A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.

1. Introduction

The impact of random factors cannot be neglected in the real world. Different kinds of random perturbations of stochastic models have been investigated in many pieces of literature. Bahar and Mao [1] discussed a stochastic delay Lotka-Volterra model, and they showed that environmental noise would suppress a potential population explosion and also made the solutions to be stochastically ultimately bounded. Almost at the same time, Mao [2] revealed that different types of environmental noise had different effects on delay population models. Meanwhile, Jiang and Shi [3] considered a randomized nonautonomous Logistic equation and represented the unique continuous global positive solution and positive T-periodic solution.

Recently, stochastic models with delay are paid more attention by many researchers. Shen et al. [4] studied stochastic Lotka-Volterra competitive models with variable delay, and they obtained the unique global positive solution, stochastically ultimate boundedness, and moment average in time of the solutions. Liu and Wang [5] considered stochastic Lotka-Volterra models with infinite delay; they replaced the intrinsic growth rate by a random perturbation term which was dependent on the difference between the population size and the equilibrium state in their model, and then the sufficient criteria for global asymptotic stability of the solution were established. Huang et al. [6] extended the conclusion of Liu and Wang [5] to a general case. Based on a general phase space
(H1)Cr≔{∥φ∥Cr=sup-∞<s≤0φ∈C((-∞,0];R++n):000000∥φ∥Cr=sup-∞<s≤0ers|φ(s)|<∞},r>0,

Xu et al. [7] and Xu [8] investigated an autonomous stochastic Lotka-Volterra model with infinite delay
(1)dx(t)=diag(x1(t),…,xn(t))×[(b+Ax(t)+B∫-∞0x(t+θ)dμ(θ))dt000+σx(t)dB(t)(b+Ax(t)+B∫-∞0x(t+θ)dμ(θ))];
they established the asymptotic pathwise properties of the solution to model (1), where x=(x1,…,xn)T, b=(b1,…,bn)T, A=(aij)n×n, B=(bij)n×n, and σ=(σij)n×n; Cr was a Banach space (see [7, 9, 10]); μ was probability measure defined on (-∞,0] such that(H2)μr≔∫-∞0e-2rθdμ(θ)<∞,r>0.

In particular, when μ(θ)=ekrθ(k>2) for θ≤0, assumption (H2) was satisfied.

Motivated by the works mentioned previously, we always assume that the intensity of white noise is dependent on the difference between the population size and the equilibrium state and also is dependent on time t in this paper. Now, we consider a more general stochastic nonautonomous Lotka-Volterra model with infinite delay
(2)dxi(t)=xi(t)[hi(t)+∑j=1naij(t)xj(t)00000000+∑j=1nbij(t)xj(t-δ(t))00000000+∑j=1ncij(t)∫-∞0xj(t+θ)dμ(θ)]dt+xi(t)[∑j=1nγij(t)∫-∞0σi(t)(xi(t)-xi*)000000000+∑j=1nαij(t)xj(t)(xi(t)-xi*)000000000+∑j=1nβij(t)xj(t-δ(t))(xi(t)-xi*)000000000+∑j=1nγij(t)∫-∞0xj(t+θ)dμ(θ)000000000×(xi(t)-xi*)∑j=1nγij(t)∫-∞0]dBi(t),i=1,2,…,n,
where hi(t)>0, aij(t) (aii(t)<0 if i=j), bij(t) and cij(t) are parameter functions; noise intensities σi(t)>0, αij(t)≥0, βij(t)≥0 and γij(t)≥0 are continuous bounded function on [0,+∞); variable delay function δ:[0,∞)→[0,τ] and δ′(t)≤0; x*=(x1*,x2*,…,xn*)T denotes an equilibrium state with respect to the deterministic part of model (2) in R++n (we always assume that such x* exists in this paper). For simplicity, model (2) can be rewritten in the following form:
(3)dxi(t)=xi(t)fidt+xi(t)gidBi(t),i=1,2,…,n.
Next, we will investigate the global positive solution and its asymptotic properties of model (2). Further, the approximation distribution of solutions to model (2) is explored. Our results will extend some classical deterministic results into the stochastic cases.

2. Preliminary

Let (Ω,ℱ,{ℱt}t≥0,P), throughout this paper unless otherwise specified, be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions, and Bi(t) (i=1,2,…,n) denote the independent 1-dimensional standard Brownian motion defined on the complete probability space. We denote the nonnegative cone and positive cone by R+n, R++n, respectively; that is, R+n={x∈Rn:xi≥0,i=1,2,…,n}, R++n={x∈Rn:xi>0,i=1,2,…,n}. If x∈Rn, its norm is denoted by |x|=(∑i=1nxi2)1/2; if A is a vector or matrix, its transpose is denoted by AT; if A is matrix, trace norm of matrix A is denoted by |A|=trace(ATA); if A is negative matrix, we denote it by A<0. Suppose that y(t) is a continuous bounded function on [0,+∞); we define yu=maxt∈[0,+∞)y(t), yl=mint∈[0,+∞)y(t), with usual assumption inf∅=∞, where ∅ denotes the empty set.

3. The Existence and Uniqueness of Global Positive Solution

In this section, we show that model (2) has a unique global solution, and the solution will remain in R++n with probability 1.

Theorem 1.

If (H1) and (H2) hold, then there is a unique solution x(t) to model (2). Moreover, x(t) remains in R++n with probability 1, where t∈R.

Proof.

Clearly, the coefficients of model (2) satisfy local Lipschitz continuous but do not satisfy the linear growth condition. To show that the solution x(t) is global, a.s., τe=∞ is needed now, where τe is the explosion time.

Let k0>0 be sufficiently large such that each component of initial data ξ(0) is lying in the interval (1/k0,k0). For each integer k≥k0, define the stopping time
(4)τk=inf{t∈[0,τe):xi∉(1k,k),00000000forsomei=1,2,…,n(1k,k)}.
Clearly, τk is increasing as k→∞. Set τ∞=limk→∞τk; hence, τ∞≤τe a.s.. If we can prove that τ∞=∞ a.s., then τe=∞ a.s.. To prove this statement, let us define a C2-function V1:R++n→R++ by V1(x)=∑i=1nxip-plogxi, where p∈(0,1). It is easy to see that V1(x)>0 for all x∈R++n. Applying the Itô formula to model (2), it leads to
(5)dV1(x(t))=LV1(x(t))dt+∑i=1np(xip(t)-1)gidBi(t),
where
(6)LV1(x)=p∑i=1n(xip-1)fi+12p∑i=1n((p-1)xip+1)gi2.
From the elementary inequality ab≤(1/2)(a2+b2) and Hölder's inequality, we have
(7)∑i=1n(xip-1)fi≤∑i=1nhiu(xip+1)+12∑i=1n∑j=1n|aij|u(xip-1)2+12∑i=1n∑j=1n|aij|uxj2+12∑i=1n∑j=1n|bij|u2(xip-1)2+12∑i=1n∑j=1nxj2(t-δ(t))+12∑i=1n∑j=1n|cij|u2(xip-1)2+12∑i=1n∑j=1n∫-∞0xj2(t+θ)dμ(θ).
By the fact that K=(∥ξ∥Cr+∥x∥Cr)2<∞, u2/ρj-v2/(ρj-1)≤(u+v)2≤u2/li+v2/(1-li), 0<li<1<ρj, and (∑i=1naibi)2≤∑i=1nai2∑i=1nbi2, it implies that
(8)gi2≤(xi-xi*)2{∑j=1n∫-∞01l1σiu2+1l2(1-l1)0000000000000×∑j=1n|αij|u2∑j=1nxj20000000000000+1l3(1-l2)(1-l1)∑j=1n|βij|u20000000000000×∑j=1nxj2(t-δ(t))0000000000000+1(1-l3)(1-l2)(1-l1)∑j=1n|γij|u20000000000000×∑j=1n∫-∞0xj2(t+θ)dμ(θ)}≤(xi-xi*)2{1l1σiu2+Kl2(1-l1)∑j=1n|αij|u20000000000000+Kl3(1-l2)(1-l1)∑j=1n|βij|u20000000000000+K(1-l3)(1-l2)(1-l1)∑j=1n|γij|u2}≔K1(xi-xi*)2,gi2≥(xi-xi*)2{σil2+(∑j=1nαij(t)xj)2000000000000+(∑j=1nβij(t)xj(t-δ(t)))2000000000000+(∑j=1nγij(t)∫-∞0xj2(t+θ)dμ(θ))2}≥σil2(xi-xi*)2∶=K2(xi-xi*)2,
where 0<K2<K1 for any li∈(0,1). Then (6) yields that
(9)LV1(x)≤∑i=1n{p(1-p)K22phiu(xip+1)000000+p2∑j=1n(|aij|u+|bij|u2+|cij|u2)(xip-1)2000000+p2∑j=1n|aij|uxj2+pK12(xi-xi*)2000000-p(1-p)K22xip(xi-xi*)2}+p2∑i=1n∑j=1nxj2(t-δ(t))+p2∑i=1n∑j=1n∫-∞0xj2(t+θ)dμ(θ).
Define V2(x(t))=V1(x(t))+(p/2)∑i=1n∑j=1n∫t-δ(t)txj2(s)ds+(p/2)∑i=1n∑j=1n∫-∞0∫t+θtxj2(s)dsdμ(θ); the proof is easily checked (details can be found at the appendix).

4. Global Asymptotic StabilityTheorem 2.

If (H1) and (H2) hold, there exist positive numbers d¯1,d¯2,…,d¯n such that D¯A¯+A¯TD¯ is negative definite, and then
(10)limt→∞xi(t)=xi*a.s.,i=1,2,…,n;
that is, x* is globally asymptotically stable a.s., where K=(∥ξ∥Cr+∥x∥Cr)2, D¯=diag(d¯1,d¯2,…,d¯n), A¯=(ηij)n×n, ηij=aij for i≠j, and
(11)ηii=(aii)u+0.5xi*[σiu2+K|∑j=1nαij|u2000000000000000+K|∑j=1nβij|u2+K|∑j=1nγij|u2|∑j=1nαij|u2]+0.5[∑j=1n|bij|u+∑j=1n|cij|u000000000+∑j=1nd¯jd¯i|bji|u+∑j=1nd¯jd¯i|cji|u].

Proof.

Define a C2-function V3:R++n→R++ by V3(x)=∑i=1nd¯i(xi-xi*-xi*log(xi/xi*)). Similar to the proof of [6] (see Theorem 2.1), we can derive
(12)dV3(x(t))≤LV3(x(t))dt+∑i=1nd¯i(xi(t)-xi*)gidBi(t)≤12(xi(t)-xi*)(D¯A¯+A¯TD¯)(xi(t)-xi*)Tdt+∑i=1nd¯i(xi(t)-xi*)gidBi(t).
Since D¯A¯+A¯TD¯ is negative definite, LV3(x)<0 is valid along trajectories in R++n except x*. The proof is complete.

5. Approximation Distribution of Solution Theorem 3.

If (H1), (H2), and the conditions of Theorem 2 hold, then each component xi(t) of solution x(t) to model (2) follows asymptotically 1-dimensional normal distribution 𝒩(xi*,+∞), where i=1,2,…,n.

Proof.

By the definition of equilibrium state xi*≠0, then fi(xi*)=0. Since x* is stable for the deterministic part to model (2), then ∂xi(fi(x)xi)|xi=xi*∶=mi<0 for i=1,2,…,n.

Linearizing the ith equation of model (2) by Taylor expansion at xi*, then we have
(13)d(xi-xi*)≈[fi(xi*)xi*+∂xi(fi(x)xi)|xi=xi*0000×(xi-xi*)+o((xi-xi*)2)]dt+[gi(xi*)xi*+∂xi(gi(x)xi)|xi=xi*0000×(xi-xi*)+o((xi-xi*)2)]dBi(t).
Denote xj(t-δ(t))=xjδ(t) and ∫-∞0xj(t+θ)dμ(θ)=xjθ(t), j=1,2,⋯,n. Since gi(xi*)=0 and ∂xi(gi(x)xi)|xi=xi*=σixi*+xi*∑j=1n(αijxj+βijxjδ+γijxjθ)|xi=xi*∶=ui≠0, (13) can be simplified as
(14)d(xi-xi*)dt≈mi(xi-xi*)+ui(xi-xi*)dBi(t)dt,i=1,2,…,n,
where mi<0, ui≠0. Then (14) implies that
(15)xi(t)-xi*=(xi(0)-xi*)emit+uiBi(t),i=1,2,…,n.
From the definition of 1-dimensional Brownian motion, we can derive that xi(t)~𝒩(xi*+(xi(0)-xi*)emit, (xi(0)-xi*)2eui2t). According to the conditions of Theorem 2, when t→+∞, one can find that ui→ui*=σixi*+xi*∑j=1n(αijxj*+βijxjδ*+γijxjθ*)≠0 a.s.; thus, limt→+∞xi(t)~𝒩(xi*+∞); in other words, xi(t) asymptotically follows 1-dimensional normal distribution. When t tends to ∞, the mean of solution x(t) to model (2) is x*; it is just consistent with the conclusion of Theorem 2. However, the deviation between solution xi(t) and mean xi* may approach infinity; it is bad information for further analysis. Now, if the variance can be evaluated, the disadvantage will be improved.

6. Moment Estimation

First, let us prove one useful moment estimation.

Theorem 4.

If (H1) and (H2) hold, then there is a positive constant G=G(p), such that limsupt→∞E|x(t)|p≤G, where p∈(0,1).

Proof.

Define a C2-function V4:R++n→R++ by V4(x)=∑i=1nxip. For any given ɛ∈(0,2r), applying the Itô formula to eɛtV4(x(t)) and taking expectation, it yields that
(16)eɛtEV4(x(t))=EV4(ξ(0))+E∫0teɛs[LV4(x(s))+ɛV4(x(s))]ds,
where
(17)LV4(x)=p∑i=1nxipfi-p(1-p)2∑i=1nxipgi2.
By fundamental inequality ab≤(1/2)(a2+b2),
(18)∑i=1nxipfi≤∑i=1nhiuxip+12∑i=1n∑j=1n|aij|uxi2p+12∑i=1n∑j=1n|aij|uxj2+12∑i=1n∑j=1n|bij|u2xi2p+12∑i=1n∑j=1n|cij|u2xi2p+12∑i=1n∑j=1nxj2(s-δ(s))+12∑i=1n∑j=1n∫-∞0xj2(s+θ)dμ(θ),∑i=1nxipgi2≥∑i=1nK2xip(xi-xi*)2.
Consequently,
(19)LV4(x(s))≤p∑i=1nhiuxip+p2∑i=1n∑j=1n(|aij|u+|bij|u2+|cij|u2)xi2p+p2∑i=1n∑j=1n|aij|uxj2+p2∑i=1n∑j=1nxj2(s-δ(s))+12∑i=1n∑j=1n∫-∞0xj2(s+θ)dμ(θ)-p(1-p)2∑i=1nK2xip(xi-xi*)2∶=H(x)-ɛV4(x)+np2[|x(s-δ(s))|2-eɛτ|x|2]+np2[∫-∞0|x(s+θ)|2dμ(θ)-μr|x|2],
where
(20)H(x)=∑i=1n(phiu+ɛ)xip+p2∑i=1n∑j=1n(|aij|u+|bij|u2+|cij|u2)xi2p+np2(eɛτ+μr)|x|2+p2∑i=1n∑j=1n|aij|uxj2-p(1-p)K22∑i=1nxip(xi-xi*)2.
Noting that H(x) is bounded in R++n; namely, K3∶=supx∈R++nH(x)<∞. Substituting this into (16), it thus follows that
(21)eɛtEV4(x(t))≤EV4(ξ(0))+1ɛK3(eɛt-1)+np2×E∫0teɛs[∫-∞0|x(s-δ(s))|2-eɛτ|x(s)|200000000+∫-∞0|x(s+θ)|2dμ(θ)0000000000-μr|x(s)|2∫-∞0|x(s-δ(s))|2]ds.
By (H1) and (H2), we have
(22)E∫0teɛs[|x(s-δ(s))|2-eɛτ|x(s)|2]ds≤E[∫-τteɛ(s+τ)|x(s)|2ds-∫0teɛ(s+τ)|x(s)|2ds]≤E∫-τ0eɛτ|x(s)|2ds,E∫0teɛs[∫-∞0|x(s+θ)|2dμ(θ)-μr|x(s)|2]ds=E{∫0teɛs[∫-∞-se2r(s+θ)|x(s+θ)|200000000000000×e-2r(s+θ)dμ(θ)∫-∞-s]ds00000000+∫0teɛs∫-s0|x(s+θ)|2dμ(θ)ds00000000-μr∫0teɛs|x(s)|2ds}≤E{∥ξ∥Cr2∫0te(ɛ-2r)sds∫-∞0e-2rθdμ(θ)00000000+∫-t0∫-θteɛs|x(s+θ)|2dsdμ(θ)00000000-μr∫0teɛs|x(s)|2ds}≤E{∥ξ∥Cr2μr∫0te(ɛ-2r)sds0000000+∫-∞0∫0teɛ(s-θ)|x(s)|2dsdμ(θ)0000000-μr∫0teɛs|x(s)|2ds}≤E{∥ξ∥Cr2μrt+∫-∞0e-ɛθdμ(θ)0000000×∫0teɛs|x(s)|2ds0000000-μr∫0teɛs|x(s)|2ds}≤E∥ξ∥Cr2μrt.
Hence, we derive
(23)eɛtEV4(x(t))≤EV4(ξ(0))+K3eɛtɛ+np2(Eμr∥ξ∥Cr2t+E∫-τ0eɛτ|x(s)|2ds).
This implies that limsupt→∞EV4(x(t))≤ɛ-1K3.

Again, by the fact that |x|2≤nmax1≤i≤nxi2, it then follows that |x|p≤np/2max1≤i≤nxip≤np/2V4(x), and we have
(24)limsupt→∞E|x(t)|p≤G(p);
the assertion follows by setting G=G(p)=np/2ε-1K3. The proof is complete.

Remark A. If (H1) and (H2) are valid, then limsupt→∞P{|x(t)|≤χ}≥1-ε; that is, solution x(t) to model (2) is stochastically ultimately bounded.

If we take p=1/2, then the result is valid by using of Theorem 4 and Chebyshev's inequality. The proof is omitted herewith.

Theorem 5.

If (H1) and (H2) hold, there exists a positive constant Q, and then
(25)limsupT→∞1T∫0TD(xi(t))dt≤Q,i=1,2,…,n.

Proof.

Rewrite (20) as
(26)H(x)=H1(x)-(np+1)|x|2+np2(eɛτ+μr)|x|2+ɛV4(x),
with
(27)H1(x)=∑i=1nphiuxip+p2∑i=1n∑j=1n(|aij|u+|bij|u2+|cij|u2)xi2p+p2∑i=1n∑j=1n|aij|uxj2+(np+1)|x|2-p(1-p)2∑i=1nK2xip(xi-xi*)2.
Clearly, H1(x) is bounded in R++n; namely, K4∶=supx∈R++nH1(x)<∞. So
(28)H(x)≤K4-(np+1)|x|2+np2(eɛτ+μr)|x|2+ɛV4(x).
By (19), we have
(29)dV4(x(t))≤{+∫-∞0|x(t+θ)|2dμ(θ)]K4-(np+1)|x(t)|2000+np2[∫-∞0|x(t-δ(t))|2000000000+∫-∞0|x(t+θ)|2dμ(θ)]}dt+p∑i=1nxip(t)gidBi(t).
Integrating both sides of (29) from 0 to T (T>0 is arbitrary) and then taking expectations, we obtain that
(30)E(V4(x(T)))≤V4(ξ(0))+K4T-E∫0T|x(t)|2dt+np2I1+np2I2.
Similar to (22), by the definition of δ(·), we can derive
(31)I1=E∫0T(|x(t-δ(t))|2-|x(t)|2)dt≤∫-τ0|x(t)|2dt,I2=E∫0T[∫-∞0|x(t+θ)|2dμ(θ)-|x(t)|2]dt≤12rμr∥ξ∥Cr2.
Substituting (31) into (30), then we have
(32)limsupT→∞1T∫0TE|x(t)|2dt≤K4∶=Q.
By the fact that D(xi)=E|xi|2-(Exi)2 and limt→∞E(x(t))=x*, the conclusion is obviously valid. The proof is complete.

7. Example and Numerical Simulation

Now, we will simulate asymptotic behaviors of solutions to model (2). Let us consider the following 1-dimensional stochastic autonomous Logistic model:
(33)dx(t)=x(t)[∫-∞0h-ax(t)-bx(t-τ)0000000-c∫-∞0x(t+θ)dμ(θ)]dt+x(t)[∫-∞0σ(x(t)-x*)+αx(t)(x(t)-x*)000000000+βx(t-τ)(x(t)-x*)000000000+γ∫-∞0x(t+θ)dμ(θ)000000000×(x(t)-x*)∫-∞0]dB(t),
with initial data ξ(θ)∈Cr, where h, a, b, c, τ, σ, α, β, and γ are all positive numbers, and the equilibrium state is x*=h/(a+b+c). By Theorem 2, if a>0.5x*(σ2+Kα2+Kβ2+Kγ2), then the equilibrium state x* of model (33) is globally asymptotically stable a.s.. According to Milstein method mentioned in Higham [11], the initial data of model (33) is given by ξ(θ)=0.32e0.5θ and μ(θ)=eθ, θ∈(-∞,0]; the difference equation is followed next
(34)x(k+1)-x(k)=x(k)[∑i=1kh-ax(k)-bx(k-τ/Δt)0000000-2c3e-kΔt-ce-kΔt∑i=1kx(i)eiΔtΔt]Δt+σx(k)(x(k)-x*)Δtξ(k)+σ22x(k)(x(k)-x*)((ξ(k))2-1)Δt+αx(k)x(k)(x(k)-x*)Δtξ(k)+α22x(k)x(k)(x(k)-x*)((ξ(k))2-1)Δt+βx(k)x(k-τ/Δt)(x(k)-x*)Δtξ(k)+β22x(k)x(k-τ/Δt)(x(k)-x*)((ξ(k))2-1)Δt+γ(2c3e-kΔt+ce-kΔt∑i=1kx(i)eiΔtΔt)x(k)×(x(k)-x*)Δtξ(k)+γ22(2c3e-kΔt+ce-kΔt∑i=1kx(i)eiΔtΔt)x(k)×(x(k)-x*)((ξ(k))2-1)Δt,
where ξ(k)(k=1,2,…,n) is the Gaussian random variable which follows the standard normal distribution 𝒩(0,1). If we fix the parameters h=0.5, a=0.6, b=0.1, c=0.3, τ=1, and Δt=0.01, then x*=h/(a+b+c)=0.5. When σ=α=β=γ=0, model (33) becomes a deterministic one; then the equilibrium state x* is globally asymptotically stable a.s. by Theorem 2 (see Figure 1). When σ=0.1 and α=β=γ=0, the intrinsic growth rate of model (33) is perturbed by white noise; if a>0.5x*σ2+b+c, the equilibrium state x* is globally asymptotically stable a.s. (see Figure 2). When σ=0.1 and α=β=γ=0.01, that is, each parameter of model (33) is fluctuated by white noise, if a>0.5x*(σ2+Kα2+Kβ2+Kγ2)+b+c and K=(0.32e(r+0.5)θ+x(k))2≤(0.32+x(k))2 are satisfied, then the equilibrium stable x* is globally asymptotically stable a.s. (see Figure 3).

Model (33) becomes a deterministic model.

The intrinsic growth rate of model (33) is perturbed by white noise.

Each parameter of model (33) is fluctuated by white noise.

Appendix

From the definition of V2(t), we obtain
(A.1)dV2(x(t))≤F(x(t))dt+∑i=1np(xip(t)-1)gidBi(t),
where
(A.2)F(x)=∑i=1n{p(1-p)K22phiu(xip+1)000000+p2∑j=1n(|aij|u+|bij|u2+|cij|u2)(xip-1)2000000+p2∑j=1n(|aij|u+2)xj2+pK12(xi-xi*)2000000-p(1-p)K22xip(xi-xi*)2}.
It is easy to check that F(x)≤K′<∞. Equation (A.1) becomes
(A.3)dV2(x(t))≤K′dt+∑i=1np(xip(t)-1)gidBi(t).
Integrating both sides of (A.3) from 0 to τk∧T (T>0 is arbitrary) and then taking expectations, it yields that
(A.4)EV1(x(τk∧T))≤EV2(x(τk∧T))≤V2(ξ(0))+K′T.
Noting that for every w∈Ωk={τk≤T}, by the definition of stopping time τk, xi(τk,w)=k or 1/k for some i=1,2,…,n, V1(x(τk∧T))≥min{kp-plogk,k-p+plogk}. It then follows that
(A.5)P{τk≤T}(kp-plogk)∧(k-p+plogk)≤E[I{τk≤T}(w)V2(x(τk,w))]≤V2(ξ(0))+K′T.

Thus, limk→∞P{τk≤T}=0. Since T>0 is arbitrary, we must have P{τ∞<∞}=0; then P{τ∞=∞}=1 as required.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 11201075), the Natural Science Foundation of Fujian Province of China (no. 2010J01005), and the Technology Innovation Platform Project of Fujian Province (no. 2009J1007).

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