughout this course we set inf ∅ =∞.) By Itô’s formula, forany t ≥ 0,
V (x(τ ∧ t), τ ∧ t) = V (x0,0) +
∫ τ∧t
0LV (x(s), s)ds
+
∫ τ∧t
0Vx(x(s), s)g(x(s), s)dB(s).
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Taking the expectation on both sides, we obtain
EV (x(τ ∧ t), τ ∧ t) = V (x0,0)+E∫ τ∧t
0LV (x(s), s)ds ≤ V (x0,0).
Noting that |x(τ ∧ t)| = |x(τ)| = r if τ ≤ t , we get
EV (x(τ ∧ t), τ ∧ t) ≥ E[Iτ≤tV (x(τ), τ)
]≥ µ(r)Pτ ≤ t.
(Throughout this course IA denotes the indicator function of setA.) We therefore obtain Pτ ≤ t ≤ ε. Letting t →∞ we getPτ <∞ ≤ ε, that is
P|x(t)| < r for all t ≥ 0 ≥ 1− ε
as required.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Theorem
Assume that there exist V ∈ C2,1(Sh × R+; R+) andµ1, µ2, µ3 ∈ K such that
µ1(|x |) ≤ V (x , t) ≤ µ2(|x |)
andLV (x , t) ≤ −µ3(|x |)
for all (x , t) ∈ Sh × R+. Then the trivial solution of the SDE isstochastically asymptotically stable.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Proof. We know from the previous theorem that the trivialsolution is stochastically stable. So we only need to show thatfor any ε ∈ (0,1), there is a δ0 = δ0(ε) > 0 such that
P limt→∞
x(t ; x0) = 0 ≥ 1− ε
whenever |x0| < δ0, or for any β ∈ (0,h/2),
Plim supt→∞
|x(t ; x0)| ≤ β ≥ 1− ε.
By the previous theorem, we can find a δ0 = δ0(ε) ∈ (0,h/2)such that
P|x(t ; x0)| < h/2 ≥ 1− ε
2. (1.1)
whenever x0 ∈ Sδ0 .
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Moreover, in the same way as the previous theorem wasproved, we can show that for any β ∈ (0,h/2), there is aα ∈ (0, β) such that
P|x(t ; x0)| < β for all t ≥ s ≥ 1− ε
2(1.2)
whenever |x(s; x0)| ≤ α and s ≥ 0. Now fix any x0 ∈ Sδ andwrite x(t ; x0) = x(t) simply. Define
τα = inft ≥ 0 : |x(t)| ≤ α
andτh = inft ≥ 0 : |x(t)| ≥ h/2.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
By Itô’s formula and the conditions, we can show that
0 ≤ V (x0,0) + E∫ τα∧τh∧t
0LV (x(s), s)ds
≤ V (x0,0)− µ3(α)E(τα ∧ τh ∧ t).
Consequently
tµ3(α)Pτα ∧ τh ≥ t ≤ E(τα ∧ τh ∧ t) ≤ V (x0,0).
This implies immediately that
Pτα ∧ τh <∞ = 1.
But, by (1.1), Pτh <∞ ≤ ε/2. Hence
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
1 = Pτα∧τh <∞ ≤ Pτα <∞+Pτh <∞ ≤ Pτα <∞+ε
2,
which yieldsPτα <∞ ≥ 1− ε
2.
We now compute, using (1.2),
Plim supt→∞
|x(t)| ≤ β
≥ Pτα <∞ and |x(t)| ≤ β for all t ≥ τα= Pτα <∞P|x(t)| ≤ β for all t ≥ τα |τα <∞ ≥ Pτα <∞(1− ε/2) ≥ (1− ε/2)2 ≥ 1− ε
as required.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Theorem
Assume that there exist V ∈ C2,1(Rd × R+; R+) andµ1, µ2 ∈ K∞ and µ3 ∈ K such that
µ1(|x |) ≤ V (x , t) ≤ µ2(|x |)
andLV (x , t) ≤ −µ3(|x |)
for all (x , t) ∈ Rd × R+. Then the trivial solution of the SDE isstochastically asymptotically stable in the large.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
Proof. Clearly, we only need to show
P limt→∞
x(t ; x0) = 0 = 1
for all x0 ∈ Rd , or for any pair of ε ∈ (0,1) and β > 0,
Plim supt→∞
|x(t ; x0)| ≤ β ≥ 1− ε.
To show this, let us fix any x0 and write x(t ; x0) = x(t) again.Let h sufficiently large for h/2 > |x0| and
µ1(h/2) ≥ 2V (x0,0)
ε.
As in the previous proof, define the stopping time
τh = inft ≥ 0 : |x(t)| ≥ h/2.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
By Itô’s formula, we can show that for any t ≥ 0,
EV (x(τh ∧ t), τh ∧ t) ≤ V (x0,0).
ButEV (x(τh ∧ t), τh ∧ t) ≥ µ1(h/2)Pτh ≤ t.
Hence Pτh ≤ t ≤ ε2 . Letting t →∞ gives Pτh <∞ ≤ ε/2.
That isP|x(t)| < h/2 for all t ≥ 0 ≥ 1− ε
2,
which is the same as (1.1). From here, we can show in thesame way as in the previous proof that
Plim supt→∞
|x(t)| ≤ β ≥ 1− ε
as desired.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Consider a scale SDE
dx(t) = f (x(t), t)dt + g(x(t), t)dB(t) on t ≥ 0
with initial value x(0) = x0 ∈ R. Assume that f : R × R+ → Rand g : R × R+ → Rm have the expansions
f (x , t) = a(t)x+o(|x |), g(x , t) = (b1(t)x , · · · ,bm(t)x)T +o(|x |).
in a neighbourhood of x = 0 uniformly with respect to t ≥ 0,where a(t), bi(t) are all bounded Borel-measurable real-valuedfunctions. We impose a condition that there is a pair of positiveconstants θ and K such that
−K ≤∫ t
0
(a(s)− 1
2
m∑i=1
b2i (s) + θ
)ds ≤ K for all t ≥ 0.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Let0 < ε <
θ
supt≥0∑m
i=1 b2i (t)
and define the Lyapunov function
V (x , t) = |x |ε exp[− ε
∫ t
0
(a(s)− 1
2
m∑i=1
b2i (s) + θ
)ds].
Then, by the condition,
|x |εe−εK ≤ V (x , t) ≤ |x |εeεK .
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Moreover, compute
LV (x , t) = ε|x |ε exp[−ε∫ t
0
(a(s)− 1
2
m∑i=1
b2i (s) + θ
)ds]
×( ε
2
m∑i=1
b2i (t)− θ
)+ o(|x |ε)
≤ −12εθe−εK |x |ε + o(|x |ε).
We hence see that LV (x , t) is negative-definite in a sufficientlysmall neighbourhood of x = 0 for t ≥ 0. We can thereforeconclude that the trivial solution of the scale SDE isstochastically asymptotically stable.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Outline
1 TheoryStochastic stabilityStochastic asymptotic stabilityStochastic asymptotic stability in the large
2 ExamplesScale SDEsMulti-dimensional SDEs
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Assume that the coefficients f and g of the underlying SDEhave the expansions
f (x , t) = F (t)x+o(|x |), g(x , t) = (G1(t)x , · · · ,Gm(t)x)+o(|x |)
in a neighbourhood of x = 0 uniformly with respect to t ≥ 0,where F (t), Gi(t) are all bounded Borel-measurabled × d-matrix-valued functions. Assume that there is asymmetric positive-definite matrix Q such that
λmax
(QF (t) + F T (t)Q +
m∑i=1
GTi (t)QGi(t)
)≤ −λ < 0
for all t ≥ 0, where (and throughout this course) λmax(A)denotes the largest eigenvalue of matrix A.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Scale SDEsMulti-dimensional SDEs
Now, define the Lyapunov function V (x , t) = xT Qx . Clearly,
λmin(Q)|x |2 ≤ V (x , t) ≤ λmax(Q)|x |2.
Moreover,
LV (x , t) = xT(
QF (t) + F T (t)Q +m∑
i=1
GTi (t)QGi(t)
)x + o(|x |2)
≤ −λ|x |2 + o(|x |2).
Hence LV (x , t) is negative-definite in a sufficiently smallneighbourhood of x = 0 for t ≥ 0. We therefore conclude thatthe trivial solution of the SDE is stochastically asymptoticallystable.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Stability of Stochastic Differential EquationsPart 3: Almost Sure Exponential Stability
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
In this part, we shall develop the classical Lyapunov method tostudy the almost sure exponential stability. In contrast to theclassical Lyapunov stability theory, we will no longer requireLV (x , t) be negative-definite but we still obtain the almost sureexponential stability making full use of the diffusion (noise)terms. It is this new feature that makes the stochastic stabilitymore interesting and more useful as well.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
To establish the theory on the almost sure exponential stability,we need prepare an important lemma. Recall that we assume,throughout this course, that both coefficients f and g obey thelocal Lipschitz condition and the linear growth condition and,moreover, f (0, t) ≡ 0, g(0, t) ≡ 0. Under these standinghypotheses, we have the following useful lemma.
Lemma
For all x0 6= 0 in Rd
Px(t ; x0) 6= 0 for all t ≥ 0 = 1.
That is, almost all the sample path of any solution starting froma non-zero state will never reach the origin with probability 1.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Proof. If the lemma were false, there would exist some x0 6= 0such that Pτ <∞ > 0, where
τ = inft ≥ 0 : x(t) = 0
in which we write x(t ; x0) = x(t) simply. So we can find a pair ofconstants T > 0 and θ > 1 sufficiently large for P(B) > 0,where
B = τ ≤ T and |x(t)| ≤ θ − 1 for all 0 ≤ t ≤ τ.
But, by the standing hypotheses, there exists a positiveconstant Kθ such that
|f (x , t)| ∨ |g(x , t)| ≤ Kθ|x | for all |x | ≤ θ, 0 ≤ t ≤ T .
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Let V (x , t) = |x |−1. Then, for 0 < |x | ≤ θ and 0 ≤ t ≤ T ,
LV (x , t) = −|x |−3xT f (x , t)
+12
(−|x |−3|g(x , t)|2 + 3|x |−5|xT g(x , t)|2
)≤ |x |−2|f (x , t)|+ |x |−3|g(x , t)|2
≤ Kθ|x |−1 + K 2θ |x |−1
= Kθ(1 + Kθ)V (x , t).
Now, for any ε ∈ (0, |x0|), define the stopping time
τε = inft ≥ 0 : |x(t)| 6∈ (ε, θ).
By Itô’s formula,
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
E[e−Kθ(1+Kθ)(τε∧T )V (x(τε ∧ T ), τε ∧ T )
]− V (x0,0)
= E∫ τε∧T
0e−Kθ(1+Kθ)s
[−(Kθ(1 + Kθ))V (x(s), s) + LV (x(s), s)
]ds
≤ 0.
Note that for ω ∈ B, τε ≤ T and |x(τε)| = ε. The aboveinequality therefore implies that
E[e−Kθ(1+Kθ)T ε−1IB
]≤ |x0|−1.
Hence P(B) ≤ ε|x0|−1eKθ(1+Kθ)T . Letting ε→ 0 yields thatP(B) = 0, but this contradicts the definition of B. The proof iscomplete.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
We will also need the well-known exponential martingaleinequality which we state here as a lemma.
Lemma
Let g = (g1, · · · ,gm) ∈ L2(R+; R1×m), and let T , α, β be anypositive numbers. Then
P
sup0≤t≤T
[∫ t
0g(s)dB(s)− α
2
∫ t
0|g(s)|2ds
]> β
≤ e−αβ.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Theorem
Assume that there exists a function V ∈ C2,1(Rd × R+; R+),and constants p > 0, c1 > 0, c2 ∈ R, c3 ≥ 0, such that for allx 6= 0 and t ≥ 0,
c1|x |p ≤ V (x , t),LV (x , t) ≤ c2V (x , t),
|Vx (x , t)g(x , t)|2 ≥ c3V 2(x , t).
Thenlim sup
t→∞
1t
log |x(t ; x0)| ≤ −c3 − 2c2
2pa.s. (1.1)
for all x0 ∈ Rd . In particular, if c3 > 2c2, then the trivial solutionof the SDE is almost surely exponentially stable.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Proof. Clearly, the assertion holds for x0 = 0 since x(t ; 0) ≡ 0.Fix any x0 6= 0 and write x(t ; x0) = x(t). By the lemma, x(t) 6= 0for all t ≥ 0 almost surely. Thus, one can apply Itô’s formulaand the condition to show that, for t ≥ 0,
log V (x(t), t) ≤ log V (x0,0) + c2t + M(t)
− 12
∫ t
0
|Vx (x(s), s)g(x(s), s)|2
V 2(x(s), s)ds, (1.2)
where
M(t) =
∫ t
0
Vx (x(s), s)g(x(s), s)
V (x(s), s)dB(s)
is a continuous martingale with initial value M(0) = 0. Assignε ∈ (0,1) arbitrarily and let n = 1,2, · · · . By the exponentialmartingale inequality,
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
P
sup0≤t≤n
[M(t)− ε
2
∫ t
0
|Vx (x(s), s)g(x(s), s)|2
V 2(x(s), s)ds]>
2ε
log n≤ 1
n2 .
Applying the Borel–Cantelli lemma we see that for almost allω ∈ Ω, there is an integer n0 = n0(ω) such that if n ≥ n0,
M(t) ≤ 2ε
log n +ε
2
∫ t
0
|Vx (x(s), s)g(x(s), s)|2
V 2(x(s), s)ds
holds for all 0 ≤ t ≤ n. Substituting this into (1.2) and thenusing the condition we obtain that
log V (x(t), t) ≤ log V (x0,0)− 12
[(1− ε)c3 − 2c2]t +2ε
log n
for all 0 ≤ t ≤ n, n ≥ n0 almost surely.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Consequently, for almost all ω ∈ Ω, if n − 1 ≤ t ≤ n and n ≥ n0,
1t
log V (x(t), t) ≤ −12
[(1− ε)c3 − 2c2] +log V (x0,0) + 2
ε log nn − 1
.
This implies
lim supt→∞
1t
log V (x(t), t) ≤ −12
[(1− ε)c3 − 2c2] a.s.
Hence
lim supt→∞
1t
log |x(t)| ≤ −(1− ε)c3 − 2c2
2pa.s.
and the required assertion follows since ε > 0 is arbitrary.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Theorem
Assume that there exists a function V ∈ C2,1(Rd × R+; R+),and constants p > 0, c1 > 0, c2 ∈ R, c3 > 0, such that for allx 6= 0 and t ≥ 0,
c1|x |p ≥ V (x , t) > 0,LV (x , t) ≥ c2V (x , t),
|Vx (x , t)g(x , t)|2 ≤ c3V 2(x , t).
Thenlim inft→∞
1t
log |x(t ; x0)| ≥ 2c2 − c3
2pa.s.
for all x0 6= 0 in Rd . In particular, if 2c2 > c3, then almost all thesample paths of |x(t ; x0)| will tend to infinity exponentially.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
Proof. Fix any x0 6= 0 and write x(t ; x0) = x(t). By Itô’s formulaand the conditions, we can easily show that for t ≥ 0,
log V (x(t), t) ≥ log V (x0,0) +12
(2c2 − c3)t + M(t), (1.3)
where
M(t) =
∫ t
0
Vx (x(s), s)g(x(s), s)
V (x(s), s)dB(s)
is a continuous martingale with the quadratic variation
〈M(t),M(t)〉 =
∫ t
0
|Vx (x(s), s)g(x(s), s)|2
V 2(x(s), s)ds ≤ c3t .
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Almost sure exponential stabilityAlmost sure exponential instability
By the strong law of large numbers for the martingales,
limt→∞
M(t)t
= 0 a.s.
It therefore follows from (1.3) that
lim inft→∞
1t
log V (x(t), t) ≥ 12
(2c2 − c3) a.s.
which implies the required assertion immediately by using thecondition.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Consider the scalar linear SDE
dx(t) = ax(t) +m∑
i=1
bix(t)dBi(t) on t ≥ 0.
It is known that it has the explicit solution
x(t) = x0 exp(
[a− 0.5m∑
i=1
b2i ]t +
m∑i=1
biBi(t)).
This implies that, for x0 6= 0,
limt→∞
1t
log |x(t)| = a− 0.5m∑
i=1
b2i a.s.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Let us now apply the stability theorem to obtain the sameconclusion. Let V (x , t) = x2. Then
LV (x , t) =(
2a +m∑
i=1
b2i
)|x |2
and, writing g(x , t) = (b1x , · · · ,bmx),
|Vx (x , t)g(x , t)|2 = 4m∑
i=1
b2i |x |4.
In other words, the conditions in the Theorems holds with
p = 2, c1 = 1, c2 = 2a +m∑
i=1
b2i , c3 = 4
m∑i=1
b2i .
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
We hence have
lim supt→∞
1t
log |x(t)| ≤ a− 12
m∑i=1
b2i a.s.
and
lim inft→∞
1t
log |x(t)| ≥ a− 12
m∑i=1
b2i a.s.
Combining these gives what we want.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Consider, for example,
dx(t) = x(t)dt + 2x(t)dB(t)
with initial value x(0) = x0 6 0, where B(t) is a one-dimensionalBrownian motion. The theory above shows that the solution ofthis linear sde obeys
lim inft→∞
1t
log |x(t)| = −1 a.s.
The following simulation shows a typical sample path of thesolution with initial value x(0) = 10.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
0 2 4 6 8 10
010
2030
4050
60
t
x(t)
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Outline
1 TheoryAlmost sure exponential stabilityAlmost sure exponential instability
2 ExamplesLinear SDEsNonlinear case
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Consider the two-dimensional SDE
dx(t) = f (x(t))dt + Gx(t)dB(t) on t ≥ 0
with initial value x(0) = x0 ∈ R2 and x0 6= 0, where B(t) is aone-dimensional Brownian motion,
f (x) =
(x2 cos x12x1 sin x2
), G =
(3 −0.3−0.3 3
)
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
Let V (x , t) = |x |2. It is easy to verify that
4.29|x |2 ≤ LV (x , t) = 2x1x2 cos x1+4x1x2 sin x2+|Gx |2 ≤ 13.89|x |2
and
29.16|x |2 ≤ |Vx (x , t)Gx |2 = |2xT Gx |2 ≤ 43.56|x |4.
Applying the Theorems we then have
−8.745 ≤ lim inft→∞
1t
log |x(t ; x0)| ≤ lim supt→∞
1t
log |x(t ; x0)| ≤ −0.345
almost surely. The following figure is a compute simulation.
Xuerong Mao FRSE Stability of SDE
TheoryExamples
Linear SDEsNonlinear case
0 2 4 6 8 10
01
23
4
t
X1(t)
0 2 4 6 8 100
24
6t
X2(t)
x1(0) = x2(0) = 1.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Stability of Stochastic Differential EquationsPart 4: Moment Exponential Stability
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Outline
1 Moment verse Almost Sure Exponential Stability
2 CriteriaNonlinear caseLinear case
3 A Case Study
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Outline
1 Moment verse Almost Sure Exponential Stability
2 CriteriaNonlinear caseLinear case
3 A Case Study
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Outline
1 Moment verse Almost Sure Exponential Stability
2 CriteriaNonlinear caseLinear case
3 A Case Study
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Generally speaking, the pth moment exponential stability andthe almost sure exponential stability do not imply each otherand additional conditions are required in order to deduce onefrom the other. The following theorem gives the conditionsunder which the pth moment exponential stability implies thealmost sure exponential stability. However we still do not knowunder what conditions the almost sure exponential stabilityimplies the pth moment exponential stability.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
TheoremAssume that there is a positive constant K such that
xT f (x , t) ∨ |g(x , t)|2 ≤ K |x |2 for all (x , t) ∈ Rd × R+.
Then the pth moment exponential stability of the trivial solutionof the SDE implies the almost sure exponential stability.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
To prove this theorem we need the Burkholder–Davis–Gundyinequality which we cite as a lemma.
Lemma
Let g ∈ L2(R+; Rd×m). Define, for t ≥ 0,
x(t) =
∫ t
0g(s)dB(s) and A(t) =
∫ t
0|g(s)|2ds.
Then for every p > 0, there exist universal positive constantscp,Cp (depending only on p), such that
cpE |A(t)|p2 ≤ E
(sup
0≤s≤t|x(s)|p
)≤ CpE |A(t)|
p2 .
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
In particular, one may take
cp = (p/2)p, Cp = (32/p)p/2 if 0 < p < 2;
cp = 1, Cp = 4 if p = 2;
cp = (2p)−p/2, Cp =[pp+1/2(p − 1)p−1]p/2 if p > 2.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Proof of the theorem. Fix any x0 6= 0 in Rd and writex(t ; x0) = x(t) simply. By the definition of the pth momentexponential stability, there is a pair of positive constants and Csuch that
E |x(t)|p ≤ Ce−λt on t ≥ 0.
Let n = 1,2, · · · . By Itô’s formula and the condition, one canshow that for n − 1 ≤ t ≤ n,
|x(t)|p ≤ |x(n − 1)|p + c1
∫ t
n−1|x(s)|pds
+
∫ t
n−1p|x(s)|p−2xT (s)g(x(s), s)dB(s),
where c1 = pK + p(1 + |p − 2|)K/2. Hence
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
E(
supn−1≤t≤n
|x(t)|p)≤ E |x(n − 1)|p + c1
∫ n
n−1E |x(s)|pds
+E(
supn−1≤t≤n
∫ t
n−1p|x(s)|p−2xT (s)g(x(s), s)dB(s)
).
On the other hand, by the well-knownBurkholder–Davis–Gundy inequality we compute
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
E(
supn−1≤t≤n
∫ t
n−1p|x(s)|p−2xT (s)g(x(s), s)dB(s)
)
≤ 4√
2E(∫ n
n−1p2|x(s)|2(p−2)|xT (s)g(x(s), s)|2ds
) 12
≤ 4√
2E(
supn−1≤s≤n
|x(s)|p∫ n
n−1p2K |x(s)|pds
) 12
≤ 12
E(
supn−1≤s≤n
|x(s)|p)
+ 16p2K∫ n
n−1E |x(s)|pds.
Substituting this into the previous inequality yields
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
E(
supn−1≤t≤n
|x(t)|p)≤ 2E |x(n − 1)|p + c2
∫ n
n−1E |x(s)|pds,
where c2 = 2c1 + 32p2K . By the property of the pth momentexponential stability, we then have
E(
supn−1≤t≤n
|x(t)|p)≤ c3e−λ(n−1),
where c3 = C(2 + c2). Now, let ε ∈ (0, λ) be arbitrary. Then
P
supn−1≤t≤n
|x(t)|p > e−(λ−ε)(n−1)≤ c3e−ε(n−1).
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
In view of the Borel–Cantelli lemma we see that for almost allω ∈ Ω,
supn−1≤t≤n
|x(t)|p ≤ e−(λ−ε)(n−1)
holds for all but finitely many n. Hence, there exists ann0 = n0(ω), for all ω ∈ Ω excluding a P-null set, for which theinequality above holds whenever n ≥ n0. Consequently, foralmost all ω ∈ Ω,
1t
log |x(t)| =1pt
log(|x(t)|p) ≤ −(λ− ε)(n − 1)
pn
if n − 1 ≤ t ≤ n, n ≥ n0.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Hencelim sup
t→∞
1t
log |x(t)| ≤ −(λ− ε)
pa.s.
Since ε > 0 is arbitrary, we must have
lim supt→∞
1t
log |x(t)| ≤ −λp
a.s.
By definition, the trivial solution of the SDE is almost surelyexponentially stable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Outline
1 Moment verse Almost Sure Exponential Stability
2 CriteriaNonlinear caseLinear case
3 A Case Study
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Theorem
Assume that there is a function V (∈ C2,1(Rd × R+; R+), andpositive constants c1–c3, such that
c1|x |p ≤ V (x , t) ≤ c2|x |p and LV (x , t) ≤ −c3V (x , t)
for all (x , t) ∈ Rd × R+. Then
E |x(t ; x0)|p ≤ c2
c1|x0|pe−c3t on t ≥ 0
for all x0 ∈ Rd .
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Proof. Fix any x0 ∈ Rd and write x(t ; x0) = x(t). For eachn ≥ |x0|, define the stopping time
τn = inft ≥ 0 : |x(t)| ≥ n.
Obviously, τn →∞ as n→∞ almost surely. By Itô’s formula,we can derive that for t ≥ 0,
E[ec3(t∧τn)V (x(t ∧ τn), t ∧ τn)
]− V (x0,0)
= E∫ t∧τn
0ec3s[c3V (x(s), s) + LV (x(s), s)
]ds ≤ 0
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Hence
c1E[ec3(t∧τn)E |x(t ∧ τn)|p
]≤ V (x0,0) ≤ c2|x0|p.
Letting n→∞ yields that
c1ec3tE |x(t)|p ≤ c2|x0|p
which implies the desired assertion.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
TheoremAssume that there exists a symmetric positive-definite d × dmatrix Q, and constants α1 ∈ R, 0 ≤ α2 < α3, such that for all(x , t) ∈ Rd × R+,
xT Qf (x , t) +12
trace[gT (x , t)Qg(x , t)] ≤ α1xT Qx
andα2xT Qx ≤ |xT Qg(x , t)| ≤ α3xT Qx .
(i) If α1 < 0, then the trivial solution of the SDE is pth momentexponentially stable provided p < 2 + 2|α1|/α2
3.(ii) If 0 ≤ α1 < α2
2, then the trivial solution of equation (1.2) ispth moment exponentially stable provided p < 2− 2α1/α
22.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Proof. Let V (x , t) = (xT Qx)p2 . Then
λp2min(Q)|x |p ≤ V (x , t) ≤ λ
p2max(Q)|x |p.
It is also easy to verify that
LV (x , t) = p(xT Qx)p2−1(
xT Qf (x , t) +12
trace[gT (x , t)Qg(x , t)])
+ p(p
2− 1)
(xT Qx)p2−2|xT Qg(x , t)|2.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
(i) Assume that α1 < 0 and p < 2 + 2|α1|/α23. Without loss of
generality, we can let p ≥ 2. Then
LV (x , t) ≤ −p[|α1| −
(p2− 1)α2
3
]V (x , t).
(ii) Assume that 0 ≤ α1 < α22 and p < 2− 2α1/α
22. Then
LV (x , t) ≤ −p[(p
2− 1)α2
2 − α1
]V (x , t).
So in both cases the stability assertion follows from theprevious theorem.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Outline
1 Moment verse Almost Sure Exponential Stability
2 CriteriaNonlinear caseLinear case
3 A Case Study
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Consider a d-dimensional linear SDE
dx(t) = Fx(t)dt +m∑
i=1
Gix(t)dBi(t),
where F , Gi ∈ Rd×d . This is of course a special case of theunderlying SDE where
f (x , t) = Fx , g(x , t) = (G1x , · · · ,Gmx).
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
CorollaryAssume that there exists a symmetric positive-definite d × dmatrix Q such that the following LMI holds:
QF + F T Q +m∑
i=1
GTi QGi < 0.
Then the trivial solution of the linear SDE is mean-squareexponentially stable as well as almost surely exponentiallystable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Proof. Let V (x , t) = xT Qx . Then
λmin(Q)|x |2 ≤ V (x , t) ≤ λmax(Q)|x |2.
Moreover
LV (x , t) = xT Qx ≤ λmax(Q)|x |2.
where Q = QF + F T Q +∑m
i=1 GTi QGi . By the condition,
λmax(Q) < 0. Hence
LV (x , t) ≤ −|λmax(Q)|λmax(Q)
V (x , t).
The assertions follow therefore from the theory establishedabove.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
In the case when
QF + F T Q +m∑
i=1
GTi QGi
is not negative-definite, the following result is useful.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
CorollaryAssume that there exists a symmetric positive-definite d × dmatrix Q, and nonnegative constants β and βi (1 ≤ i ≤ m),such that β <
∑mi=1 βi ,
QF + F T Q +m∑
i=1
GTi QGi − βQ ≤ 0,
and, moreover, for each i = 1, · · · ,m,
either QGi + GTi Q −
√2βiQ ≥ 0 or QGi + GT
i Q +√
2βiQ ≤ 0.
If 0 < p < 2− 2β/(∑m
i=1 βi), then the trivial solution of thelinear SDE is pth moment exponentially stable, whence it isalso almost surely exponentially stable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
Proof. We will use the 2nd theorem established above to showthis corollary. We first have that
xT Qf (x , t) +12
trace[gT (x , t)Qg(x , t)]
= 0.5xT(
QF + F T Q +m∑
i=1
GTi QGi
)x ≤ 0.5βxT Qx .
We also observe from the condition that for each i ,
|xT QGix |2 = 0.25|xT (QGi + GTi Q)x |2 ≥ 0.5βi(xT Qx)2.
Hence
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
|xT Qg(x , t)| =
√√√√ m∑i=1
|xT QGix |2 ≥
√√√√0.5m∑
i=1
βi xT Qx .
Applying the theorem with
α1 = 0.5β, α2 =
√√√√0.5m∑
i=1
βi ,
we can therefore conclude that the trivial solution of the linearSDE is pth moment exponentially stable if0 < p < 2− 2β/(
∑mi=1 βi). This implies that the trivial solution
of the linear SDE is also almost surely exponentially stable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
As an even more special case, let us consider the scalar linearSDE
dx(t) = ax(t)dt +m∑
i=1
bix(t)dBi(t),
where a, bi are all real numbers. Using the corollaries above,we can conclude:
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
If 2a +∑m
i=1 b2i < 0, then the trivial solution of this scalar
linear SDE is mean-square exponentially stable as well asalmost surely exponentially stable.If 0 ≤ 2a +
∑mi=1 b2
i < 2∑m
i=1 b2i , then the trivial solution of
this scalar linear SDE is pth moment exponentially stableprovided
0 < p < 1− 2a∑mi=1 b2
i,
whence it is also almost surely exponentially stable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Nonlinear caseLinear case
If 2a +∑m
i=1 b2i < 0, then the trivial solution of this scalar
linear SDE is mean-square exponentially stable as well asalmost surely exponentially stable.If 0 ≤ 2a +
∑mi=1 b2
i < 2∑m
i=1 b2i , then the trivial solution of
this scalar linear SDE is pth moment exponentially stableprovided
0 < p < 1− 2a∑mi=1 b2
i,
whence it is also almost surely exponentially stable.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
This example is from the satellite dynamics. Sagirow in 1970derived the equation
y(t) + β(1 + αB(t))y(t) + (1 + αB(t))y(t)− γ sin(2y(t)) = 0
in the study of the influence of a rapidly fluctuating density ofthe atmosphere of the earth on the motion of a satellite in acircular orbit. Here B(t) is a scalar white noise, α is a constantrepresenting the intensity of the disturbance, and β, γ are twopositive constants.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Introducing x = (x1, x2)T = (y , y)T , we can write this equationas the two-dimensional SDE
dx1(t) = x2(t)dt ,dx2(t) = [−x1(t) + γ sin(2x1(t))− βx2(t)]dt
−α[x1(t) + βx2(t)]dB(t).
For the Lyapunov function, we try an expression consisting of aquadratic form and integral of the nonlinear component:
V (x , t) = ax21 + bx1x2 + x2
2 + c∫ x1
0sin(2y)dy
= ax21 + bx1x2 + x2
2 + c sin2 x1.
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
This yields
LV (x , t) = −(b − α2)x21 + bγx1 sin(2x1)− (2β − b − α2β2)x2
2
+ (2a− bβ − 2 + 2α2β)x1x2 + (c + 2γ)x2 sin(2x1).
Setting 2a− bβ − 2 + 2α2β = 0 and c + 2γ = 0 we obtain
V (x , t) =12
(bβ + 2− 2α2β)x21 + bx1x2 + x2
2 − 2γ sin2 x1
and
LV (x , t) = −(b − α2)x21 + bγx1 sin(2x1)− (2β − b − α2β2)x2
2 .
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
Note that
V (x , t) ≥ 12
(bβ + 2− 2α2β − 4γ)x21 + bx1x2 + x2
2 .
So V (x , t) ≥ ε|x |2 for some ε > 0 if
2(bβ + 2− 2α2β − 4γ) ≥ b2
or equivalently
β −√β2 + 4− 8γ − 4α2β < b < β +
√β2 + 4− 8γ − 4α2β.
Note also that
LV (x , t) ≤ −(b − α2 − 2bγ)x21 − (2β − b − α2β2)x2
2 .
Xuerong Mao FRSE Stability of SDE
Moment verse Almost Sure Exponential StabilityCriteria
A Case Study
So LV (x , t) ≤ −ε|x |2 for some ε > 0 provided bothb − α2 − 2bγ > 0 and 2β − b − α2β2 > 0, that is
2γ < 1 and α2/(1− 2γ) < b < 2β − α2β2.
We can therefore conclude that if γ < 1/2 and
maxα2/(1− 2γ), β −
√β2 + 4− 8γ − 4α2β
< min
2β − α2β2, β +
√β2 + 4− 8γ − 4α2β
then the trivial solution of the SDE is exponentially stable inmean square.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
Stability of Stochastic Differential EquationsPart 5: Stochastic Stabilization and Destabilization
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
It is not surprising that noise can destabilize a stable system.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Consider a 2-dimensional ODE
y(t) = −y(t) on t ≥ 0, y(0) = y0 ∈ R2.
This is an exponentially stable system. Perturb it by noise andassume the stochastically perturbed system is described by anSDE
dx(t) = −x(t)dt + Gx(t)dB(t) on t ≥ 0, x(0) = y0 ∈ R2,
where B(t) is a scalar Brownian motion and
G =
(0 −22 0
)
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
The SDE has the explicit solution
x(t) = exp[(−I − 0.5G2)t + GB(t)]x(0) = exp[It + GB(t)]x(0),
where I is the 2× 2 identity matrix. Consequently
limt→∞
1t
log(|x(t)|) = 1 a.s.
That is, the stochastically perturbed system has becomeunstable with probability one.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
0.0 0.5 1.0 1.5 2.0
−4−2
02
46
8
t
x1(t)
or y
1(t)
x1(t)y1(t)
0.0 0.5 1.0 1.5 2.0
05
10
t
x2(t)
or y
2(t)
x2(t)y2(t)
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Do you believe that noise can also stabilize an unstablesystem?
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Consider the scalar ODE
y(t) = y(t) on t ≥ 0, y(0) = y0 ∈ R.
The solution isy(t) = y(0)et .
So |y(t)| → ∞ if y(0) 6= 0. That is, the ODE is an exponentiallyunstable system. Perturb it by noise and assume thestochastically perturbed system is described by an SDE
dx(t) = x(t)dt + σx(t)dB(t) on t ≥ 0, x(0) = y0 ∈ R,
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
The SDE has the explicit solution
x(t) = x(0) exp[(1− 0.5σ2)t + σB(t)].
Consequentlyx(t)→ 0 a.s. if σ >
√2.
That is, the stochastically perturbed system has become stablewith probability one.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
0 2 4 6 8 10
01
23
45
67
t
x(t) o
r y(t)
x(t)y(t)
σ = 2
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Of course, if the noise is not strong enough, it will not be able tostabilize the system.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
0 2 4 6 8 10
010
020
030
040
050
0
t
x(t) o
r y(t)
x(t)y(t)
σ = 0.5
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
0 2 4 6 8 10
01
23
45
67
t
x(t) o
r y(t)
x(t)y(t)
σ =√
2
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
DestabilizationStabilizationA brief history
Has’minskii (1969): The pioneering work where two whitenoise sources were used to stabilize a particular system.Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Anylinear system x(t) = Ax(t) with trace(A) < 0 can bestabilized by one real noise source.Scheutzow (1993): Stochastic stabilization for two specialnonlinear systems.Mao (1994): The general theory on stochastic stabilizationfor nonlinear SDEs.Mao (1996): Design a stochastic control that canself-stabilize the underlying system.Caraballo, Liu and Mao (2001): Stochastic stabilization forpartial differential equations (PDEs).Appleby and Mao (2004): Stochastic stabilization forfunctional differential equations.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
Suppose that the given system is described by a nonlinear ODE
y(t) = f (y(t), t) on t ≥ 0, y(0) = x0 ∈ Rd .
Here f : Rd × R+ → Rd is a locally Lipschitz continuousfunction and particularly, for some K > 0,
|f (x , t)| ≤ K |x | for all (x , t) ∈ Rd × R+.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
We now use the m-dimensional Brownian motionB(t) = (B1(t), · · · ,Bm(t))T as the source of noise to perturb thegiven system. For simplicity, suppose the stochasticperturbation is of a linear form, that is the stochasticallyperturbed system is described by the semi-linear Itô equation
dx(t) = f (x(t), t)dt+m∑
i=1
Gix(t)dBi(t) on t ≥ 0, x(0) = x0 ∈ Rd ,
where Gi , 1 ≤ i ≤ m, are all d × d matrices.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
TheoremAssume that there are two constants λ > 0 and ρ ≥ 0 such that
m∑i=1
|Gix |2 ≤ λ|x |2 andm∑
i=1
|xT Gix |2 ≥ ρ|x |4
for all x ∈ Rd . Then
lim supt→∞
1t
log |x(t)| ≤ −(ρ− K − λ
2
)a.s.
for all x0 ∈ Rd . In particular, if ρ > K + 12λ, then the
stochastically perturbed system is almost surely exponentiallystable.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
ExampleLet Gi = σi I for 1 ≤ i ≤ m, where I is the d × d identity matrixand σi a constant. Then the SDE becomes
dx(t) = f (x(t), t)dt +m∑
i=1
σix(t)dBi(t).
Moreover,
m∑i=1
|Gix |2 =m∑
i=1
σ2i |x |2 and
m∑i=1
|xT Gix |2 =m∑
i=1
σ2i |x |4.
The solution of the SDE has the property
lim supt→∞
1t
log |x(t)| ≤ −(1
2
m∑i=1
σ2i − K
)a.s.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
Therefore, The SDE is almost surely exponentially stableprovided
12
m∑i=1
σ2i > K .
An even simpler case is that when σi = 0 for 2 ≤ i ≤ m, i.e. theSDE
dx(t) = f (x(t), t)dt + σ1x(t)dB1(t).
This SDE is almost surely exponentially stable provided12σ
21 > K . These show that if we add a strong enough stochastic
perturbation to the given ODE, then the system is stabilized.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
The Theorem above ensures that there are many choices forthe matrices Gi in order to stabilize a given system and ofcourse the above choices are just the simplest ones. Forillustration, we give one more example here.
For each i , choose a positive-definite matrix Di such that
xT Dix ≥√
32||Di || |x |2.
Obviously, there are many such matrices. Let σ be a constantand Gi = σDi . Then
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
m∑i=1
|Gix |2 ≤ σ2m∑
i=1
||Di ||2|x |2
andm∑
i=1
|xT Gix |2 ≥3σ2
4
m∑i=1
||Di ||2|x |3.
By the Theorem, the solution of the SDE satisfies
lim supt→∞
1t
log |x(t)| ≤ −(σ2
4
m∑i=1
||Di ||2 − K)
a.s.
The SDE is therefore almost surely exponentially stable if
σ2 >4K∑m
i=1 ||Di ||2.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
TheoremAny nonlinear system y(t) = f (y(t), t) can be stabilized byBrownian motions provided the following condition is fulfilled
|f (x , t)| ≤ K |x | for all (x , t) ∈ Rd × R+.
Moreover, one can even use only a scalar Brownian motion tostabilize the system.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
ExampleGiven an unstable 2-dimensional ODE
y(t) = f (y(t), t),
where
f (y , t) =
(y1 cos(t) + y2 sin(y1)y2 sin(t) + y1 cos(y2)
).
It is easy to see
|f (y , t)| ≤ 2|y | ∀(y , t) ∈ R2 × R+.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
Perturbing this ODE by a scalar Brownian motion results in anSDE
dx(t) = f (x(t), t)dt + σ1x(t)dB1(t).
The Theorem above shows that this SDE is almost surelyexponentially stable provided
σ1 > 2.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
0 1 2 3 4 5
01
23
45
6
t
x1(t)
or y1
(t)
x1(t)y1(t)
0 1 2 3 4 50
12
34
t
x2(t)
or y2
(t)
x2(t)y2(t)
σ1 = 3
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
0 1 2 3 4 5
01
23
45
67
t
x1(t)
or y1
(t)
x1(t)y1(t)
0 1 2 3 4 50
12
34
t
x2(t)
or y2
(t)
x2(t)y2(t)
σ1 = 2.5
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
0 1 2 3 4 5
02
46
8
t
x1(t)
or y1
(t)
x1(t)y1(t)
0 1 2 3 4 50
24
68
t
x2(t)
or y2
(t)
x2(t)y2(t)
σ1 = 2
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryExamples and Simulations
0 1 2 3 4 5
1.01.5
2.02.5
3.03.5
t
x1(t)
or y1
(t)
x1(t)y1(t)
0 1 2 3 4 51
23
4
t
x2(t)
or y2
(t)
x2(t)y2(t)
σ1 = 1.5
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
TheoremAssume that there are two positive constants λ and ρ such that
m∑i=1
|Gix |2 ≥ λ|x |2 andm∑
i=1
|xT Gix |2 ≤ ρ|x |4
for all x ∈ Rd . Then
lim inft→∞
1t
log |x(t)| ≥(λ
2− K − ρ
)a.s.
for all x0 6= 0. In particular, if λ > 2(K + ρ), then the SDE isalmost surely exponentially unstable.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Can we find the matrices Gi as described in the theorem abovein order to destabilize the given ODE?
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Outline
1 Motivating Examples and HistoryDestabilizationStabilizationA brief history
2 Stochastic StabilizationTheoryExamples and Simulations
3 Stochastic DestabilizationTheoryCase study and simulations
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Case 1. d ≥ 3
Choose the dimension of the Brownian motion m = d . Let σ bea constant. For each i = 1,2, · · · ,d − 1, define the d × d matrixGi = (g i
uv ) by g iuv = σ if u = i and v = i + 1 or otherwise
g iuv = 0. Moreover, define Gd = (gd
uv ) by gduv = σ if u = d and
v = 1 or otherwise gduv = 0. Then SDE becomes
dx(t) = f (x(t), t)dt + σ
x2(t)dB1(t)
...xd(t)dBd−1(t)x1(t)dBd(t)
.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Compute thatm∑
i=1
|Gix |2 =m∑
i=1
(σxi+1)2 = σ2|x |2
andm∑
i=1
|xT Gix |2 = σ2m∑
i=1
x2i x2
i+1,
where we use xd+1 = x1. Notingm∑
i=1
x2i x2
i+1 ≤12
m∑i=1
(x4i + x4
i+1) =m∑
i=1
x4i ,
we have
3m∑
i=1
x2i x2
i+1 ≤ 2m∑
i=1
x2i x2
i+1 +m∑
i=1
x4i ≤ |x |4.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Thereforem∑
i=1
|xT Gix |2 ≤σ2
3|x |4.
By the theorem, the solution of the SDE has the property that
lim inft→∞
1t
log |x(t)| ≥(σ2
2− K − σ2
3
)=σ2
6− K a.s.
for any x0 6= 0. If σ2 > 6K , then the SDE will be almost surelyexponentially unstable.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Example
Given a stable 3-dimensional ODE
y(t) = f (y(t), t),
where
f (y , t) =
−2y1 + sin(y2)−2y2 + sin(y3)−2y3 + sin(y1)
.It is easy to see
|f (y , t)| ≤ 3|y | ∀(y , t) ∈ R3 × R+.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Perturbing this ODE by a 3-dimensional Brownian motionresults in an SDE
dx(t) = f (x(t), t)dt + σ
x2(t)dB1(t)x3(t)dB2(t)x1(t)dB3(t)
.This SDE is almost surely exponentially unstable provided
σ >√
18.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0.0 0.2 0.4
−10
12
t
x1(t)y1(t)
0.0 0.2 0.4
−10
12
3
t
x2(t)y2(t)
0.0 0.2 0.4
−3−2
−10
12
3
t
x3(t)y3(t)
σ = 5
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0 2 4 6 8 10
0e+0
01e
+13
2e+1
33e
+13
4e+1
3
t
x1(t)y1(t)
0 2 4 6 8 10
0e+0
01e
+13
2e+1
33e
+13
t
x2(t)y2(t)
0 2 4 6 8 10
−1e+
130e
+00
1e+1
32e
+13
3e+1
3
t
x3(t)y3(t)
σ = 4
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0 1 2 3 4 5
0.00.5
1.01.5
t
x1(t)y1(t)
0 1 2 3 4 5
−0.5
0.00.5
1.0
t
x2(t)y2(t)
0 1 2 3 4 5
−0.5
0.00.5
1.0
t
x3(t)y3(t)
σ = 2
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Case 2. d is an even number
Let d = 2k(k ≥ 1) and σ be a constant. Define
G1 =
0 σ−σ 0
0
. . .
00 σ−σ 0
but set Gi = 0 for 2 ≤ i ≤ m. So the SDE becomes
dx(t) = f (x(t), t)dt + σ
x2(t)−x1(t)
...x2k (t)−x2k−1(t)
dB1(t).
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
In this case we have
m∑i=1
|Gix |2 = σ2|x |2 andm∑
i=1
|xT Gix |2 = 0
Hence, the solution of SDE obeys
lim inft→∞
1t
log |x(t)| ≥ σ2
2− K a.s.
for any x0 6= 0. If σ2 > 2K , then the SDE will be almost surelyexponentially unstable.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Example
Given a stable 4-dimensional ODE
y(t) = f (y(t), t),
where
f (y , t) =
−2y1 + sin(y2)−2y2 + sin(y3)−2y3 + sin(y4)−2y4 + sin(y1)
.It is easy to see
|f (y , t)| ≤ 3|y | ∀(y , t) ∈ R4 × R+.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Perturbing this ODE by a scale Brownian motion results in anSDE
dx(t) = f (x(t), t)dt + σ
x2(t)−x1(t)x3(t)−x4(t)
dB1(t).
This SDE is almost surely exponentially unstable provided
σ >√
6.
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0.0 0.5 1.0 1.5 2.0
−6−2
24
6
t
x1(t)y1(t)
0.0 0.5 1.0 1.5 2.0
−22
46
8
t
x2(t)y2(t)
0.0 0.5 1.0 1.5 2.0
−6−2
24
6
t
x3(t)y3(t)
0.0 0.5 1.0 1.5 2.0
−22
46
8
t
x4(t)y4(t)
σ = 2.5
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0.0 0.5 1.0 1.5 2.0
−8−4
02
4
t
x1(t)y1(t)
0.0 0.5 1.0 1.5 2.0
−40
24
6
t
x2(t)y2(t)
0.0 0.5 1.0 1.5 2.0
−8−4
02
4
t
x3(t)y3(t)
0.0 0.5 1.0 1.5 2.0
−40
24
6
t
x4(t)y4(t)
σ =√
6
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
0 1 2 3 4 5
−0.2
0.20.6
1.0
t
x1(t)y1(t)
0 1 2 3 4 5
0.00.4
0.8
t
x2(t)y2(t)
0 1 2 3 4 5
−0.2
0.20.6
1.0
t
x3(t)y3(t)
0 1 2 3 4 5
0.00.4
0.8
t
x4(t)y4(t)
σ = 1
Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
Case 3. d = 1
Consider a stable scale linear ODE
y(t) = −ay(t) (a > 0),
and its perturbed linear SDE
dx(t) = −ax(t) +m∑
i=1
bix(t)dBi(t).
It is known that
limt→∞
1t
log |x(t)| = −a− 12
m∑i=1
b2i < 0 a.s.
That is, the perturbed system remains stable.
Noise does not destabilize the given system in this case.Xuerong Mao FRSE Stability of SDE
Motivating Examples and HistoryStochastic Stabilization
Stochastic Destabilization
TheoryCase study and simulations
TheoremAny d-dimensional nonlinear system y(t) = f (y(t), t) can bedestabilized by Brownian motions provided the dimension ofthe state d ≥ 2 and
|f (x , t)| ≤ K |x | for all (x , t) ∈ Rd × R+.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of Stochastic Differential EquationsPart 6: New Developments
Xuerong Mao FRSE
Department of Mathematics and StatisticsUniversity of Strathclyde
Glasgow, G1 1XH
December 2010
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
The Lyapunov method has been developed and applied bymany authors during the past century. One of the importantdevelopments in this direction is the LaSalle theorem forlocating limit sets of nonautonomous ODE established by
LaSalle, J.P., Stability theory of ordinary differentialequations, J. Differential Equations 4 (1968), 57–65.
The first LaSalle-type theorem for SDEs established by
Mao, X., Stochastic versions of the LaSalle theorem, J.Differential Equations 153 (1999), 175–195.
is stated below:
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Theorem
Assume that there are functions V ∈ C2,1(Rn × R+; R+),γ ∈ L1(R+; R+) and w ∈ C(Rn; R+) such that
lim|x |→∞
inf0≤t<∞
V (x , t) =∞
andLV (x , t) ≤ γ(t)− w(x), (x , t) ∈ Rn × R+.
Moreover, for each initial value x0 ∈ Rn there is a p > 2 suchthat
sup0≤t<∞
E |x(t ; x0)|p <∞.
Then, for every x0 ∈ Rn, limt→∞ V (x(t ; x0), t) exists and is finitealmost surely, and moreover, limt→∞w(x(t ; x0)) = 0 a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
This LaSalle-type theorem for SDEs has been developedsignificantly for the past 10 years. In this course we willhighlight a couple of developments.
Although the boundedness of the pth moment of the solution
sup0≤t<∞
E |x(t ; x0)|p <∞
has its own right, it is somehow too restrictive. Can we removethis condition?
The answer is yes. To state the improved LaSalle-typetheorem, let us introduce a few more notations.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
L1(R+;R+): the family of all continuous functions γ : R+ → R+
such that∫∞
0 γ(t)dt <∞.
d(x ,A) = infy∈A |x − y | for x ∈ Rn and set A ⊂ Rn.
If µ ∈ K, its inverse function is denoted by µ−1 with domain[0, µ(∞)).
If w ∈ C(Rd ; R+), then Ker(w) = x ∈ Rd : w(x) = 0.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Theorem
Assume that there are functions V ∈ C2,1(Rn × R+; R+),γ ∈ L1(R+; R+) and w ∈ C(Rn; R+) such that
lim|x |→∞
inf0≤t<∞
V (x , t) =∞ and LV (x , t) ≤ γ(t)− w(x)
for (x , t) ∈ Rn × R+. Then, for every initial value x0 ∈ Rd , thesolution x(t ; x0) = x(t) of the SDE has the following properties:∫∞
0 Ew(x(t))dt <∞.∫∞0 w(x(t))dt <∞ a.s.
lim supt→∞ V (x(t), t) <∞ a.s.Ker(w) 6= ∅ and limt→∞ d(x(t),Ker(w)) = 0 a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
The proof of the theorem is very technical so is omitted in thiscourse.
To see the powerfulness of this theorem, let us demonstratethat many classical stability results follow from this theorem. Infact, under the conditions of the theorem, we have:
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
If w(x) = 0 iff x = 0, then the SDE is almost surelyasymptotically stable in the sense that limt→∞ x(t) = 0 a.s.If V (x , t) ≥ eλt |x |p on (x , t) ∈ Rn × R+ for some λ > 0 andp > 0, then the SDE is almost surely exponentially stable.If V (x , t) ≥ (1 + t)λ|x |p on (x , t) ∈ Rn × R+ for some λ > 0and p > 0, then the SDE is almost surely polynomiallystable in the sense that
lim supt→∞
log(|x(t)|)log t
≤ −λp
a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
If there is a positive constant p and a convex functionµ ∈ K∞ such that
lim suph→0+
µ−1(h)
h<∞ (1.1)
andw(x) ≥ µ(|x |p) ∀x ∈ Rn, (1.2)
then ∫ ∞0
E|x(t)|pdt <∞; (1.3)
and ∫ ∞0|x(t)|pdt <∞ a.s. (1.4)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
All items except the last one are obvious. To show the last item,we observe that there is a constant C > 0 such that∫ t
0Eµ(|x(s)|p)ds ≤ C, ∀t ≥ 0.
Since µ convex, we may apply the Jensen inequality to obtain
tµ(1
t
∫ t
0E |x(s)|pds
)≤ C, ∀t ≥ 0.
This implies∫ t
0E |x(s)|pds ≤ tµ−1(C/t) = C
µ−1(C/t)C/t
, ∀t ≥ 0.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Letting t →∞ yiels ∫ t
0E |x(s)|pds <∞.
By the Fubini theorem, we also have
E∫ t
0|x(s)|pds <∞.
Hence ∫ t
0|x(s)|pds <∞ a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
This improved LaSalle-type theorem can also be used tohandle the problem of partially asymptotic stability. Let1 ≤ n ≤ n and 1 ≤ i1 < i2 < · · · < in ≤ n be all integers. Letx = (xi1 , xi2 , · · · , xin ) be the partial coordinates of x , which can
be regarded as in Rn with the norm |x | =√
x2i1
+ · · ·+ x2in
.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Under the assumptions of the theorem, we have:
If w(x) = 0 iff x = 0, then limt→∞ x(t) = 0 a.s.If there is a positive constant p and a convex functionµ ∈ K∞ such that
lim suph→0+
µ−1(h)
h<∞
andw(x) ≥ µ(|x |p) ∀x ∈ Rn,
then∫ ∞0
E|x(t)|pdt <∞ and∫ ∞
0|x(t)|pdt <∞ a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
In many practical situations, we require the solutions of theSDEs to stay in some regime of the state space Rd . Forexample, the SDEs used in finance or population systemsrequire the solutions remain in the positive conex ∈ Rd : xi > 0, 1 ≤ i ≤ d.
To develop the LaSalle-type theorems to cope with thesecases, let us recall the definition of an invariant set with respectto the solutions of the SDE.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Definition
An open subset G of Rd is said to be invariant with respect tothe solutions of the SDE if
Px(t ; x0) ∈ G for all t ≥ 0 = 1 for every x0 ∈ G,
that is, the solutions starting in G will remain in G.
Under our standing hypotheses, we know that Rd − 0 is aninvariant set of the underlying SDE. As another example,consider the one-dimensional equation
dx(t) = − sin(x(t))dt + sin(x(t))dB(t)
where B(t) is a scalar Brownian motion. The open interval(0, π) is an invariant set of this equation.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Theorem
Let G be an invariant set and G be its closure. Assume thatthere are functions V ∈ C2,1(G × R+; R+), γ ∈ L1(R+; R+) andw ∈ C(G; R+) such that
LV (x , t) ≤ γ(t)− w(x), (x , t) ∈ G × R+.
If G is bounded; or otherwise if
limx∈G,|x |→∞
inf0≤t<∞
V (x , t) =∞,
then, for every initial value x0 ∈ G, the solution x(t ; x0) = x(t) ofthe SDE has the following properties:
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Theorem ∫ ∞0
Ew(x(t))dt <∞,∫ ∞0
w(x(t))dt <∞ a.s.
lim supt→∞
V (x(t), t) <∞ a.s.
KerG(w) := x ∈ G : w(x) = 0 6= ∅,
limt→∞
d(x(t),KerG(w)) = 0 a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
In the following examples we will let B(t) be a scalar Brownianmotion.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Example 1Let α and β be bounded functions and consider a scalar SDE
dx(t) = α(t)x(t)dt + β(t)x(t)dB(t), t ≥ 0.
It is known that G = R − 0 is an invariant set. Assume thatthere is a δ ∈ (0,1) such that
ε := inf0≤t<∞
(1− δ2
β2(t)− α(t))> 0.
Let V (x , t) = |x |δ for x 6= 0 and t ≥ 0. Then
LV (x , t) = −δ(1− δ
2β2(t)− α(t)
)|x |δ ≤ −δε|x |δ.
We can therefore conclude that limt→∞ x(t) = 0 a.s.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Example 2Consider
dx(t) = − sin(x(t))dt + sin(x(t))dB(t), t ≥ 0.
It is known that G = (0, π) is an invariant set. Let V (x , t) = |x |2for x ∈ (0, π) and t ≥ 0. Then
LV (x , t) = − sin(x)[2x − sin(x)] ≤ 0.
Hence, for any x0 ∈ (0, π), almost every sample path of x(t ; x0)will tend to either 0 or π.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
Example 3Let α be a function on R+ such that 0 < δ1 ≤ α(t) ≤ δ2 <∞.Consider a stochastic oscillator
y(t) +[α(t) +
√α(t)B(t)
]y(t) + y(t) = 0, t ≥ 0.
Introducing a new variable x = (x1, x2)T = (y , y)T , thisoscillator can be written as an Itô equation
dx(t) =
[x2(t)
−x1(t)− α(t)x2(t)
]dt +
[0
−√α(t)x2(t)
]dB(t).
Let 2 < p < 3 and define V (x , t) = |x |p for (x , t) ∈ R2 × R+.Then
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Original theoremImproved resultsExamples
LV (x , t) ≤ p|x |p−2x1x2 + p|x |p−2x2(−x1 − α(t)x2)
+p(p − 1)
2α(t)|x |p−2x2
2
= −p(3− p)
2α(t)|x |p−2x2
2
≤ −p(3− p)
2δ1|x |p−2x2
2 .
We can therefore conclude that we see that
limt→∞
x2(t ; x0)→ 0 a.s.
and limt→∞ x1(t ; x0) exists and is finite almost surely.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Two key questions
Q1. Given a stable SDE, for what choices of stepsize does thenumerical method reproduce the stability property of thetest equation?
Q2. Given that the numerical solution to an SDE is stable for asufficiently small stepsize, can we conclude confidentlythat the SDE is stable?
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Mitsui, Saito et al. (93, 94, 95, 96), Higham (00) discussedQ1 on the mean-square exponential stability for scalarlinear SDEs.Higham, Mao and Stuart (2003) discussed Q1 and Q2 onthe mean-square exponential stability for multi-dimensionalnonlinear SDEs under the global Lipschitz condition.Some results answer Q1 on the almost sure exponentialstability.No results answer Q2 on the almost sure exponentialstability yet.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
SDE in Rd :
dx(t) = f (x(t))dt + g(x(t))dB(t), t ≥ 0, x(0) = x0.
The Euler-Maruyama (EM) discrete approximation:
X0 = x0,
Xk+1 = Xk + ∆f (Xk ) + g(Xk )∆Bk , ∀k ≥ 0,
where ∆Bk = B((k + 1)∆)− B(k∆).The EM continuous approximation:
X (t) = Xk + f (Xk )(t −∆) + g(Xk )(B(t)− B(k∆)),
for t ∈ [k∆, (k + 1)∆], k = 0,1,2, · · · .
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
The SDE is said to be exponentially stable in mean square ifthere is a pair of positive constants λ and M such that withinitial data x0 ∈ Rd
E |x(t)|2 ≤ M|x0|2e−λt , ∀t ≥ 0. (2.1)
We refer to as the rate constant and M as the growth constant.
For a given step size ∆ > 0, the Euler-Maruyama solution issaid to be exponentially stable in mean square on the SDE ifthere is a pair of positive constants γ and N such that with initialdata x0 ∈ Rd
E |Xk |2 ≤ N|x0|2e−γk∆, ∀k ≥ 0. (2.2)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
Hypothesis (H1):
|f (x)− f (y)|2 ≤ K1|x − y |2, ∀x , y ∈ Rd ,
|g(x)− g(y)|2 ≤ K2|x − y |2, ∀x , y ∈ Rd ,
f (0) = 0, g(0) = 0.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
TheoremUnder (H1) the SDE is exponentially stable in mean square ifand only if the Euler-Maruyama solution is exponentially stablein mean square for some step size ∆ with the rate constant γand the growth constant N satisfying
CeγT (∆ +√
∆) + 1 +√
∆ ≤ e14γT ,
where T = 1 + 4 log(N)/γ and C > 0 is a constant whichdepends only on T ,K1 and K2 (but not on ∆ and ξ) such that
sup0≤t≤2T
E|x(t)− X (t)|2 ≤
(sup
0≤t≤2TE|X (t)|2
)C∆.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Stability of numerical methodsThe Euler-Maruyama methodStability of the EM method
In practice, the constant C can be computed by
C = 4T (TK1 + K2)(1 + K1)e4(T +K1+K2),
though this may not be optimal.We emphasize that this Theorem is an “if and only if ”result, and hence has important practical implications. Ifcareful numerical simulations indicate exponential stabilityin mean square, then we may confidently infer that theunderlying SDE has the same property.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Considerdx(t) = αx(t)dt + σx(t)dB(t) (3.1)
on t ≥ 0 with initial value x(0) = x0 ∈ R, where α and σ are realnumbers. If x0 6= 0, then
limt→∞
1t
log(|x(t)|) = α− 12σ
2 a.s.
That is, the linear SDE is almost surely exponential stable if andonly if α− 1
2σ2 < 0.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
The Euler-Maruyama (EM) method
Given a step size ∆ > 0, the EM method is to compute thediscrete approximations Xk ≈ x(k∆) by setting X0 = x0 andforming
Xk+1 = Xk (1 + α∆ + σ∆Bk ), (3.2)
for k = 0,1, · · · , where ∆Bk = B((k + 1)∆)− B(k∆).
Question: If α− 12σ
2 < 0, is the EM method almost surelyexponentially stable for sufficiently small ∆?
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Theorem
If α− 12σ
2 < 0, then for any ε ∈ (0,1) there is a ∆1 ∈ (0,1)such that for any ∆ < ∆1, the EM approximate solution has theproperty that
limk→∞
1k∆
log(|Xk |) ≤ (1− ε)(α− 12σ
2) < 0 a.s. (3.3)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
ExampleConsider
dx(t) = x(t)dt + 2x(t)dB(t), t ≥ 0.
The following 4 simulations are carried out using ∆ = 0.001with the initial value x(0) = 10. These simulations show clearlythat the EM method reproduces the almost sure exponentialstability of the linear SDE.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 2 4 6 8 10
010
3050
70
t
X(t)
or x
(t)
true solnEM soln
0 2 4 6 8 10
02
46
810
t
X(t)
or x
(t)
true solnEM soln
0 2 4 6 8 10
05
1015
2025
t
X(t)
or x
(t)
true solnEM soln
0 2 4 6 8 10
02
46
810
t
X(t)
or x
(t)
true solnEM soln
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Consider the nonlinear SDEdx(t) = f (x(t))dt + g(x(t))dB(t), t ≥ 0,x(0) = x0 ∈ Rd ,
(3.4)
where f ,g : Rd → Rd . As before, we assume thatf ,g : Rd → Rd are smooth enough so that the SDE (3.4) has aunique global solution x(t) on [0,∞). The following stabilityresult can be proved in a similar way as we did in Part 3.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Theorem
If
−λ := supx∈Rd ,x 6=0
(〈x , f (x)〉+ 1
2 |g(x)|2
|x |2− 〈x ,g(x)〉2
|x |4
)< 0, (3.5)
then the solution of the SDE (3.4) obeys
lim supt→∞
1t
log(|x(t)|) ≤ −λ a.s. (3.6)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Question: Under condition (3.5), can the EM reproduce thealmost sure exponential stability?
Unlike the linear case, the answer is in general no. However,under the following additional condition, the answer is yes.
AssumptionAssume that there is a K > 0 such that
|f (x)| ∨ |g(x)| ≤ K |x |, ∀x ∈ Rd . (3.7)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
The Euler-Maruyama method
Recall that given a step size ∆ > 0, the Euler-Maruyamamethod is to compute the discrete approximations Xk ≈ x(k∆)by setting X0 = x0 and forming
Xk+1 = Xk + f (Xk )∆ + g(Xk )∆Bk , (3.8)
for k = 0,1, · · · , where ∆Bk = B((k + 1)∆)− B(k∆) as before.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Theorem
Let (3.7) and (3.5) hold. Then for any ε ∈ (0, λ), there is a∆∗ ∈ (0,1) such that for any ∆ < ∆∗, the EM approximatesolution has the property that
lim supk→∞
1k∆
log(|Xk |) ≤ −(λ− ε) a.s. (3.9)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
ExampleConsider the two-dimensional SDE
dx(t) = f (x(t))dt + Gx(t)dB(t) on t ≥ 0
with initial value x(0) = x0 ∈ R2 and x0 6= 0, where B(t) is aone-dimensional Brownian motion,
f (x) =
(x2 cos x12x1 sin x2
), G =
(3 −0.3−0.3 3
)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
It is easy to verify that
−λ = supx∈Rd ,x 6=0
(〈x , f (x)〉+ 1
2 |g(x)|2
|x |2− 〈x ,g(x)〉2
|x |4
)≤ −0.345.
Hencelim sup
t→∞
1t
log |x(t ; x0)| ≤ −0.345 a.s.
which is the same as we obtained in Part 3.
The following simulation uses ∆ = 0.001 and x(0) = (1,1)T . Itshows clearly that the EM method reproduces the a.s.exponential stability.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t
X1(t)
0 1 2 3 4 5
01
23
45
t
X2(t)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Stable SDE without the linear growth condition:
dx(t) = (x(t)− x3(t))dt + 2x(t)dB(t). (3.10)
Although the coefficients f (x) = x − x3 and g(x) = 2x dosatisfy (3.5) as
supx∈R,x 6=0
(〈x , f (x)〉+ 1
2 |g(x)|2
|x |2− 〈x ,g(x)〉2
|x |4
)
= supx∈R,x 6=0
(x2 − x4 + 2x2
x2 − 4x4
x4
)≤ −1.
An application of the theorem shows that its solution obeys
lim supt→∞
1t
log(|x(t)|) ≤ −1 a.s. (3.11)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
We observe that the shift coefficient f (x) does not obey thelinear growth condition (3.7). We may therefore wonder:
Question: If the EM method is applied to the SDE (3.10), will itrecover the property of almost surely exponential stability?
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Counter example
Applying the EM to the SDE
dx(t) = (x(t)− x3(t))dt + 2x(t)dB(t).
givesXk+1 = Xk (1 + ∆− X 2
k ∆ + 2∆Bk ).
LemmaGiven any initial value X0 6= 0 and any ∆ > 0,
P(
limk→∞
|Xk | =∞)> 0.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
The following 2 simulations show that the EM method does notreproduce the almost sure exponential stability of this nonlinearSDE. Both use ∆ = 0.001 and the first one uses the initial valuex(0) = 30 while the 2nd one uses x(0) = 50. In particular, the2nd one shows that the EM method could blow up very quickly.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 1 2 3 4 5
05
1015
2025
30
t
X(t)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 1 2 3 4 5
−1.2
e+26
5−8
.0e+
264
−4.0
e+26
40.
0 e+
00
t
X(t)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Outline
1 The LaSalle-Type TheoremsOriginal theoremImproved resultsExamples
2 Mean-Square Exponential Stability of Numerical MethodsStability of numerical methodsThe Euler-Maruyama methodStability of the EM method
3 Almost Sure Exponential Stability of Numerical MethodsLinear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Definition of the BEM
Given a step size ∆ > 0, set Z0 = x0 and compute
Zk+1 = Zk + f (Zk+1)∆ + g(Zk )∆Bk (3.12)
for k = 0,1,2, · · · .
The BE method is implicit as for every step given Zk , equation(3.12) needs to be solved for Zk+1. For this purpose, someconditions need to be imposed on f .
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
The one-side Lipschitz condition:There is a constant µ ∈ R such that
〈x − y , f (x)− f (y)〉 ≤ µ|x − y |2, ∀x , y ∈ Rd . (3.13)
Under this condition, it is known that equation (3.12) can besolved uniquely for Zk+1 given Zk as long as the step size∆ < 1/(1 + 2|µ|).
We also need a condition on g: there is a K > 0 such that
|g(x)| ≤ K |x |, ∀x ∈ Rd . (3.14)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Theorem
Let (3.14) and (3.13) hold and f (0) = 0. Assume also that
β := supx∈Rd ,x 6=0
( |g(x)|2
|x |2− 2〈x ,g(x)〉2
|x |4)<∞ (3.15)
If µ+ 12β < 0, then for any ε ∈ (0, |µ+ 1
2β|), there is a∆∗ ∈ (0,1/(1 + 2|µ|)) such that for any ∆ < ∆∗,
lim supk→∞
1k∆
log(|Zk |) ≤ µ+ 12β + ε < 0 a.s. (3.16)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
Let us return to the SDE (3.10). By setting f (x) = x − x3 andg(x) = 2x for x ∈ R, we have
〈x − y , f (x)− f (y)〉 ≤ |x − y |2
which gives µ = 1 while
β := supx∈R,x 6=0
( |g(x)|2
|x |2− 2〈x ,g(x)〉2
|x |4)
= −4,
whence µ+ 12β = −1. Thus, for any ε ∈ (0,1), there is a
∆∗ > 0 sufficiently small so that if ∆ < ∆∗, then
lim supk→∞
1k∆
log(|Zk |) ≤ −1 + ε a.s.
which recovers property (3.11) very well indeed.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
The following 2 simulations show that the BEM method DOreproduce the almost sure exponential stability of the nonlinearSDE (3.10). Both use ∆ = 0.001 and the first one uses theinitial value x(0) = 30 while the 2nd one uses x(0) = 50.
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 1 2 3 4 5
05
1015
2025
30
t
Z(t)
Xuerong Mao FRSE Stability of SDE
The LaSalle-Type TheoremsMean-Square Exponential Stability of Numerical Methods
Almost Sure Exponential Stability of Numerical Methods
Linear scalar SDEsMulti-Dimensional SDEsThe Backward Euler-Maruyama (BEM) Method
0 1 2 3 4 5
010
2030
4050
t
Z(t)
Xuerong Mao FRSE Stability of SDE