On the practical global uniform asymptotic stability of stochastic differential equations

On the practical global uniform asymptotic stability of stochastic differential equations∗ Tom´as CaraballobMohamed Ali HammamiaLassaad Mchiria aUniversity of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia E.mail: MohamedAli.Ha[email protected]u.tn bUniversidad de Sevilla, Depto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Apdo. de Correos 1160 41080-Sevilla (Spain) Abstract The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of stochastic differential equations based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result. Keywords: Stochastic differential equations, Lyapunov techniques, practical asymptotic stability. ∗The research of T. Caraballo has been partially supported by FEDER and the Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411, and Junta de Andaluc´ıa (Spain) under Proyecto de Excelencia P12-FQM-1492 and the Ayudas de consolidaci´on for the research group FQM314. 1 1 Introduction The main aim of this paper is to establish some criteria for the global practical uniform exponential stability of a class of nonlinear stochastic differential equations of the form dx(t) = f(x(t), t)dt +g(x(t), t)dW(t).(1.1) As it is well known, stochastic differential equations are useful for modeling physical, technical, biological and economical dynamical systems in which significant uncertainly is present. To investigate stability for stochastic differential equations, we usually study the stability and attractivity of solutions in a neighborhood of the origin considered as an equilibrium point. The equations are worth being investigated since the solutions with different initial values have similar large-time properties. When f(0, t) = g(0, t) = 0, the stability and the exponential stability of equation (1.1) have received very much attention in the literature. We would like to mention here the references [5]-[14], among others. Several interesting and important variants to Lyapunov’s original concepts of practical stability were proposed in [1]-[4]. When the origin is not necessarily an equilibrium point, we can study the asymptotic stability of solutions with respect to a small neighborhood of the origin. The goal is to analyze systems whose desired behavior is asymptotic stability about a small ball of the state space or a close approximation to this. Almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. One also wishes that the state approaches the origin (or some sufficiently small neighborhood of it) in a sufficiently fast manner, for instance, exponentially. This property is referred to as practical stability. Therefore, in this paper we state some sufficient conditions to ensure the almost sure practical global uniform asymptotic stability and the exponential stability of the solution of equation (1.1). The content of the paper is as follows. In Section 2, we introduce some auxiliary facts and results. In Section 3, we prove the main theorem about the global practical uniform exponential stability of stochastic differential equation and exhibit an example as an application of this study. 2 Auxiliary Facts and Results Consider the following d-dimensional stochastic differential equation (SDE): dx(t) = f(x(t), t)dt +g(x(t), t)dWt,∀t≥0,(2.1) where f:Rd×R+−→ Rdand g:Rd×R+−→ Rd×m,x(t) = (x1(t), ..., xd(t))T∈Rdand Wt= (W1(t), ...., Wm(t))Tis an m-dimensional Brownian motion defined on a complete probability space (Ω,F,P). The Euclidian norm of a vector x∈Rdis denoted by |x|. We assume that both fand gsatisfy the following conditions: |f(x, t)| ≤ C1 + |x|;|g(x, t)| ≤ C1 + |x|,for all t≥0, x ∈Rd,(2.2) 2 |f(x, t)−f(y, t)| ≤ C|x−y|;|g(x, t)−g(y, t)| ≤ C|x−y|,for all t≥0, x, y ∈Rd.(2.3) Hence, for any given initial value x0∈Rd, the SDE has a unique global solution denoted by x(t, t0, x0), such that x(t0) = x0, and defined for all t≥t0≥0. In what follows we use x(t, t0, x0), or simply x(t) if no confusion is possible, to denote the solution of (2.1) with initial value x0at the initial time t0. We suppose that there exists tsuch that f(0, t)6= 0 and g(0, t)6= 0. In this case, the SDE does not possess the trivial solution x(t, t0,0) = 0. Therefore, we will study the asymptotic stability of the SDE when 0 in not an equilibrium point, but in a small neighborhood of the origin in terms of convergence of solution in probability to a small ball Br:= {x∈Rd:|x| ≤ r},r > 0. Definition 2.1. The solution of (2.1) is said to be globally uniformly bounded in probability if, for each α > 0, there exists c=c(α)>0 (independent of t0) such that, for every t0≥0,and all x0∈Rdwith |x0| ≤ α, sup t≥t0 |x(t, t0, x0)| ≤ c(α),a.s. (2.4) Definition 2.2. (i) The ball Bris said to be uniformly stable in probability if, for each ǫ∈]0,1[ and k > r, there exists δ=δ(ǫ, k)>0 such that P|x(t, t0, x0)|< k, ∀t≥t0≥0≥1−ǫfor all |x0|< δ. (2.5) (ii) The ball Bris said to be globally uniformly stable in probability if it is uniformly stable in probability and the solution of (2.1) is globally uniformly bounded in probability. Definition 2.3. The ball Bris said to be globally uniformly attractive in probability if for each ǫ∈]0,1[, k > r and c > 0 (independent of t0), there exists T=T(ǫ, c)>0 such that, for all t0≥0, it holds P|x(t, t0, x0)|< k, ∀t≥t0+T≥1−ǫfor all x0∈Rdsuch that |x0|< c. (2.6) Let us now denote by C2,1(Rd×R+,R+) the family of all nonnegative functions V(x, t) defined on Rd×R+which are twice continuously differentiable in xand once in t. Thanks to the Itˆo formula it follows dV (x(t), t) = LV (x(t), t)dt +Vx(x(t), t)g(x(t), t)dWt, where LV (x, t) = Vt(x, t) + Vx(x, t)f(x, t) + 1 2trace[g(x, t)TVxx(x, t)g(x, t)], Vt(x, t) = ∂V ∂t (x, t) ; Vx(x, t) = ( ∂V ∂x1 (x, t), ..., ∂V ∂xd (x, t)); Vxx(x, t) = ∂2V ∂xi∂xj (x, t)d×d. Let us also recall that by Kit is usual to denote the family of all continuous nondecreasing functions µ:R+→R+such that µ(0) = 0, and by K∞the class of all functions µ∈ K such that µ(r)→+∞,r→+∞. Now we can prove our first result. 3 Theorem 2.1. Assume that there exist V∈C2,1(Rd×R+,R+),µ1, µ2∈ K∞, with µ1:R∗ +→R∗ +and µ3∈ K,M > 0, such that for all t≥0, and all x∈Rd, µ1(|x|)≤V(x, t)≤µ2(|x|), LV (x, t)≤(t)−µ3(|x|), where (t)is a continuous nonnegative function with lim t→+∞(t) = 0, and Z+∞ 0 (t)dt ≤M < +∞. Then, system (2.1) is globally uniformly practically stable in probability. Proof. Let ǫ∈]0,1[, and take a small positive real number r, and k > r. Assume that there exists δ=δ(ǫ, k)∈]0, k[ such that, for x0∈Sδ,Sδ={x∈Rd/|x|< δ}and µ1(k)>M ǫ, we deduce V(x0,0) ǫ≤µ1(k)−M ǫ. Let us write x(t, t0, x0) = x(t) for simplicity. Define τ= inf{t≥0; |x(t)| ≥ k}(Throughout this paper we set inf ∅=∞). By the Itˆo formula, for any t≥0, we have V(x(τ∧t), τ ∧t) = V(x0,0) + Zτ∧t 0 LV (x(s), s)ds +Zτ∧t 0 Vx(x(s), s)g(x(s), s)dWs. Then, 0 ≤EV(x(τ∧t), τ ∧t)=V(x0,0) + EZτ∧t 0 LV (x(s), s)dsand EV(x(τ∧t), τ ∧t)≤V(x0,0) + EZτ∧t 0 (s)−µ3(|x(s)|)ds ≤V(x0,0) + EZτ∧t 0 (s)ds ≤V(x0,0) + EZ+∞ 0 (s)ds ≤V(x0,0) + M. 4 Notice that |x(τ∧t)|=|x(τ)|=kif τ≤t. Therefore, we can obtain EV(x(τ∧t), τ ∧t)≥E1{τ≤t}V(x(τ), τ) ≥E1{τ≤t}µ1(|x(τ)|) ≥E1{τ≤t}µ1(k) =µ1(k)E1{τ≤t} =µ1(k)Pτ≤t. Then, µ1(k)Pτ≤t≤V(x0,0) + M≤ǫµ1(k). Thus, Pτ≤t≤ǫ. Letting t→+∞, we arrive at Pτ < +∞≤ǫ. Thus, for all ǫ∈]0,1[, r≥0 and k > r, there exists δ=δ(ǫ, k)∈]0, k[, such that P|x(t)|< k, ∀t≥t0≥0≥1−ǫ, for all |x0|< δ. (2.7) In particular, for a small r∈]0, δ[ and |x0|< r, we can take k=k(r)>0, such that for all t≥t0≥0, we have |x(t)|< k, a.s., for all |x0|< r, and thus, sup t≥t0 |x(t)|< k for all |x0|< r a.s. Therefore, Bris globally uniformly stable in probability. Let us now prove the uniform attractivity of Br. To this end, it is sufficient to prove that, for each ǫ∈]0,1[, h > r and c > 0,there exists T=T(ǫ, c)>0 (independent of t0), such that for all t≥t0+T, we have P|x(t)|< h≥1−ǫ 2,for all |x0|< c. (2.8) Pick ǫ∈]0,1[, h > r large enough and c > 0 (independent of t0), such that |x0|< c, µ1(h)>2M ǫand µ1(h)−2M ǫ≥2V(x0,0) ǫ. Suppose now that there exists T=T(ǫ, c)>0 such that, for all t0≥0, τh= inf{t≥t0+T;|x(t)| ≥ h}. 5 Noting that, |x(τh∧t)|=|x(τh)|=hwhen τh≤t, and applying the Itˆo formula, we obtain, for all t≥t0+T, EV(x(τh∧t), τh∧t)≤V(x0,0) + EZτh∧t 0 ((s)−µ3(|x(s)|)ds, ≤V(x0,0) + M, ≤ǫ 2µ1(h)−M+M, ≤ǫ 2µ1(h). We also have EV(x(τh∧t), τh∧t)≥E1{τh≤t}V(x(τh), τh) ≥E1{τh≤t}µ1(|x(τh)|) ≥E1{τh≤t}µ1(h) =µ1(h)E1{τh≤t} =µ1(h)Pτh≤t. Therefore, µ1(h)Pτh≤t≤ǫ 2µ1(h), which implies that Pτh≤t≤ǫ 2. Then, for any ǫ∈]0,1[, h > r and c > 0 (independent of t0), for a fixed T=T(ǫ, c)>0, we have, for all t≥t0+T,Pτh≤t≤ǫ 2, for all |x0|< c. Letting t→+∞, we arrive at Pτh<+∞≤ǫ 2. Thus, P|x(t)|< h;∀t≥t0+T≥1−ǫ 2,for all |x0|< c. Then, Bris globally uniformly attractive in probability, as desired. 2 Remark 2.2. Note that if r→0 or (t)→0 when t→+∞, we then recover the classical stability results when 0 is an equilibrium point. 6 3 The global practical uniform exponential stability of stochastic differential equation Let us start by recalling a technical lemma which will be useful in our analysis. Lemma 3.1. For all x0∈Rdsuch that x06= 0 it holds Px(t, t0, x0)6= 0,∀t≥0= 1.(3.1) Proof. Arguing by contradiction, if the lemma were false, there would exist some x06= 0 such that P{τ < ∞} >0, where τ= inf{t≥0; x(t) = 0}, where we write x(t, t0, x0) = x(t) for simplicity. Thus, we can find a pair of constants T > 0 and θ > 1, sufficiently large, such that P(B)>0, where B={τ≤Tand |x(t)| ≤ θ−1,for all 0 ≤t≤τ}. But, by the standing hypotheses, there exist positive constants Kθ,r, and a small positive constant csuch that |f(x, t)| ∨ |g(x, t)| ≤ Kθ|x|+rfor all 0 < c ≤ |x| ≤ θ, 0≤t≤T. Let V(x, t) = |x|−1. Then, for 0 < c ≤ |x| ≤ θand 0 ≤t≤T, LV (x, t) = −|x|−3xTf(x, t), +1 2− |x|−3|g(x, t)|2+ 3|x|−5|xTg(x, t)|2 ≤ |x|−2|f(x, t)|+|x|−3|g(x, t)|2 ≤ |x|−2Kθ|x|+r+|x|−3Kθ|x|+r2 ≤Kθ(1 + Kθ)V(x, t) + r|x|−1|x|−1+ 2Kθ|x|−1+r|x|−2 ≤Kθ(1 + Kθ)V(x, t) + rV (x, t)1 c+2Kθ c+r c2 ≤Kθ(1 + Kθ) + βV(x, t), where β=r1 c+2Kθ c+r c2. Now, for any ǫ∈]0,|x0|[, define the stopping time τǫ= inf{t≥0; |x(t)| ∈]− ∞, ǫ]∪[θ, +∞[}. 7 Again by the Itˆo formula, Ehexp n−Kθ(1 + Kθ) + β(τǫ∧T)oVx(τǫ∧T), τǫ∧Ti−V(x0,0), =EhZτǫ∧T 0 exp n−Kθ(1 + Kθ) + βsoh−Kθ(1 + Kθ) + βV(x(s), s) + LV (x(s), s)ids, ≤0. Note that for ω∈B, it follows that τǫ≤Tand |x(τǫ)|=ǫ. The above inequality therefore implies that Ehexp n−Kθ(1 + Kθ) + βToǫ−11Bi≤ |x0|−1. Hence P(B)≤ǫ|x0|−1exp nKθ(1 + Kθ) + βTo. Letting ǫ−→ 0 yields that P(B) = 0, but this contradicts the definition of B. The proof is complete. 2 The next lemma, whose proof can be found in [15], will be crucial in the proof of our main result. Lemma 3.2. Let g= (g1, ...., gm)∈L2(R+,Rm), and let T,α,βbe any positive numbers. Then Psup 0≤t≤TZt 0 g(s)dWs−α 2Zt 0 |g(s)|2ds> β≤exp(−αβ). Definition 3.1. The ball Bris said to be almost surely globally practically uniformly exponentially stable if, for any x0such that 0 <|x(t, t0, x0)| − r, for all t≥t0≥0, it holds that lim t→∞ sup 1 tln(|x(t, t0, x0)| − r)<0,a.s. (3.2) System (2.1) is said to be almost surely globally practically uniformly exponentially stable if there exists r > 0 such that Bris almost surely globally practically uniformly exponentially stable. Now, we can establish and prove our main result in this paper. Theorem 3.3. Assume that there exist a function V∈C2,1(Rd×R+,R∗+)and constants p∈N∗, c1≥1,≥c1,γ≥0and c2∈R,c3≥0such that for all t≥t0≥0, and x∈Rd, the following conditions hold: c1|x|p≤V(x, t), LV (x, t)≤c2V(x, t) + , 8 |Vx(x, t)g(x, t)|2≥c3V2(x, t) + γ. Then lim t→+∞sup 1 tln |x(t, t0, x0)| − ( c1 )1 p≤ −c3−2(c2+ 1) 2, a.s., for all x0∈Rd In particular, if c3>2(c2+ 1), then the solution of (2.1) is almost surely globally practically uniformly exponentially stable with r= (  c1 )1 p. Proof. Fix x06= 0 in Rdand write x(t, t0, x0) = x(t). By Lemma 3.1, x(t)6= 0, for all t≥0 almost surely. Observe that, for any x∈Rd, we have c1|x|p−=c1|x|p− c1, =c1|x|p−( c1 )1 pp, =c1|x| − ( c1 )1 p|x|p−1+|x|p−2( c1 )1 p+... + (  c1 )p−1 p, ≥c1|x| − ( c1 )1 p. Since c1≥1, we obtain V(x, t)≥c1|x|p≥c1|x|p−≥|x| − ( c1 )1 p. Therefore, |x| − ( c1)1 p≤V(x, t) and ln |x| − ( c1)1 p≤ln V(x, t), for all t≥0, x ∈Rd. Applying the Itˆo formula once more and taking into account the assumptions, we obtain that, for all t≥0 dln(V(x(t), t))=LV (x(t), t) V(x(t), t)dt +Vx(x(t), t)g(x(t), t) V(x(t), t)dWt, −1 2 |Vx(x(t), t)g(x(t), t)|2 V2(x(t), t)dt, Zt 0 dln(V(x(s), s))ds =Zt 0 LV (x(s), s) V(x(s), s)ds +Zt 0 Vx(x(s), s)g(x(s), s) V(x(s), s)dWs, −1 2Zt 0 |Vx(x(s), s)g(x(s), s)|2 V2(x(s), s)ds, 9