Stability of Stochastic Differential Equations

Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

Outline Lyapunov stability theory for ODEs s 1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method 2 s SDEs Definition of stochastic stability Diffusion operator

Outline Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method 1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method 2 s SDEs Definition of stochastic stability Diffusion operator

Lyapunov stability theory for ODEs s Concept of stability 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.

Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method Consider a d-dimensional ordinary differential equation (ODE) dx(t) dt = f (x(t), t) on t 0, where f = (f 1,, f d ) T : R d R + R d. Assume that for every initial value x(0) = x 0 R d, there exists a unique global solution which is denoted by x(t; x 0 ). 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.

Lyapunov stability theory for ODEs s Concept of stability 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; x 0 ) < ε for all t 0. whenever x 0 < 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 δ 0 > 0 such that whenever x 0 < δ 0. lim x(t; x 0) = 0 t

Lyapunov stability theory for ODEs s Concept of stability 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.

Notation Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method 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 lim r µ(r) =. For h > 0, let S h = {x R d : x < h}. Let C 1,1 (S h R + ; R + ) denote the family of all continuous functions V (x, t) from S h R + to R + with continuous first partial derivatives with respect to every component of x and to t.

Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method Basic ideas of the Lyapunov method Let x(t) be a solution of the ODE and V C 1,1 (S h R + ; R + ). Then v(t) = V (x(t), t) represents a function of t with the derivative dv(t) = dt V (x(t), t), where V (x, t) = V t (x, t) + (V x1 (x, t),, V xd (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.

Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method Theorem Assume that there exist V C 1,1 (S h 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) S h R +. Then the trivial solution of the ODE is stable.

Lyapunov stability theory for ODEs s Concept of stability The Lyapunov method Theorem Assume that there exist V C 1,1 (S h R + ; R + ) and µ 1, µ 2, µ 3 K such that µ 1 ( x ) V (x, t) µ 2 ( x ) and V (x, t) µ 3 ( x ) for all (x, t) S h R +. Then the trivial solution of the ODE is asymptotically stable.

Outline Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability Diffusion operator 1 Lyapunov stability theory for ODEs Concept of stability The Lyapunov method 2 s SDEs Definition of stochastic stability Diffusion operator

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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, where f : R d R + R d and g : R d R + R d m, and B(t) = (B 1 (t),, B m (t)) 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. Hence, for any given initial value x(0) = x 0 R d, the SDE has a unique global solution denoted by x(t; x 0 ). 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.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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?

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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; x 0 ) < r for all t 0} 1 ε whenever x 0 < δ. Otherwise, it is said to be stochastically unstable.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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 δ 0 = δ 0 (ε) > 0 such that whenever x 0 < δ 0. P{ lim t x(t; x 0 ) = 0} 1 ε

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability Diffusion operator Definition The trivial solution is said to be stochastically asymptotically stable in the large if it is stochastically stable and, moreover, for all x 0 R d P{ lim t x(t; x 0 ) = 0} = 1.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability Diffusion operator Definition The trivial solution is said to be almost surely exponentially stable if for all x 0 R d lim sup t 1 t log( x(t; x 0) ) < 0 a.s. It is said to be pth moment exponentially stable if for all x 0 R d lim sup t 1 t log(e x(t; x 0) p ) < 0.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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 V C 2,1 (S h R + ; R + ), which denotes the family of all nonnegative functions V (x, t) defined on S h R + such that they are continuously twice differentiable in x and once in t.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability Diffusion operator By the Itô formula, we have dv (x(t), t) = LV (x(t), t)dt + V x (x(t), t)g(x(t), t)db(t), where LV (x, t) = V t (x, t)+v x (x, t)f (x, t)+ 1 ] [g 2 trace T (x, t)v xx (x, t)g(x, t), in which V x = (V x1,, V xd ) and V xx = (V xi x j ) d d.

Lyapunov stability theory for ODEs s SDEs Definition of stochastic stability 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.

Theory Examples Stability of Stochastic Differential Equations Part 2: Stability in Probability Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

Outline Theory Examples 1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large 2 Examples Scale SDEs Multi-dimensional SDEs

Theory Examples Stochastic stability Stochastic asymptotic stability 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.

Outline Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large 1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large 2 Examples Scale SDEs Multi-dimensional SDEs

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large Theorem Assume that there exist V C 2,1 (S h 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) S h R +. Then the trivial solution of the SDE is stochastically stable.

Theory Examples Stochastic stability Stochastic asymptotic stability 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 ε sup x S δ V (x, 0) µ(r). Now fix any x 0 S δ and write x(t; x 0 ) = x(t) simply. Define τ = inf{t 0 : x(t) S r }. (Throughout this course we set inf =.) By Itô s formula, for any t 0, τ t V (x(τ t), τ t) = V (x 0, 0) + LV (x(s), s)ds + τ t 0 0 V x (x(s), s)g(x(s), s)db(s).

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large Taking the expectation on both sides, we obtain EV (x(τ t), τ t) = V (x 0, 0)+E τ t 0 LV (x(s), s)ds V (x 0, 0). Noting that x(τ t) = x(τ) = r if τ t, we get [ ] EV (x(τ t), τ t) E I {τ t} V (x(τ), τ) µ(r)p{τ t}. (Throughout this course I A denotes the indicator function of set A.) We therefore obtain P{τ t} ε. Letting t we get P{τ < } ε, that is as required. P{ x(t) < r for all t 0} 1 ε

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large Theorem Assume that there exist V C 2,1 (S h R + ; R + ) and µ 1, µ 2, µ 3 K such that and µ 1 ( x ) V (x, t) µ 2 ( x ) LV (x, t) µ 3 ( x ) for all (x, t) S h R +. Then the trivial solution of the SDE is stochastically asymptotically stable.

Theory Examples Stochastic stability Stochastic asymptotic stability 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 δ 0 = δ 0 (ε) > 0 such that P{ lim t x(t; x 0 ) = 0} 1 ε whenever x 0 < δ 0, or for any β (0, h/2), P{lim sup x(t; x 0 ) β} 1 ε. t By the previous theorem, we can find a δ 0 = δ 0 (ε) (0, h/2) such that P{ x(t; x 0 ) < h/2} 1 ε 2. (1.1) whenever x 0 S δ0.

Theory Examples Stochastic stability Stochastic asymptotic stability 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 P{ x(t; x 0 ) < β for all t s} 1 ε 2 (1.2) whenever x(s; x 0 ) α and s 0. Now fix any x 0 S δ and write x(t; x 0 ) = x(t) simply. Define τ α = inf{t 0 : x(t) α} and τ h = inf{t 0 : x(t) h/2}.

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large By Itô s formula and the conditions, we can show that Consequently 0 V (x 0, 0) + E τα τ h t 0 LV (x(s), s)ds V (x 0, 0) µ 3 (α)e(τ α τ h t). tµ 3 (α)p{τ α τ h t} E(τ α τ h t) V (x 0, 0). This implies immediately that P{τ α τ h < } = 1. But, by (1.1), P{τ h < } ε/2. Hence

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large 1 = P{τ α τ h < } P{τ α < }+P{τ h < } P{τ α < }+ ε 2, which yields P{τ α < } 1 ε 2. We now compute, using (1.2), P{lim sup x(t) β} t P{τ α < and x(t) β for all t τ α } = P{τ α < }P{ x(t) β for all t τ α τ α < } as required. P{τ α < }(1 ε/2) (1 ε/2) 2 1 ε

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large Theorem Assume that there exist V C 2,1 (R d R + ; R + ) and µ 1, µ 2 K and µ 3 K such that and µ 1 ( x ) V (x, t) µ 2 ( x ) LV (x, t) µ 3 ( x ) for all (x, t) R d R +. Then the trivial solution of the SDE is stochastically asymptotically stable in the large.

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large Proof. Clearly, we only need to show P{ lim t x(t; x 0 ) = 0} = 1 for all x 0 R d, or for any pair of ε (0, 1) and β > 0, P{lim sup x(t; x 0 ) β} 1 ε. t To show this, let us fix any x 0 and write x(t; x 0 ) = x(t) again. Let h sufficiently large for h/2 > x 0 and µ 1 (h/2) 2V (x 0, 0). ε As in the previous proof, define the stopping time τ h = inf{t 0 : x(t) h/2}.

Theory Examples Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large By Itô s formula, we can show that for any t 0, But EV (x(τ h t), τ h t) V (x 0, 0). EV (x(τ h t), τ h t) µ 1 (h/2)p{τ h t}. Hence P{τ h t} ε 2. Letting t gives P{τ h < } ε/2. That is P{ x(t) < h/2 for all t 0} 1 ε 2, which is the same as (1.1). From here, we can show in the same way as in the previous proof that as desired. P{lim sup x(t) β} 1 ε t

Outline Theory Examples Scale SDEs Multi-dimensional SDEs 1 Theory Stochastic stability Stochastic asymptotic stability Stochastic asymptotic stability in the large 2 Examples Scale SDEs Multi-dimensional SDEs

Theory Examples Scale SDEs 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) = x 0 R. Assume that f : R R + R and g : R R + R m have the expansions f (x, t) = a(t)x+o( x ), g(x, t) = (b 1 (t)x,, b m (t)x) T +o( x ). in a neighbourhood of x = 0 uniformly with respect to t 0, where a(t), b i (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 K t 0 ( a(s) 1 2 m bi )ds 2 (s) + θ K for all t 0. i=1

Theory Examples Scale SDEs Multi-dimensional SDEs Let 0 < ε < and define the Lyapunov function θ sup t 0 m i=1 b2 i (t) [ t ( V (x, t) = x ε exp ε a(s) 1 0 2 Then, by the condition, m i=1 x ε e εk V (x, t) x ε e εk. ] bi )ds 2 (s) + θ.

Theory Examples Scale SDEs Multi-dimensional SDEs Moreover, compute [ LV (x, t) = ε x ε exp ε ( ε 2 t 0 ( a(s) 1 2 m i=1 m ) bi 2 (t) θ + o( x ε ) i=1 1 2 εθe εk x ε + o( x ε ). ] bi )ds 2 (s) + θ 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.

Theory Examples Scale SDEs 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) = (G 1 (t)x,, G m (t)x)+o( x ) in a neighbourhood of x = 0 uniformly with respect to t 0, where F(t), G i (t) are all bounded Borel-measurable d d-matrix-valued functions. Assume that there is a symmetric positive-definite matrix Q such that λ max (QF(t) + F T (t)q + m i=1 ) Gi T (t)qg i (t) λ < 0 for all t 0, where (and throughout this course) λ max (A) denotes the largest eigenvalue of matrix A.

Theory Examples Scale SDEs Multi-dimensional SDEs Now, define the Lyapunov function V (x, t) = x T Qx. Clearly, Moreover, λ min (Q) x 2 V (x, t) λ max (Q) x 2. LV (x, t) = x T ( QF (t) + F T (t)q + λ x 2 + o( x 2 ). m i=1 ) Gi T (t)qg i (t) x + 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.

Theory Examples Stability of Stochastic Differential Equations Part 3: Almost Sure Exponential Stability Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

Outline Theory Examples 1 Theory Almost sure exponential stability Almost sure exponential instability 2 Examples Linear SDEs Nonlinear case

Theory Examples Almost sure exponential stability 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.

Theory Examples Almost sure exponential stability 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 For all x 0 0 in R d P{x(t; x 0 ) 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.

Theory Examples Almost sure exponential stability Almost sure exponential instability Proof. If the lemma were false, there would exist some x 0 0 such that P{τ < } > 0, where τ = inf{t 0 : x(t) = 0} in which we write x(t; x 0 ) = 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 K θ such that f (x, t) g(x, t) K θ x for all x θ, 0 t T.

Theory Examples Almost sure exponential stability Almost sure exponential instability Let V (x, t) = x 1. Then, for 0 < x θ and 0 t T, LV (x, t) = x 3 x T f (x, t) + 1 ( x 3 g(x, t) 2 + 3 x 5 x T g(x, t) 2) 2 x 2 f (x, t) + x 3 g(x, t) 2 K θ x 1 + Kθ 2 x 1 = K θ (1 + K θ )V (x, t). Now, for any ε (0, x 0 ), define the stopping time τ ε = inf{t 0 : x(t) (ε, θ)}. By Itô s formula,

Theory Examples Almost sure exponential stability Almost sure exponential instability = E 0. [ ] E e K θ(1+k θ )(τ ε T ) V (x(τ ε T ), τ ε T ) τε T 0 V (x 0, 0) [ ] e K θ(1+k θ )s (K θ (1 + K θ ))V (x(s), s) + LV (x(s), s) ds Note that for ω B, τ ε T and x(τ ε ) = ε. The above inequality therefore implies that ] E [e K θ(1+k θ )T ε 1 I B x 0 1. Hence P(B) ε x 0 1 e K θ(1+k θ )T. Letting ε 0 yields that P(B) = 0, but this contradicts the definition of B. The proof is complete.

Theory Examples Almost sure exponential stability Almost sure exponential instability We will also need the well-known exponential martingale inequality which we state here as a lemma. Lemma Let g = (g 1,, g m ) L 2 (R + ; R 1 m ), and let T, α, β be any positive numbers. Then { [ t P sup g(s)db(s) α t ] } g(s) 2 ds > β e αβ. 0 t T 0 2 0

Outline Theory Examples Almost sure exponential stability Almost sure exponential instability 1 Theory Almost sure exponential stability Almost sure exponential instability 2 Examples Linear SDEs Nonlinear case

Theory Examples Almost sure exponential stability Almost sure exponential instability Theorem Assume that there exists a function V C 2,1 (R d R + ; R + ), and constants p > 0, c 1 > 0, c 2 R, c 3 0, such that for all x 0 and t 0, Then lim sup t c 1 x p V (x, t), LV (x, t) c 2 V (x, t), V x (x, t)g(x, t) 2 c 3 V 2 (x, t). 1 t log x(t; x 0) c 3 2c 2 2p a.s. (1.1) for all x 0 R d. In particular, if c 3 > 2c 2, then the trivial solution of the SDE is almost surely exponentially stable.

Theory Examples Almost sure exponential stability Almost sure exponential instability Proof. Clearly, the assertion holds for x 0 = 0 since x(t; 0) 0. Fix any x 0 0 and write x(t; x 0 ) = x(t). By the lemma, x(t) 0 for all t 0 almost surely. Thus, one can apply Itô s formula and the condition to show that, for t 0, where log V (x(t), t) log V (x 0, 0) + c 2 t + M(t) M(t) = 1 2 t 0 t 0 V x (x(s), s)g(x(s), s) 2 V 2 ds, (1.2) (x(s), s) V x (x(s), s)g(x(s), s) db(s) 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,

Theory Examples Almost sure exponential stability Almost sure exponential instability { [ P sup M(t) ε t 0 t n 2 0 V x (x(s), s)g(x(s), s) 2 V 2 (x(s), s) ] ds > 2ε } log n 1 n 2. Applying the Borel Cantelli lemma we see that for almost all ω Ω, there is an integer n 0 = n 0 (ω) such that if n n 0, M(t) 2 ε log n + ε 2 t 0 V x (x(s), s)g(x(s), s) 2 V 2 ds (x(s), s) holds for all 0 t n. Substituting this into (1.2) and then using the condition we obtain that log V (x(t), t) log V (x 0, 0) 1 2 [(1 ε)c 3 2c 2 ]t + 2 ε log n for all 0 t n, n n 0 almost surely.

Theory Examples Almost sure exponential stability Almost sure exponential instability Consequently, for almost all ω Ω, if n 1 t n and n n 0, 1 t log V (x(t), t) 1 2 [(1 ε)c 3 2c 2 ] + log V (x 0, 0) + 2 ε log n. n 1 This implies lim sup t 1 t log V (x(t), t) 1 2 [(1 ε)c 3 2c 2 ] a.s. Hence lim sup t 1 t log x(t) (1 ε)c 3 2c 2 2p a.s. and the required assertion follows since ε > 0 is arbitrary.

Theory Examples Almost sure exponential stability Almost sure exponential instability Theorem Assume that there exists a function V C 2,1 (R d R + ; R + ), and constants p > 0, c 1 > 0, c 2 R, c 3 > 0, such that for all x 0 and t 0, Then c 1 x p V (x, t) > 0, LV (x, t) c 2 V (x, t), V x (x, t)g(x, t) 2 c 3 V 2 (x, t). lim inf t 1 t log x(t; x 0) 2c 2 c 3 2p a.s. for all x 0 0 in R d. In particular, if 2c 2 > c 3, then almost all the sample paths of x(t; x 0 ) will tend to infinity exponentially.

Theory Examples Almost sure exponential stability Almost sure exponential instability Proof. Fix any x 0 0 and write x(t; x 0 ) = x(t). By Itô s formula and the conditions, we can easily show that for t 0, where log V (x(t), t) log V (x 0, 0) + 1 2 (2c 2 c 3 )t + M(t), (1.3) M(t) = t 0 V x (x(s), s)g(x(s), s) db(s) V (x(s), s) is a continuous martingale with the quadratic variation M(t), M(t) = t 0 V x (x(s), s)g(x(s), s) 2 V 2 ds c 3 t. (x(s), s)

Theory Examples Almost sure exponential stability 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 lim inf t 1 t log V (x(t), t) 1 2 (2c 2 c 3 ) a.s. which implies the required assertion immediately by using the condition.

Outline Theory Examples Linear SDEs Nonlinear case 1 Theory Almost sure exponential stability Almost sure exponential instability 2 Examples Linear SDEs Nonlinear case

Theory Examples Linear SDEs Nonlinear case Consider the scalar linear SDE dx(t) = ax(t) + m b i x(t)db i (t) on t 0. i=1 It is known that it has the explicit solution ( x(t) = x 0 exp [a 0.5 This implies that, for x 0 0, m m bi 2 ]t + i=1 i=1 ) b i B i (t). 1 m lim t t log x(t) = a 0.5 i=1 b 2 i a.s.

Theory Examples Linear SDEs Nonlinear case Let us now apply the stability theorem to obtain the same conclusion. Let V (x, t) = x 2. Then LV (x, t) = ( 2a + m i=1 and, writing g(x, t) = (b 1 x,, b m x), V x (x, t)g(x, t) 2 = 4 b 2 i ) x 2 m bi 2 x 4. In other words, the conditions in the Theorems holds with p = 2, c 1 = 1, c 2 = 2a + i=1 i=1 m m bi 2, c 3 = 4 bi 2. i=1

Theory Examples Linear SDEs Nonlinear case We hence have and lim sup t lim inf t 1 t log x(t) a 1 2 1 t log x(t) a 1 2 Combining these gives what we want. m i=1 m i=1 b 2 i b 2 i a.s. a.s.

Theory Examples Linear SDEs Nonlinear case Consider, for example, dx(t) = x(t)dt + 2x(t)dB(t) with initial value x(0) = x 0 0, where B(t) is a one-dimensional Brownian motion. The theory above shows that the solution of this linear sde obeys lim inf t 1 t log x(t) = 1 a.s. The following simulation shows a typical sample path of the solution with initial value x(0) = 10.

Theory Examples Linear SDEs Nonlinear case x(t) 0 10 20 30 40 50 60 0 2 4 6 8 10 t

Outline Theory Examples Linear SDEs Nonlinear case 1 Theory Almost sure exponential stability Almost sure exponential instability 2 Examples Linear SDEs Nonlinear case

Theory Examples Linear SDEs Nonlinear case Consider the two-dimensional SDE dx(t) = f (x(t))dt + Gx(t)dB(t) on t 0 with initial value x(0) = x 0 R 2 and x 0 0, where B(t) is a one-dimensional Brownian motion, ( ) ( ) x2 cos x f (x) = 1 3 0.3, G = 2x 1 sin x 2 0.3 3

Theory Examples Linear SDEs Nonlinear case Let V (x, t) = x 2. It is easy to verify that 4.29 x 2 LV (x, t) = 2x 1 x 2 cos x 1 +4x 1 x 2 sin x 2 + Gx 2 13.89 x 2 and 29.16 x 2 V x (x, t)gx 2 = 2x T Gx 2 43.56 x 4. Applying the Theorems we then have 8.745 lim inf t 1 t log x(t; x 1 0) lim sup t t log x(t; x 0) 0.345 almost surely. The following figure is a compute simulation.

Theory Examples Linear SDEs Nonlinear case X1(t) 0 1 2 3 4 X2(t) 0 2 4 6 0 2 4 6 8 10 t 0 2 4 6 8 10 t x 1 (0) = x 2 (0) = 1.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Stability of Stochastic Differential Equations Part 4: Moment Exponential Stability Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

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

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.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Theorem Assume that there is a positive constant K such that x T f (x, t) g(x, t) 2 K x 2 for all (x, t) R d R +. Then the pth moment exponential stability of the trivial solution of the SDE implies the almost sure exponential stability.

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 Let g L 2 (R + ; R d m ). Define, for t 0, x(t) = t 0 g(s)db(s) and A(t) = t 0 g(s) 2 ds. Then for every p > 0, there exist universal positive constants c p, C p (depending only on p), such that ( c p E A(t) p 2 E sup x(s) p) C p E A(t) p 2. 0 s t

Moment verse Almost Sure Exponential Stability Criteria A Case Study In particular, one may take c p = (p/2) p, C p = (32/p) p/2 if 0 < p < 2; c p = 1, C p = 4 if p = 2; c p = (2p) p/2, C p = [ p p+1 /2(p 1) p 1] p/2 if p > 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Proof of the theorem. Fix any x 0 0 in R d and write x(t; x 0 ) = 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, x(t) p x(n 1) p + c 1 t + t n 1 n 1 x(s) p ds p x(s) p 2 x T (s)g(x(s), s)db(s), where c 1 = pk + p(1 + p 2 )K /2. Hence

Moment verse Almost Sure Exponential Stability Criteria A Case Study ( E sup n 1 t n ( +E x(t) p) E x(n 1) p + c 1 n sup n 1 t n t n 1 n 1 E x(s) p ds ) p x(s) p 2 x T (s)g(x(s), s)db(s). On the other hand, by the well-known Burkholder Davis Gundy inequality we compute

Moment verse Almost Sure Exponential Stability Criteria A Case Study ( E sup n 1 t n 4 ( n 2E 4 ( 2E 1 2 E ( t n 1 ) p x(s) p 2 x T (s)g(x(s), s)db(s) p 2 x(s) 2(p 2) x T (s)g(x(s), s) 2 ds n 1 n sup x(s) p p 2 K x(s) p ds n 1 s n n 1 sup n 1 s n x(s) p) + 16p 2 K n n 1 Substituting this into the previous inequality yields ) 1 2 E x(s) p ds. ) 1 2

Moment verse Almost Sure Exponential Stability Criteria A Case Study ( E sup n 1 t n x(t) p) 2E x(n 1) p + c 2 n n 1 E x(s) p ds, where c 2 = 2c 1 + 32p 2 K. By the property of the pth moment exponential stability, we then have ( E x(t) p) c 3 e λ(n 1), sup n 1 t n where c 3 = C(2 + c 2 ). Now, let ε (0, λ) be arbitrary. Then { P x(t) p > e (λ ε)(n 1)} c 3 e ε(n 1). sup n 1 t n

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 n 0 = n 0 (ω), for all ω Ω excluding a P-null set, for which the inequality above holds whenever n n 0. Consequently, for almost all ω Ω, 1 t log x(t) = 1 (λ ε)(n 1) pt log( x(t) p ) pn if n 1 t n, n n 0.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Hence lim sup t 1 t log x(t) (λ ε) p Since ε > 0 is arbitrary, we must have a.s. lim sup t 1 t log x(t) λ p a.s. By definition, the trivial solution of the SDE is almost surely exponentially stable.

Outline Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case 1 Moment verse Almost Sure Exponential Stability 2 Criteria Nonlinear case Linear case 3 A Case Study

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Theorem Assume that there is a function V ( C 2,1 (R d R + ; R + ), and positive constants c 1 c 3, such that c 1 x p V (x, t) c 2 x p and LV (x, t) c 3 V (x, t) for all (x, t) R d R +. Then E x(t; x 0 ) p c 2 c 1 x 0 p e c 3t on t 0 for all x 0 R d.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Proof. Fix any x 0 R d and write x(t; x 0 ) = x(t). For each n x 0, define the stopping time τ n = inf{t 0 : x(t) n}. Obviously, τ n as n almost surely. By Itô s formula, we can derive that for t 0, [ ] E e c 3(t τ n) V (x(t τ n ), t τ n ) V (x 0, 0) = E t τn 0 e c 3s [ c 3 V (x(s), s) + LV (x(s), s) ] ds 0

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Hence [ c 1 E e c 3(t τ n) E x(t τ n ) p] V (x 0, 0) c 2 x 0 p. Letting n yields that c 1 e c 3t E x(t) p c 2 x 0 p which implies the desired assertion.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Theorem Assume that there exists a symmetric positive-definite d d matrix Q, and constants α 1 R, 0 α 2 < α 3, such that for all (x, t) R d R +, and x T Qf (x, t) + 1 2 trace[gt (x, t)qg(x, t)] α 1 x T Qx α 2 x T Qx x T Qg(x, t) α 3 x T Qx. (i) If α 1 < 0, then the trivial solution of the SDE is pth moment exponentially stable provided p < 2 + 2 α 1 /α3 2. (ii) If 0 α 1 < α2 2, then the trivial solution of equation (1.2) is pth moment exponentially stable provided p < 2 2α 1 /α2 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Proof. Let V (x, t) = (x T Qx) p 2. Then It is also easy to verify that λ p 2 min (Q) x p V (x, t) λ p 2 max (Q) x p. LV (x, t) = p(x T Qx) p 1( 2 x T Qf (x, t) + 1 ) 2 trace[gt (x, t)qg(x, t)] ( p ) + p 2 1 (x T Qx) p 2 2 x T Qg(x, t) 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case (i) Assume that α 1 < 0 and p < 2 + 2 α 1 /α3 2. Without loss of generality, we can let p 2. Then [ ( p ) ] LV (x, t) p α 1 2 1 α3 2 V (x, t). (ii) Assume that 0 α 1 < α 2 2 and p < 2 2α 1/α 2 2. Then [( p ) ] LV (x, t) p 2 1 α2 2 α 1 V (x, t). So in both cases the stability assertion follows from the previous theorem.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Consider a d-dimensional linear SDE dx(t) = Fx(t)dt + m G i x(t)db i (t), where F, G i R d d. This is of course a special case of the underlying SDE where i=1 f (x, t) = Fx, g(x, t) = (G 1 x,, G m x).

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Corollary Assume that there exists a symmetric positive-definite d d matrix Q such that the following LMI holds: QF + F T Q + m Gi T QG i < 0. Then the trivial solution of the linear SDE is mean-square exponentially stable as well as almost surely exponentially stable. i=1

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Proof. Let V (x, t) = x T Qx. Then Moreover λ min (Q) x 2 V (x, t) λ max (Q) x 2. LV (x, t) = x T Qx λmax ( Q) x 2. where Q = QF + F T Q + m i=1 GT i QG i. By the condition, λ max ( Q) < 0. Hence LV (x, t) λ max( Q) V (x, t). λ max (Q) The assertions follow therefore from the theory established above.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case In the case when m QF + F T Q + Gi T QG i i=1 is not negative-definite, the following result is useful.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Corollary Assume that there exists a symmetric positive-definite d d matrix Q, and nonnegative constants β and β i (1 i m), such that β < m i=1 β i, QF + F T Q + m Gi T QG i βq 0, i=1 and, moreover, for each i = 1,, m, either QG i + G T i Q 2β i Q 0 or QG i + G T i Q + 2β i Q 0. If 0 < p < 2 2β/( m i=1 β i), then the trivial solution of the linear SDE is pth moment exponentially stable, whence it is also almost surely exponentially stable.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case Proof. We will use the 2nd theorem established above to show this corollary. We first have that x T Qf (x, t) + 1 2 trace[gt (x, t)qg(x, t)] = 0.5x T ( QF + F T Q + m Gi T QG i )x 0.5βx T Qx. i=1 We also observe from the condition that for each i, Hence x T QG i x 2 = 0.25 x T (QG i + G T i Q)x 2 0.5β i (x T Qx) 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case x T Qg(x, t) = m x T QG i x 2 m 0.5 β i x T Qx. i=1 Applying the theorem with α 1 = 0.5β, α 2 = m 0.5 β i, we can therefore conclude that the trivial solution of the linear SDE is pth moment exponentially stable if 0 < p < 2 2β/( m i=1 β i). This implies that the trivial solution of the linear SDE is also almost surely exponentially stable. i=1 i=1

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case As an even more special case, let us consider the scalar linear SDE m dx(t) = ax(t)dt + b i x(t)db i (t), where a, b i are all real numbers. Using the corollaries above, we can conclude: i=1

Moment verse Almost Sure Exponential Stability Criteria A Case Study Nonlinear case Linear case If 2a + m i=1 b2 i < 0, then the trivial solution of this scalar linear SDE is mean-square exponentially stable as well as almost surely exponentially stable. If 0 2a + m i=1 b2 i < 2 m i=1 b2 i, then the trivial solution of this scalar linear SDE is pth moment exponentially stable provided 0 < p < 1 2a m, i=1 b2 i whence it is also almost surely exponentially stable.

Moment verse Almost Sure Exponential Stability Criteria A Case Study This example is from the satellite dynamics. Sagirow in 1970 derived the equation ÿ(t) + β(1 + αḃ(t))ẏ(t) + (1 + αḃ(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 Ḃ(t) is a scalar white noise, α is a constant representing the intensity of the disturbance, and β, γ are two positive constants.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Introducing x = (x 1, x 2 ) T = (y, ẏ) T, we can write this equation as the two-dimensional SDE dx 1 (t) = x 2 (t)dt, dx 2 (t) = [ x 1 (t) + γ sin(2x 1 (t)) βx 2 (t)]dt α[x 1 (t) + βx 2 (t)]db(t). For the Lyapunov function, we try an expression consisting of a quadratic form and integral of the nonlinear component: x1 V (x, t) = ax1 2 + bx 1x 2 + x2 2 + c sin(2y)dy = ax 2 1 + bx 1x 2 + x 2 2 + c sin2 x 1. 0

Moment verse Almost Sure Exponential Stability Criteria A Case Study This yields LV (x, t) = (b α 2 )x1 2 + bγx 1 sin(2x 1 ) (2β b α 2 β 2 )x2 2 + (2a bβ 2 + 2α 2 β)x 1 x 2 + (c + 2γ)x 2 sin(2x 1 ). Setting 2a bβ 2 + 2α 2 β = 0 and c + 2γ = 0 we obtain and V (x, t) = 1 2 (bβ + 2 2α2 β)x 2 1 + bx 1x 2 + x 2 2 2γ sin2 x 1 LV (x, t) = (b α 2 )x 2 1 + bγx 1 sin(2x 1 ) (2β b α 2 β 2 )x 2 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study Note that V (x, t) 1 2 (bβ + 2 2α2 β 4γ)x1 2 + bx 1x 2 + x2 2. So V (x, t) ε x 2 for some ε > 0 if 2(bβ + 2 2α 2 β 4γ) b 2 or equivalently β β 2 + 4 8γ 4α 2 β < b < β + β 2 + 4 8γ 4α 2 β. Note also that LV (x, t) (b α 2 2bγ)x1 2 (2β b α2 β 2 )x2 2.

Moment verse Almost Sure Exponential Stability Criteria A Case Study So LV (x, t) ε x 2 for some ε > 0 provided both b α 2 2bγ > 0 and 2β b α 2 β 2 > 0, that is 2γ < 1 and α 2 /(1 2γ) < b < 2β α 2 β 2. We can therefore conclude that if γ < 1/2 and { max α 2 /(1 2γ), β } β 2 + 4 8γ 4α 2 β { < min 2β α 2 β 2, β + } β 2 + 4 8γ 4α 2 β then the trivial solution of the SDE is exponentially stable in mean square.

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Stability of Stochastic Differential Equations Part 5: Stochastic Stabilization and Destabilization Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

Outline Motivating Examples and History Stochastic Stabilization Stochastic Destabilization 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

Outline Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history 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

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history It is not surprising that noise can destabilize a stable system.

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history Consider a 2-dimensional ODE ẏ(t) = y(t) on t 0, y(0) = y 0 R 2. This is an exponentially stable system. Perturb it by noise and assume the stochastically perturbed system is described by an SDE dx(t) = x(t)dt + Gx(t)dB(t) on t 0, x(0) = y 0 R 2, where B(t) is a scalar Brownian motion and ( ) 0 2 G = 2 0

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history The SDE has the explicit solution x(t) = exp[( I 0.5G 2 )t + GB(t)]x(0) = exp[it + GB(t)]x(0), where I is the 2 2 identity matrix. Consequently 1 lim t t log( x(t) ) = 1 a.s. That is, the stochastically perturbed system has become unstable with probability one.

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history x1(t) or y1(t) 4 2 0 2 4 6 8 x1(t) y1(t) x2(t) or y2(t) 0 5 10 x2(t) y2(t) 0.0 0.5 1.0 1.5 2.0 t 0.0 0.5 1.0 1.5 2.0 t

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history Do you believe that noise can also stabilize an unstable system?

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history Consider the scalar ODE The solution is ẏ(t) = y(t) on t 0, y(0) = y 0 R. y(t) = y(0)e t. So y(t) if y(0) 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) = y 0 R,

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history The SDE has the explicit solution Consequently x(t) = x(0) exp[(1 0.5σ 2 )t + σb(t)]. x(t) 0 a.s. if σ > 2. That is, the stochastically perturbed system has become stable with probability one.

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history x(t) or y(t) 0 1 2 3 4 5 6 7 x(t) y(t) 0 2 4 6 8 10 t σ = 2

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history Of course, if the noise is not strong enough, it will not be able to stabilize the system.

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history x(t) or y(t) 0 100 200 300 400 500 x(t) y(t) 0 2 4 6 8 10 t σ = 0.5

Motivating Examples and History Stochastic Stabilization Stochastic Destabilization Destabilization Stabilization A brief history x(t) or y(t) 0 1 2 3 4 5 6 7 x(t) y(t) 0 2 4 6 8 10 t σ = 2