Stability of Stochastic Differential Equations Part 1: Introduction

Home , Exponential stability, Lyapunov stability

Lyapunov stability theory for ODEs Stability of SDEs

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Stability of SDEs Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Stability of SDEs Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

In 1892, A.M. Lyapunov introduced the concept of stability of a dynamic system. Roughly speaking, the stability means insensitivity of the state of the system to small changes in the initial state or the parameters of the system. For a stable system, the trajectories which are “close" to each other at a speciﬁc instant should therefore remain close to each other at all subsequent instants.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

Consider a d-dimensional ordinary differential equation (ODE)

dx(t) = f (x(t), t) on t ≥ 0, dt

T d d where f = (f1, ··· , fd ) : R × R+ → R . Assume that for every d initial value x(0) = x0 ∈ R , there exists a unique global solution 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 the initial value x(0) = 0. This solution is called the trivial solution or equilibrium position.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

Deﬁnition The trivial solution is said to be stable if, for every ε > 0, there exists 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 is stable and if there moreover exists a δ0 > 0 such that

lim x(t; x0) = 0 t→∞ whenever |x0| < δ0.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

If the ODE can be solved explicitly, it would be rather easy to determine whether the trivial solution is stable or not. However, the ODE can only be solved explicitly in some special cases. Fortunately, Lyapunov in 1892 developed a method for determining stability without solving the equation. This method is now known as the method of Lyapunov functions or the Lyapunov method.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method Notation

Let K denote the family of all continuous nondecreasing functions 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 that limr→∞ µ(r) = ∞. d For h > 0, let Sh = {x ∈ R : |x| < h}. 1,1 Let C (Sh × R+; R+) denote the family of all continuous functions V (x, t) from Sh × R+ to R+ with continuous ﬁrst partial derivatives with respect to every component of x and to t.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method Basic ideas of the Lyapunov method

1,1 Let x(t) be a solution of the ODE and V ∈ C (Sh × R+; R+). Then v(t) = V (x(t), t) represents a function of t with the derivative dv(t) = V˙ (x(t), t), dt ˙ 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 the distance will decrease to zero, that is x(t) → 0.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method Basic ideas of the Lyapunov method

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

Theorem 1,1 Assume that there exist V ∈ C (Sh × R+; R+) and µ ∈ K such that V (0, t) = 0, µ(|x|) ≤ V (x, t) and V˙ (x, t) ≤ 0 for all (x, t) ∈ Sh × R+. Then the trivial solution of the ODE is stable.

Xuerong Mao FRSE Stability of SDE Lyapunov stability theory for ODEs Concept of stability Stability of SDEs The Lyapunov method

Theorem 1,1 Assume that there exist V ∈ C (Sh × R+; R+) and µ1, µ2, µ3 ∈ K such that

µ1(|x|) ≤ V (x, t) ≤ µ2(|x|) and ˙ V (x, t) ≤ −µ3(|x|) for all (x, t) ∈ Sh × R+. Then the trivial solution of the ODE is asymptotically stable.

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion 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,

d d d d×m where f : R × R+ → R and g : R × R+ → R , and T B(t) = (B1(t), ··· , Bm(t)) is an m-dimensional Brownian motion. As a standing hypothesis in this course, we assume that both f and g obey the local Lipschitz condition and the linear growth condition. d Hence, for any given initial value x(0) = x0 ∈ R , the SDE has a unique global solution denoted by x(t; x0). Assume furthermore 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 SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

When we try to carry over the principles of the Lyapunov stability 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) be replaced? What conditions should a stochastic Lyapunov function satisfy?

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

It turns out that there are various different types of stochastic stability. 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 SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

Deﬁnition The trivial solution of the SDE is said to be stochastically stable or 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 stochastically unstable.

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

Deﬁnition The trivial solution is said to be stochastically asymptotically stable if it is stochastically stable and, moreover, for every ε ∈ (0, 1), there exists a δ0 = δ0(ε) > 0 such that

P{ lim x(t; x0) = 0} ≥ 1 − ε t→∞ whenever |x0| < δ0.

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

Deﬁnition The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically stable and, moreover, for d all x0 ∈ R P{ lim x(t; x0) = 0} = 1. t→∞

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

Deﬁnition The trivial solution is said to be almost surely exponentially d stable if for all x0 ∈ R 1 lim sup log(|x(t; x0)|) < 0 a.s. t→∞ t

d It is said to be pth moment exponentially stable if for all x0 ∈ R

1 p lim sup log(E|x(t; x0)| ) < 0. t→∞ t

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator Outline

1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method

2 Stability of SDEs SDEs Deﬁnition of stochastic stability Diffusion operator

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

To ﬁgure out with what the derivative dv(t)/dt or V˙ (x, t) should be replaced, we naturally consider the Itô differential of the process V (x(t), t), where x(t) is a solution of the SDE and V is a Lyapunov function.

According to the Itô formula, we of course require 2,1 V ∈ C (Sh × R+; R+), which denotes the family of all nonnegative functions V (x, t) deﬁned on Sh × R+ such that they are continuously twice differentiable in x and once in t.

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion 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 1 h i LV (x, t) = V (x, t)+V (x, t)f (x, t)+ trace gT (x, t)V (x, t)g(x, t) , t x 2 xx in which Vx = (Vx1 , ··· , Vxd ) and Vxx = (Vxi xj )d×d .

Xuerong Mao FRSE Stability of SDE SDEs Lyapunov stability theory for ODEs Deﬁnition of stochastic stability Stability of SDEs Diffusion operator

We shall see that V˙ (x, t) will be replaced by the diffusion operator LV (x, t) in the study of stochastic stability. For example, the inequality V˙ (x, t) ≤ 0 will sometimes be replaced by LV (x, t) ≤ 0 to get the stochastic stability. However, it is not necessary 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 than the classical stability of ODEs. Let us begin to explore this wonderful world of stochastic stability.

Xuerong Mao FRSE Stability of SDE Theory Examples

Stability of Stochastic Differential Equations Part 2: Stability in Probability

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE Theory Examples Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Theory Examples Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

In this part, we shall see how the classical Lyapunov method is developed to study stochastic stability in such a similar way that the results in this part are natural generalizations of the Lyapunov stability theory for ODEs. Of course, such results may not be surprising but we will see some surprising results in the next part.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Theorem 2,1 Assume that there exist V ∈ C (Sh × R+; R+) and µ ∈ K such that V (0, t) = 0, µ(|x|) ≤ V (x, t) and LV (x, t) ≤ 0 for all (x, t) ∈ Sh × R+. Then the trivial solution of the SDE is stochastically stable.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large Proof. Let ε ∈ (0, 1) and r ∈ (0, h) be arbitrary. Clearly, we can ﬁnd a δ = δ(ε, r) ∈ (0, r) such that 1 sup V (x, 0) ≤ µ(r). ε x∈Sδ

Now ﬁx any x0 ∈ Sδ and write x(t; x0) = x(t) simply. Deﬁne

τ = inf{t ≥ 0 : x(t) 6∈ Sr }.

(Throughout this course we set inf ∅ = ∞.) By Itô’s formula, for any t ≥ 0,

Z τ∧t V (x(τ ∧ t), τ ∧ t) = V (x0, 0) + LV (x(s), s)ds 0 Z τ∧t + Vx (x(s), s)g(x(s), s)dB(s). 0

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Taking the expectation on both sides, we obtain

Z τ∧t EV (x(τ ∧t), τ ∧t) = V (x0, 0)+E LV (x(s), s)ds ≤ V (x0, 0). 0 Noting that |x(τ ∧ t)| = |x(τ)| = r if τ ≤ t, we get h i EV (x(τ ∧ t), τ ∧ t) ≥ E I{τ≤t}V (x(τ), τ) ≥ µ(r)P{τ ≤ t}.

(Throughout this course IA denotes the indicator function of set A.) We therefore obtain P{τ ≤ t} ≤ ε. Letting t → ∞ we get P{τ < ∞} ≤ ε, that is

P{|x(t)| < r for all t ≥ 0} ≥ 1 − ε as required.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Theorem 2,1 Assume that there exist V ∈ C (Sh × R+; R+) and µ1, µ2, µ3 ∈ K such that

µ1(|x|) ≤ V (x, t) ≤ µ2(|x|) and LV (x, t) ≤ −µ3(|x|) for all (x, t) ∈ Sh × R+. Then the trivial solution of the SDE is stochastically asymptotically stable.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Proof. We know from the previous theorem that the trivial solution is stochastically stable. So we only need to show that for any ε ∈ (0, 1), there is a δ0 = δ0(ε) > 0 such that

P{ lim x(t; x0) = 0} ≥ 1 − ε t→∞ whenever |x0| < δ0, or for any β ∈ (0, h/2),

P{lim sup |x(t; x0)| ≤ β} ≥ 1 − ε. t→∞

By the previous theorem, we can ﬁnd a δ0 = δ0(ε) ∈ (0, h/2) such that ε P{|x(t; x )| < h/2} ≥ 1 − . (1.1) 0 2 whenever x0 ∈ Sδ0 .

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Moreover, in the same way as the previous theorem was proved, we can show that for any β ∈ (0, h/2), there is a α ∈ (0, β) such that ε P{|x(t; x )| < β for all t ≥ s} ≥ 1 − (1.2) 0 2 whenever |x(s; x0)| ≤ α and s ≥ 0. Now ﬁx any x0 ∈ Sδ and write x(t; x0) = x(t) simply. Deﬁne

τα = inf{t ≥ 0 : |x(t)| ≤ α} and τh = inf{t ≥ 0 : |x(t)| ≥ h/2}.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

By Itô’s formula and the conditions, we can show that

Z τα∧τh∧t 0 ≤ V (x0, 0) + E LV (x(s), s)ds 0 ≤ V (x0, 0) − µ3(α)E(τα ∧ τh ∧ t).

Consequently

tµ3(α)P{τα ∧ τh ≥ t} ≤ E(τα ∧ τh ∧ t) ≤ V (x0, 0).

This implies immediately that

P{τα ∧ τh < ∞} = 1.

But, by (1.1), P{τh < ∞} ≤ ε/2. Hence

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

ε 1 = P{τ ∧τ < ∞} ≤ P{τ < ∞}+P{τ < ∞} ≤ P{τ < ∞}+ , α h α h α 2 which yields ε P{τ < ∞} ≥ 1 − . α 2 We now compute, using (1.2),

P{lim sup |x(t)| ≤ β} t→∞ ≥ P{τα < ∞ and |x(t)| ≤ β for all t ≥ τα}

= P{τα < ∞}P{|x(t)| ≤ β for all t ≥ τα |τα < ∞ } 2 ≥ P{τα < ∞}(1 − ε/2) ≥ (1 − ε/2) ≥ 1 − ε as required.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large

Theorem 2,1 d Assume that there exist V ∈ C (R × R+; R+) and µ1, µ2 ∈ K∞ and µ3 ∈ K such that

µ1(|x|) ≤ V (x, t) ≤ µ2(|x|) and LV (x, t) ≤ −µ3(|x|) d for all (x, t) ∈ R × R+. Then the trivial solution of the SDE is stochastically asymptotically stable in the large.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic asymptotic stability in the large Proof. Clearly, we only need to show

P{ lim x(t; x0) = 0} = 1 t→∞

d for all x0 ∈ R , or for any pair of ε ∈ (0, 1) and β > 0,

P{lim sup |x(t; x0)| ≤ β} ≥ 1 − ε. t→∞

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 2V (x , 0) µ (h/2) ≥ 0 . 1 ε As in the previous proof, define the stopping time

τh = inf{t ≥ 0 : |x(t)| ≥ h/2}.

Xuerong Mao FRSE Stability of SDE Stochastic stability Theory Stochastic asymptotic stability Examples Stochastic 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).

But EV (x(τh ∧ t), τh ∧ t) ≥ µ1(h/2)P{τh ≤ t}. ε Hence P{τh ≤ t} ≤ 2 . Letting t → ∞ gives P{τh < ∞} ≤ ε/2. That is ε P{|x(t)| < h/2 for all t ≥ 0} ≥ 1 − , 2 which is the same as (1.1). From here, we can show in the same way as in the previous proof that

P{lim sup |x(t)| ≤ β} ≥ 1 − ε t→∞ as desired.

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-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+ → R m and g : R × R+ → R have the expansions

T f (x, t) = a(t)x+o(|x|), g(x, t) = (b1(t)x, ··· , bm(t)x) +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-valued functions. We impose a condition that there is a pair of positive constants θ and K such that

m Z t 1 X −K ≤ a(s) − b2(s) + θ ds ≤ K for all t ≥ 0. 2 i 0 i=1

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs

Let θ 0 < ε < Pm 2 supt≥0 i=1 bi (t) and deﬁne the Lyapunov function

m h Z t 1 X i V (x, t) = |x|ε exp − ε a(s) − b2(s) + θ ds . 2 i 0 i=1 Then, by the condition,

|x|εe−εK ≤ V (x, t) ≤ |x|εeεK .

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs

Moreover, compute

m Z t 1 X LV (x, t) = ε|x|ε exp −ε a(s) − b2(s) + θ ds 2 i 0 i=1 m ε X × b2(t) − θ + o(|x|ε) 2 i i=1 1 ≤ − εθe−εK |x|ε + o(|x|ε). 2 We hence see that LV (x, t) is negative-deﬁnite in a sufﬁciently small neighbourhood of x = 0 for t ≥ 0. We can therefore conclude that the trivial solution of the scale SDE is stochastically asymptotically stable.

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs Outline

1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large

2 Examples Scale SDEs Multi-dimensional SDEs

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs

Assume that the coefﬁcients f and g of the underlying SDE have 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-measurable d × d-matrix-valued functions. Assume that there is a symmetric positive-deﬁnite matrix Q such that

m T X T λmax QF(t) + F (t)Q + Gi (t)QGi (t) ≤ −λ < 0 i=1 for all t ≥ 0, where (and throughout this course) λmax(A) denotes the largest eigenvalue of matrix A.

Xuerong Mao FRSE Stability of SDE Theory Scale SDEs Examples Multi-dimensional SDEs

Now, deﬁne the Lyapunov function V (x, t) = xT Qx. Clearly,

2 2 λmin(Q)|x| ≤ V (x, t) ≤ λmax(Q)|x| .

Moreover,

m T T X T 2 LV (x, t) = x QF(t) + F (t)Q + Gi (t)QGi (t) x + o(|x| ) i=1 ≤ −λ|x|2 + o(|x|2).

Hence LV (x, t) is negative-deﬁnite in a sufﬁciently small neighbourhood of x = 0 for t ≥ 0. We therefore conclude that the trivial solution of the SDE is stochastically asymptotically stable.

Xuerong Mao FRSE Stability of SDE Theory Examples

Stability of Stochastic Differential Equations Part 3: Almost Sure Exponential Stability

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE Theory Examples Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Examples Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

In this part, we shall develop the classical Lyapunov method to study the almost sure exponential stability. In contrast to the classical Lyapunov stability theory, we will no longer require LV (x, t) be negative-deﬁnite but we still obtain the almost sure exponential stability making full use of the diffusion (noise) terms. It is this new feature that makes the stochastic stability more interesting and more useful as well.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost 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 the local Lipschitz condition and the linear growth condition and, moreover, f (0, t) ≡ 0, g(0, t) ≡ 0. Under these standing hypotheses, we have the following useful lemma.

Lemma d For all x0 6= 0 in R

P{x(t; x0) 6= 0 for all t ≥ 0} = 1.

That is, almost all the sample path of any solution starting from a non-zero state will never reach the origin with probability 1.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

Proof. If the lemma were false, there would exist some x0 6= 0 such that P{τ < ∞} > 0, where

τ = inf{t ≥ 0 : x(t) = 0} in which we write x(t; x0) = x(t) simply. So we can ﬁnd a pair of constants 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 positive constant Kθ such that

|f (x, t)| ∨ |g(x, t)| ≤ Kθ|x| for all |x| ≤ θ, 0 ≤ t ≤ T .

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost 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) 1 + −|x|−3|g(x, t)|2 + 3|x|−5|xT g(x, t)|2 2 ≤ |x|−2|f (x, t)| + |x|−3|g(x, t)|2 −1 2 −1 ≤ Kθ|x| + Kθ |x| = Kθ(1 + Kθ)V (x, t).

Now, for any ε ∈ (0, |x0|), deﬁne the stopping time

τε = inf{t ≥ 0 : |x(t)| 6∈ (ε, θ)}.

By Itô’s formula,

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

h i −Kθ(1+Kθ)(τε∧T ) E e V (x(τε ∧ T ), τε ∧ T ) − V (x0, 0) Z τε∧T h i −Kθ(1+Kθ)s = E e −(Kθ(1 + Kθ))V (x(s), s) + LV (x(s), s) ds 0 ≤ 0.

Note that for ω ∈ B, τε ≤ T and |x(τε)| = ε. The above inequality therefore implies that h i −Kθ(1+Kθ)T −1 −1 E e ε IB ≤ |x0| .

−1 Kθ(1+Kθ)T Hence P(B) ≤ ε|x0| e . Letting ε → 0 yields that P(B) = 0, but this contradicts the deﬁnition of B. The proof is complete.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

We will also need the well-known exponential martingale inequality which we state here as a lemma.

Lemma 2 1×m Let g = (g1, ··· , gm) ∈ L (R+; R ), and let T , α, β be any positive numbers. Then

Z t α Z t P sup g(s)dB(s) − |g(s)|2ds > β ≤ e−αβ. 0≤t≤T 0 2 0

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

Theorem 2,1 d Assume that there exists a function V ∈ C (R × R+; R+), and constants p > 0, c1 > 0, c2 ∈ R, c3 ≥ 0, such that for all x 6= 0 and t ≥ 0,

p c1|x| ≤ V (x, t),

LV (x, t) ≤ c2V (x, t), 2 2 |Vx (x, t)g(x, t)| ≥ c3V (x, t).

Then 1 c3 − 2c2 lim sup log |x(t; x0)| ≤ − a.s. (1.1) t→∞ t 2p d for all x0 ∈ R . In particular, if c3 > 2c2, then the trivial solution of the SDE is almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost 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= 0 for all t ≥ 0 almost surely. Thus, one can apply Itô’s formula and the condition to show that, for t ≥ 0,

log V (x(t), t) ≤ log V (x0, 0) + c2t + M(t) 1 Z t |V (x(s), s)g(x(s), s)|2 − x , 2 ds (1.2) 2 0 V (x(s), s) where Z t V (x(s), s)g(x(s), s) M(t) = x dB(s) 0 V (x(s), s) is a continuous martingale with initial value M(0) = 0. Assign ε ∈ (0, 1) arbitrarily and let n = 1, 2, ··· . By the exponential martingale inequality,

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

ε Z t |V (x(s), s)g(x(s), s)|2 2 1 ( )− x > ≤ . P sup M t 2 ds log n 2 0≤t≤n 2 0 V (x(s), s) ε n

Applying the Borel–Cantelli lemma we see that for almost all ω ∈ Ω, there is an integer n0 = n0(ω) such that if n ≥ n0,

2 ε Z t |V (x(s), s)g(x(s), s)|2 ( ) ≤ + x M t log n 2 ds ε 2 0 V (x(s), s) holds for all 0 ≤ t ≤ n. Substituting this into (1.2) and then using the condition we obtain that

1 2 log V (x(t), t) ≤ log V (x , 0) − [(1 − ε)c − 2c ]t + log n 0 2 3 2 ε for all 0 ≤ t ≤ n, n ≥ n0 almost surely.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

Consequently, for almost all ω ∈ Ω, if n − 1 ≤ t ≤ n and n ≥ n0,

1 1 log V (x , 0) + 2 log n log V (x(t), t) ≤ − [(1 − ε)c − 2c ] + 0 ε . t 2 3 2 n − 1 This implies

1 1 lim sup log V (x(t), t) ≤ − [(1 − ε)c3 − 2c2] a.s. t→∞ t 2 Hence 1 (1 − ε)c − 2c lim sup log |x(t)| ≤ − 3 2 a.s. t→∞ t 2p and the required assertion follows since ε > 0 is arbitrary.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

Theorem 2,1 d Assume that there exists a function V ∈ C (R × R+; R+), and constants p > 0, c1 > 0, c2 ∈ R, c3 > 0, such that for all x 6= 0 and t ≥ 0,

p c1|x| ≥ V (x, t) > 0,

LV (x, t) ≥ c2V (x, t), 2 2 |Vx (x, t)g(x, t)| ≤ c3V (x, t).

Then 1 2c2 − c3 lim inf log |x(t; x0)| ≥ a.s. t→∞ t 2p d for all x0 6= 0 in R . In particular, if 2c2 > c3, then almost all the sample paths of |x(t; x0)| will tend to inﬁnity exponentially.

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

Proof. Fix any x0 6= 0 and write x(t; x0) = x(t). By Itô’s formula and the conditions, we can easily show that for t ≥ 0,

1 log V (x(t), t) ≥ log V (x , 0) + (2c − c )t + M(t), (1.3) 0 2 2 3 where Z t V (x(s), s)g(x(s), s) M(t) = x dB(s) 0 V (x(s), s) is a continuous martingale with the quadratic variation

Z t |V (x(s), s)g(x(s), s)|2 h ( ), ( )i = x ≤ . M t M t 2 ds c3t 0 V (x(s), s)

Xuerong Mao FRSE Stability of SDE Theory Almost sure exponential stability Examples Almost sure exponential instability

By the strong law of large numbers for the martingales,

M(t) lim = 0 a.s. t→∞ t It therefore follows from (1.3) that

1 1 lim inf log V (x(t), t) ≥ (2c2 − c3) a.s. t→∞ t 2 which implies the required assertion immediately by using the condition.

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case

Consider the scalar linear SDE

m X dx(t) = ax(t) + bi x(t)dBi (t) on t ≥ 0. i=1 It is known that it has the explicit solution

m m X 2 X x(t) = x0 exp [a − 0.5 bi ]t + bi Bi (t) . i=1 i=1

This implies that, for x0 6= 0,

m 1 X 2 lim log |x(t)| = a − 0.5 bi a.s. t→∞ t i=1

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case

Let us now apply the stability theorem to obtain the same conclusion. Let V (x, t) = x2. Then

m X 2 2 LV (x, t) = 2a + bi |x| i=1 and, writing g(x, t) = (b1x, ··· , bmx),

m 2 X 2 4 |Vx (x, t)g(x, t)| = 4 bi |x| . i=1 In other words, the conditions in the Theorems holds with

m m X 2 X 2 p = 2, c1 = 1, c2 = 2a + bi , c3 = 4 bi . i=1 i=1

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case

We hence have

m 1 1 X 2 lim sup log |x(t)| ≤ a − bi a.s. →∞ t 2 t i=1 and m 1 1 X 2 lim inf log |x(t)| ≥ a − bi a.s. t→∞ t 2 i=1 Combining these gives what we want.

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear 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-dimensional Brownian motion. The theory above shows that the solution of this linear sde obeys

1 lim inf log |x(t)| = −1 a.s. t→∞ t The following simulation shows a typical sample path of the solution with initial value x(0) = 10.

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case 60 50 40 30 x(t) 20 10 0

0 2 4 6 8 10

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case Outline

1 Theory Almost sure exponential stability Almost sure exponential instability

2 Examples Linear SDEs Nonlinear case

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case

Consider the two-dimensional SDE

dx(t) = f (x(t))dt + Gx(t)dB(t) on t ≥ 0

2 with initial value x(0) = x0 ∈ R and x0 6= 0, where B(t) is a one-dimensional Brownian motion,

x cos x 3 −0.3 f (x) = 2 1 , G = 2x1 sin x2 −0.3 3

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case

Let V (x, t) = |x|2. It is easy to verify that

2 2 2 4.29|x| ≤ LV (x, t) = 2x1x2 cos x1+4x1x2 sin x2+|Gx| ≤ 13.89|x| and

2 2 T 2 4 29.16|x| ≤ |Vx (x, t)Gx| = |2x Gx| ≤ 43.56|x| .

Applying the Theorems we then have

1 1 −8.745 ≤ lim inf log |x(t; x0)| ≤ lim sup log |x(t; x0)| ≤ −0.345 t→∞ t t→∞ t almost surely. The following ﬁgure is a compute simulation.

Xuerong Mao FRSE Stability of SDE Theory Linear SDEs Examples Nonlinear case 4 6 3 4 X1(t) X2(t) 2 2 1 0 0

0 2 4 6 8 10 0 2 4 6 8 10

t t

x1(0) = x2(0) = 1.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Stability of Stochastic Differential Equations Part 4: Moment Exponential Stability

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study Outline

1 Moment verse Almost Sure Exponential Stability

2 Criteria Nonlinear case Linear case

3 A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study Outline

1 Moment verse Almost Sure Exponential Stability

2 Criteria Nonlinear case Linear case

3 A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study Outline

1 Moment verse Almost Sure Exponential Stability

2 Criteria Nonlinear case Linear case

3 A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Generally speaking, the pth moment exponential stability and the almost sure exponential stability do not imply each other and additional conditions are required in order to deduce one from the other. The following theorem gives the conditions under which the pth moment exponential stability implies the almost sure exponential stability. However we still do not know under what conditions the almost sure exponential stability implies the pth moment exponential stability.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Theorem Assume that there is a positive constant K such that

T 2 2 d x f (x, t) ∨ |g(x, t)| ≤ K |x| for all (x, t) ∈ R × R+.

Then the pth moment exponential stability of the trivial solution of the SDE implies the almost sure exponential stability.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

To prove this theorem we need the Burkholder–Davis–Gundy inequality which we cite as a lemma. Lemma 2 d×m Let g ∈ L (R+; R ). Deﬁne, for t ≥ 0,

Z t Z t x(t) = g(s)dB(s) and A(t) = |g(s)|2ds. 0 0 Then for every p > 0, there exist universal positive constants cp, Cp (depending only on p), such that

p p p cpE|A(t)| 2 ≤ E sup |x(s)| ≤ CpE|A(t)| 2 . 0≤s≤t

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

In particular, one may take

p p/2 cp = (p/2) , Cp = (32/p) if 0 < p < 2;

cp = 1, Cp = 4 if p = 2; −p/2 p+1 p−1p/2 cp = (2p) , Cp = p /2(p − 1) if p > 2.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

d Proof of the theorem. Fix any x0 6= 0 in R and write x(t; x0) = x(t) simply. By the deﬁnition of the pth moment exponential stability, there is a pair of positive constants and C such that E|x(t)|p ≤ Ce−λt on t ≥ 0. Let n = 1, 2, ··· . By Itô’s formula and the condition, one can show that for n − 1 ≤ t ≤ n,

Z t p p p |x(t)| ≤ |x(n − 1)| + c1 |x(s)| ds n−1 Z t + p|x(s)|p−2xT (s)g(x(s), s)dB(s), n−1 where c1 = pK + p(1 + |p − 2|)K /2. Hence

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Z n p p p E sup |x(t)| ≤ E|x(n − 1)| + c1 E|x(s)| ds n−1≤t≤n n−1 Z t +E sup p|x(s)|p−2xT (s)g(x(s), s)dB(s) . n−1≤t≤n n−1

On the other hand, by the well-known Burkholder–Davis–Gundy inequality we compute

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Z t E sup p|x(s)|p−2xT (s)g(x(s), s)dB(s) n−1≤t≤n n−1 1 √ Z n 2 ≤ 4 2E p2|x(s)|2(p−2)|xT (s)g(x(s), s)|2ds n−1 1 √ Z n 2 ≤ 4 2E sup |x(s)|p p2K |x(s)|pds n−1≤s≤n n−1 1 Z n ≤ E sup |x(s)|p + 16p2K E|x(s)|pds. 2 n−1≤s≤n n−1

Substituting this into the previous inequality yields

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Z n p p p E sup |x(t)| ≤ 2E|x(n − 1)| + c2 E|x(s)| ds, n−1≤t≤n n−1

2 where c2 = 2c1 + 32p K . By the property of the pth moment exponential stability, we then have

p −λ(n−1) E sup |x(t)| ≤ c3e , n−1≤t≤n where c3 = C(2 + c2). Now, let ε ∈ (0, λ) be arbitrary. Then

n p −(λ−ε)(n−1)o −ε(n−1) P sup |x(t)| > e ≤ c3e . n−1≤t≤n

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

In view of the Borel–Cantelli lemma we see that for almost all ω ∈ Ω, sup |x(t)|p ≤ e−(λ−ε)(n−1) n−1≤t≤n holds for all but ﬁnitely many n. Hence, there exists an n0 = n0(ω), for all ω ∈ Ω excluding a P-null set, for which the inequality above holds whenever n ≥ n0. Consequently, for almost all ω ∈ Ω, 1 1 (λ − ε)(n − 1) log |x(t)| = log(|x(t)|p) ≤ − t pt pn if n − 1 ≤ t ≤ n, n ≥ n0.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Hence 1 (λ − ε) lim sup log |x(t)| ≤ − a.s. t→∞ t p Since ε > 0 is arbitrary, we must have

1 λ lim sup log |x(t)| ≤ − a.s. t→∞ t p By deﬁnition, the trivial solution of the SDE is almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study Outline

1 Moment verse Almost Sure Exponential Stability

2 Criteria Nonlinear case Linear case

3 A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Theorem 2,1 d Assume that there is a function V (∈ C (R × R+; R+), and positive constants c1–c3, such that

p p c1|x| ≤ V (x, t) ≤ c2|x| and LV (x, t) ≤ −c3V (x, t)

d for all (x, t) ∈ R × R+. Then c p 2 p −c3t E|x(t; x0)| ≤ |x0| e on t ≥ 0 c1

d for all x0 ∈ R .

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

d Proof. Fix any x0 ∈ R and write x(t; x0) = x(t). For each n ≥ |x0|, deﬁne the stopping time

τn = inf{t ≥ 0 : |x(t)| ≥ n}.

Obviously, τn → ∞ as n → ∞ almost surely. By Itô’s formula, we can derive that for t ≥ 0, h i c3(t∧τn) E e V (x(t ∧ τn), t ∧ τn) − V (x0, 0)

Z t∧τn c3s = E e c3V (x(s), s) + LV (x(s), s) ds ≤ 0 0

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Hence h i c3(t∧τn) p p c1E e E|x(t ∧ τn)| ≤ V (x0, 0) ≤ c2|x0| .

Letting n → ∞ yields that

c3t p p c1e E|x(t)| ≤ c2|x0| which implies the desired assertion.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Theorem Assume that there exists a symmetric positive-deﬁnite d × d matrix Q, and constants α1 ∈ R, 0 ≤ α2 < α3, such that for all d (x, t) ∈ R × R+, 1 xT Qf (x, t) + trace[gT (x, t)Qg(x, t)] ≤ α xT Qx 2 1 and T T T α2x Qx ≤ |x Qg(x, t)| ≤ α3x Qx.

(i) If α1 < 0, then the trivial solution of the SDE is pth moment 2 exponentially stable provided p < 2 + 2|α1|/α3. 2 (ii) If 0 ≤ α1 < α2, then the trivial solution of equation (1.2) is 2 pth moment exponentially stable provided p < 2 − 2α1/α2.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

T p Proof. Let V (x, t) = (x Qx) 2 . Then

p p 2 p 2 p λmin(Q)|x| ≤ V (x, t) ≤ λmax(Q)|x| . It is also easy to verify that

T p −1 T 1 T LV (x, t) = p(x Qx) 2 x Qf (x, t) + trace[g (x, t)Qg(x, t)] 2 p T p −2 T 2 + p − 1 (x Qx) 2 |x Qg(x, t)| . 2

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

2 (i) Assume that α1 < 0 and p < 2 + 2|α1|/α3. Without loss of generality, we can let p ≥ 2. Then h p i LV (x, t) ≤ −p |α | − − 1 α2 V (x, t). 1 2 3

2 2 (ii) Assume that 0 ≤ α1 < α2 and p < 2 − 2α1/α2. Then hp i LV (x, t) ≤ −p − 1 α2 − α V (x, t). 2 2 1 So in both cases the stability assertion follows from the previous theorem.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study Outline

1 Moment verse Almost Sure Exponential Stability

2 Criteria Nonlinear case Linear case

3 A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Consider a d-dimensional linear SDE

m X dx(t) = Fx(t)dt + Gi x(t)dBi (t), i=1

d×d where F, Gi ∈ R . This is of course a special case of the underlying SDE where

f (x, t) = Fx, g(x, t) = (G1x, ··· , Gmx).

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Corollary Assume that there exists a symmetric positive-deﬁnite d × d matrix Q such that the following LMI holds:

m T X T QF + F Q + Gi QGi < 0. i=1 Then the trivial solution of the linear SDE is mean-square exponentially stable as well as almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Proof. Let V (x, t) = xT Qx. Then

2 2 λmin(Q)|x| ≤ V (x, t) ≤ λmax(Q)|x| .

Moreover

T ¯ ¯ 2 LV (x, t) = x Qx ≤ λmax(Q)|x| .

¯ T Pm T where Q = QF + F Q + i=1 Gi QGi . By the condition, ¯ λmax(Q) < 0. Hence

|λ (Q¯ )| LV (x, t) ≤ − max V (x, t). λmax(Q) The assertions follow therefore from the theory established above.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

In the case when

m T X T QF + F Q + Gi QGi i=1 is not negative-deﬁnite, the following result is useful.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Corollary Assume that there exists a symmetric positive-deﬁnite d × d matrix Q, and nonnegative constants β and βi (1 ≤ i ≤ m), Pm such that β < i=1 βi ,

m T X T QF + F Q + Gi QGi − βQ ≤ 0, i=1 and, moreover, for each i = 1, ··· , m,

T p T p either QGi + Gi Q − 2βi Q ≥ 0 or QGi + Gi Q + 2βi Q ≤ 0. Pm If 0 < p < 2 − 2β/( i=1 βi ), then the trivial solution of the linear SDE is pth moment exponentially stable, whence it is also almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Proof. We will use the 2nd theorem established above to show this corollary. We ﬁrst have that

1 xT Qf (x, t) + trace[gT (x, t)Qg(x, t)] 2

m T T X T T = 0.5x QF + F Q + Gi QGi x ≤ 0.5βx Qx. i=1 We also observe from the condition that for each i,

T 2 T T 2 T 2 |x QGi x| = 0.25|x (QGi + Gi Q)x| ≥ 0.5βi (x Qx) .

Hence

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

v v u m u m T uX T 2 u X T |x Qg(x, t)| = t |x QGi x| ≥ t0.5 βi x Qx. i=1 i=1 Applying the theorem with v u m u X α1 = 0.5β, α2 = t0.5 βi , i=1 we can therefore conclude that the trivial solution of the linear SDE is pth moment exponentially stable if Pm 0 < p < 2 − 2β/( i=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 Stability Nonlinear case Criteria Linear case A Case Study

As an even more special case, let us consider the scalar linear SDE m X dx(t) = ax(t)dt + bi x(t)dBi (t), i=1 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 Stability Nonlinear case Criteria Linear case A Case Study

Pm 2 If 2a + i=1 bi < 0, then the trivial solution of this scalar linear SDE is mean-square exponentially stable as well as almost surely exponentially stable. Pm 2 Pm 2 If 0 ≤ 2a + i=1 bi < 2 i=1 bi , then the trivial solution of this scalar linear SDE is pth moment exponentially stable provided 2a 0 < p < 1 − , Pm 2 i=1 bi whence it is also almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Nonlinear case Criteria Linear case A Case Study

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

This example is from the satellite dynamics. Sagirow in 1970 derived 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 of the atmosphere of the earth on the motion of a satellite in a circular orbit. Here B˙ (t) is a scalar white noise, α is a constant representing the intensity of the disturbance, and β, γ are two positive constants.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

T T Introducing x = (x1, x2) = (y, y˙ ) , we can write this equation as 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 a quadratic form and integral of the nonlinear component:

Z x1 2 2 V (x, t) = ax1 + bx1x2 + x2 + c sin(2y)dy 0 2 2 2 = ax1 + bx1x2 + x2 + c sin x1.

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

This yields

2 2 2 2 2 LV (x, t) = −(b − α )x1 + bγx1 sin(2x1) − (2β − b − α β )x2 2 + (2a − bβ − 2 + 2α β)x1x2 + (c + 2γ)x2 sin(2x1).

Setting 2a − bβ − 2 + 2α2β = 0 and c + 2γ = 0 we obtain

1 V (x, t) = (bβ + 2 − 2α2β)x2 + bx x + x2 − 2γ sin2 x 2 1 1 2 2 1 and

2 2 2 2 2 LV (x, t) = −(b − α )x1 + bγx1 sin(2x1) − (2β − b − α β )x2 .

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

Note that 1 V (x, t) ≥ (bβ + 2 − 2α2β − 4γ)x2 + bx x + x2. 2 1 1 2 2 So V (x, t) ≥ ε|x|2 for some ε > 0 if

2(bβ + 2 − 2α2β − 4γ) ≥ b2 or equivalently p p β − β2 + 4 − 8γ − 4α2β < b < β + β2 + 4 − 8γ − 4α2β.

Note also that

2 2 2 2 2 LV (x, t) ≤ −(b − α − 2bγ)x1 − (2β − b − α β )x2 .

Xuerong Mao FRSE Stability of SDE Moment verse Almost Sure Exponential Stability Criteria A Case Study

So LV (x, t) ≤ −ε¯|x|2 for some ε¯ > 0 provided both b − 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 n p o max α2/(1 − 2γ), β − β2 + 4 − 8γ − 4α2β

n p o < min 2β − α2β2, β + β2 + 4 − 8γ − 4α2β then the trivial solution of the SDE is exponentially stable in mean square.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Stochastic Stabilization Stochastic Destabilization

Stability of Stochastic Differential Equations Part 5: Stochastic Stabilization and Destabilization

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

It is not surprising that noise can destabilize a stable system.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Consider a 2-dimensional ODE

2 y˙ (t) = −y(t) on t ≥ 0, y(0) = y0 ∈ R .

This is an exponentially stable system. Perturb it by noise and assume the stochastically perturbed system is described by an SDE

2 dx(t) = −x(t)dt + Gx(t)dB(t) on t ≥ 0, x(0) = y0 ∈ R , where B(t) is a scalar Brownian motion and

0 −2 G = 2 0

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A 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

1 lim log(|x(t)|) = 1 a.s. t→∞ t That is, the stochastically perturbed system has become unstable with probability one.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

x1(t) x2(t) 8

y1(t) 10 y2(t) 6 4 5 2 x1(t) or y1(t) x2(t) or y2(t) 0 0 −2 −4

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

t t

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Do you believe that noise can also stabilize an unstable system?

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Consider the scalar ODE

y˙ (t) = y(t) on t ≥ 0, y(0) = y0 ∈ R.

The solution is y(t) = y(0)et . So |y(t)| → ∞ if y(0) 6= 0. That is, the ODE is an exponentially unstable system. Perturb it by noise and assume the stochastically 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 History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

The SDE has the explicit solution

x(t) = x(0) exp[(1 − 0.5σ2)t + σB(t)].

Consequently √ x(t) → 0 a.s. if σ > 2. That is, the stochastically perturbed system has become stable with probability one.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history 7 x(t) y(t) 6 5 4 3 x(t) or y(t) 2 1 0

0 2 4 6 8 10

σ = 2

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Of course, if the noise is not strong enough, it will not be able to stabilize the system.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

500 x(t) y(t) 400 300 x(t) or y(t) 200 100 0

0 2 4 6 8 10

σ = 0.5

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

7 x(t) y(t) 6 5 4 3 x(t) or y(t) 2 1 0

0 2 4 6 8 10

t √ σ = 2

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Has’minskii (1969): The pioneering work where two white noise sources were used to stabilize a particular system. Arnold, Crauel & Wihstutz (1983) and Arnold (1990): Any linear system x˙ (t) = Ax(t) with trace(A) < 0 can be stabilized by one real noise source. Scheutzow (1993): Stochastic stabilization for two special nonlinear systems. Mao (1994): The general theory on stochastic stabilization for nonlinear SDEs. Mao (1996): Design a stochastic control that can self-stabilize the underlying system. Caraballo, Liu and Mao (2001): Stochastic stabilization for partial differential equations (PDEs). Appleby and Mao (2004): Stochastic stabilization for functional differential equations.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Destabilization Stochastic Stabilization Stabilization Stochastic Destabilization A brief history

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Suppose that the given system is described by a nonlinear ODE

d y˙ (t) = f (y(t), t) on t ≥ 0, y(0) = x0 ∈ R .

d d Here f : R × R+ → R is a locally Lipschitz continuous function and particularly, for some K > 0,

d |f (x, t)| ≤ K |x| for all (x, t) ∈ R × R+.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

We now use the m-dimensional Brownian motion T B(t) = (B1(t), ··· , Bm(t)) as the source of noise to perturb the given system. For simplicity, suppose the stochastic perturbation is of a linear form, that is the stochastically perturbed system is described by the semi-linear Itô equation

m X d dx(t) = f (x(t), t)dt+ Gi x(t)dBi (t) on t ≥ 0, x(0) = x0 ∈ R , i=1 where Gi , 1 ≤ i ≤ m, are all d × d matrices.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Theorem Assume that there are two constants λ > 0 and ρ ≥ 0 such that

m m X 2 2 X T 2 4 |Gi x| ≤ λ|x| and |x Gi x| ≥ ρ|x| i=1 i=1 for all x ∈ Rd . Then 1 λ lim sup log |x(t)| ≤ − ρ − K − a.s. t→∞ t 2

d 1 for all x0 ∈ R . In particular, if ρ > K + 2 λ, then the stochastically perturbed system is almost surely exponentially stable.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization Example Let Gi = σi I for 1 ≤ i ≤ m, where I is the d × d identity matrix and σi a constant. Then the SDE becomes

m X dx(t) = f (x(t), t)dt + σi x(t)dBi (t). i=1 Moreover,

m m m m X 2 X 2 2 X T 2 X 2 4 |Gi x| = σi |x| and |x Gi x| = σi |x| . i=1 i=1 i=1 i=1 The solution of the SDE has the property

m 1 1 X 2 lim sup log |x(t)| ≤ − σi − K a.s. →∞ t 2 t i=1

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Therefore, The SDE is almost surely exponentially stable provided m 1 X σ2 > K . 2 i i=1

An even simpler case is that when σi = 0 for 2 ≤ i ≤ m, i.e. the SDE dx(t) = f (x(t), t)dt + σ1x(t)dB1(t). This SDE is almost surely exponentially stable provided 1 2 2 σ1 > 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 History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

The Theorem above ensures that there are many choices for the matrices Gi in order to stabilize a given system and of course the above choices are just the simplest ones. For illustration, we give one more example here.

For each i, choose a positive-deﬁnite matrix Di such that √ 3 xT D x ≥ ||D || |x|2. i 2 i Obviously, there are many such matrices. Let σ be a constant and Gi = σDi . Then

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

m m X 2 2 X 2 2 |Gi x| ≤ σ ||Di || |x| i=1 i=1 and m m X 3σ2 X |xT G x|2 ≥ ||D ||2|x|3. i 4 i i=1 i=1 By the Theorem, the solution of the SDE satisﬁes

2 m 1 σ X 2 lim sup log |x(t)| ≤ − ||Di || − K a.s. →∞ t 4 t i=1 The SDE is therefore almost surely exponentially stable if 4K σ2 > . Pm 2 i=1 ||Di ||

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Theorem Any nonlinear system y˙ (t) = f (y(t), t) can be stabilized by Brownian motions provided the following condition is fulﬁlled

d |f (x, t)| ≤ K |x| for all (x, t) ∈ R × R+.

Moreover, one can even use only a scalar Brownian motion to stabilize the system.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Example Given an unstable 2-dimensional ODE

y˙ (t) = f (y(t), t), where y cos(t) + y sin(y ) f (y, t) = 1 2 1 . y2 sin(t) + y1 cos(y2) It is easy to see

2 |f (y, t)| ≤ 2|y| ∀(y, t) ∈ R × R+.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

Perturbing this ODE by a scalar Brownian motion results in an SDE dx(t) = f (x(t), t)dt + σ1x(t)dB1(t). The Theorem above shows that this SDE is almost surely exponentially stable provided

σ1 > 2.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

x1(t) 4 x2(t) 6 y1(t) y2(t) 5 3 4 2 3 x1(t) or y1(t) x2(t) or y2(t) 2 1 1 0 0

0 1 2 3 4 5 0 1 2 3 4 5

t t

σ1 = 3

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization 4 7 x1(t) x2(t) y1(t) y2(t) 6 3 5 4 2 3 x1(t) or y1(t) x2(t) or y2(t) 2 1 1 0 0

0 1 2 3 4 5 0 1 2 3 4 5

t t

σ1 = 2.5

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

x1(t) x2(t) 8 y1(t) 8 y2(t) 6 6 4 4 x1(t) or y1(t) x2(t) or y2(t) 2 2 0 0

0 1 2 3 4 5 0 1 2 3 4 5

t t

σ1 = 2

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Examples and Simulations Stochastic Destabilization

x1(t) x2(t) 3.5 y1(t) 4 y2(t) 3.0 3 2.5 2.0 2 x1(t) or y1(t) x2(t) or y2(t) 1.5 1 1.0

0 1 2 3 4 5 0 1 2 3 4 5

t t

σ1 = 1.5

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Theorem Assume that there are two positive constants λ and ρ such that

m m X 2 2 X T 2 4 |Gi x| ≥ λ|x| and |x Gi x| ≤ ρ|x| i=1 i=1 for all x ∈ Rd . Then 1 λ lim inf log |x(t)| ≥ − K − ρ a.s. t→∞ t 2 for all x0 6= 0. In particular, if λ > 2(K + ρ), then the SDE is almost surely exponentially unstable.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Can we ﬁnd the matrices Gi as described in the theorem above in order to destabilize the given ODE?

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization Outline

1 Motivating Examples and History Destabilization Stabilization A brief history

2 Stochastic Stabilization Theory Examples and Simulations

3 Stochastic Destabilization Theory Case study and simulations

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Case 1. d ≥ 3 Choose the dimension of the Brownian motion m = d. Let σ be a constant. For each i = 1, 2, ··· , d − 1, deﬁne the d × d matrix i i Gi = (guv ) by guv = σ if u = i and v = i + 1 or otherwise i d d guv = 0. Moreover, deﬁne Gd = (guv ) by guv = σ if u = d and d v = 1 or otherwise guv = 0. Then SDE becomes   x2(t)dB1(t)  .   .  dx(t) = f (x(t), t)dt + σ   . xd (t)dBd−1(t) x1(t)dBd (t)

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization Compute that m m X 2 X 2 2 2 |Gi x| = (σxi+1) = σ |x| i=1 i=1 and m m X T 2 2 X 2 2 |x Gi x| = σ xi xi+1, i=1 i=1 where we use xd+1 = x1. Noting m m m X 1 X X x2x2 ≤ (x4 + x4 ) = x4, i i+1 2 i i+1 i i=1 i=1 i=1 we have m m m X 2 2 X 2 2 X 4 4 3 xi xi+1 ≤ 2 xi xi+1 + xi ≤ |x| . i=1 i=1 i=1

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Therefore m X σ2 |xT G x|2 ≤ |x|4. i 3 i=1 By the theorem, the solution of the SDE has the property that

1 σ2 σ2 σ2 lim inf log |x(t)| ≥ − K − = − K a.s. t→∞ t 2 3 6

2 for any x0 6= 0. If σ > 6K , then the SDE will be almost surely exponentially unstable.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Example Given a stable 3-dimensional ODE

y˙ (t) = f (y(t), t), where   −2y1 + sin(y2) f (y, t) = −2y2 + sin(y3) . −2y3 + sin(y1) It is easy to see

3 |f (y, t)| ≤ 3|y| ∀(y, t) ∈ R × R+.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Perturbing this ODE by a 3-dimensional Brownian motion results in an SDE   x2(t)dB1(t) dx(t) = f (x(t), t)dt + σ 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 History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

x1(t) x2(t) x3(t) y1(t) y2(t) y3(t) 3 3 2 2 1 2 1 0 1 −1 0 0 −2 −1 −3 −1

0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4

t t t σ = 5

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

x1(t) x2(t) x3(t) y1(t) y2(t) y3(t) 3e+13 4e+13 3e+13 2e+13 3e+13 2e+13 1e+13 2e+13 1e+13 0e+00 1e+13 0e+00 −1e+13 0e+00

0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

t t t σ = 4

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

x1(t) x2(t) x3(t) y1(t) y2(t) y3(t) 1.5 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 −0.5 −0.5

0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5

t t t σ = 2

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization Case 2. d is an even number Let d = 2k(k ≥ 1) and σ be a constant. Deﬁne  0 σ  0 −σ 0     ..  G1 =  .   0 σ  0  −σ 0 but set Gi = 0 for 2 ≤ i ≤ m. So the SDE becomes   x2(t)  −x1(t)     .  dx(t) = f (x(t), t)dt + σ  .  dB1(t).    x2k (t)  −x2k−1(t)

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

In this case we have

m m X 2 2 2 X T 2 |Gi x| = σ |x| and |x Gi x| = 0 i=1 i=1 Hence, the solution of SDE obeys

1 σ2 lim inf log |x(t)| ≥ − K a.s. t→∞ t 2

2 for any x0 6= 0. If σ > 2K , then the SDE will be almost surely exponentially unstable.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Example Given a stable 4-dimensional ODE

y˙ (t) = f (y(t), t), where   −2y1 + sin(y2) −2y2 + sin(y3) f (y, t) =   . −2y3 + sin(y4) −2y4 + sin(y1) It is easy to see

4 |f (y, t)| ≤ 3|y| ∀(y, t) ∈ R × R+.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Perturbing this ODE by a scale Brownian motion results in an SDE   x2(t) −x1(t) dx(t) = f (x(t), t)dt + σ   dB1(t).  x3(t)  −x4(t) This SDE is almost surely exponentially unstable provided √ σ > 6.

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization 6 6 x1(t) x3(t) 4 y1(t) 4 y3(t) 2 2 −2 −2 −6 −6

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

t t

x2(t) x4(t) y2(t) y4(t) 8 8 6 6 4 4 2 2 −2 −2

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

t t σ = 2.5

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization 4 4 x1(t) x3(t) 2 y1(t) 2 y3(t) 0 0 −4 −4 −8 −8

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

t t 6 6 x2(t) x4(t) 4 y2(t) 4 y4(t) 2 2 0 0 −4 −4

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

t t √ σ = 6

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

x1(t) x3(t) 1.0 y1(t) 1.0 y3(t) 0.6 0.6 0.2 0.2 −0.2 −0.2 0 1 2 3 4 5 0 1 2 3 4 5

t t

x2(t) x4(t) y2(t) y4(t) 0.8 0.8 0.4 0.4 0.0 0.0

0 1 2 3 4 5 0 1 2 3 4 5

t t σ = 1

Xuerong Mao FRSE Stability of SDE Motivating Examples and History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization Case 3. d = 1 Consider a stable scale linear ODE

y˙ (t) = −ay(t)(a > 0), and its perturbed linear SDE

m X dx(t) = −ax(t) + bi x(t)dBi (t). i=1 It is known that m 1 1 X 2 lim log |x(t)| = −a − bi < 0 a.s. t→∞ t 2 i=1 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 History Theory Stochastic Stabilization Case study and simulations Stochastic Destabilization

Theorem Any d-dimensional nonlinear system y˙ (t) = f (y(t), t) can be destabilized by Brownian motions provided the dimension of the state d ≥ 2 and

d |f (x, t)| ≤ K |x| for all (x, t) ∈ R × R+.

Xuerong Mao FRSE Stability of SDE The LaSalle-Type Theorems Mean-Square Exponential Stability of Numerical Methods Almost Sure Exponential Stability of Numerical Methods

Stability of Stochastic Differential Equations Part 6: New Developments

Xuerong Mao FRSE

Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH

December 2010

Xuerong Mao FRSE Stability of SDE The LaSalle-Type Theorems Mean-Square Exponential Stability of Numerical Methods Almost Sure Exponential Stability of Numerical Methods Outline

1 The LaSalle-Type Theorems Original theorem Improved results Examples

2 Mean-Square Exponential Stability of Numerical Methods Stability of numerical methods The Euler-Maruyama method Stability of the EM method

3 Almost Sure Exponential Stability of Numerical Methods Linear scalar SDEs Multi-Dimensional SDEs The Backward Euler-Maruyama (BEM) Method