Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation

Computers and Mathematics with Applications 66 (2013) 2220–2230

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Almost sure instability of the equilibrium solution of a Milstein-type stochastic difference equation Cónall Kelly, Peter Palmer, Alexandra Rodkina ∗ Department of Mathematics, The University of the West Indies, Mona Campus, Kingston, Jamaica

article

abstract

info

Keywords: Stochastic difference equations A.s. instability Discrete Itô formula Multiplicative noise Martingale convergence

We derive a condition guaranteeing the almost sure instability of the equilibrium of a stochastic difference equation with a structure motivated by the Euler–Milstein discretisation of an Itô stochastic differential equation. Our analysis relies upon the convergence of non-negative martingale sequences coupled with a discrete form of the Itô formula and requires a distinct variant of this formula for each of the linear and nonlinear cases. The conditions developed in this article appear to be quite sharp. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Consider the nonlinear Itô stochastic differential equation given by dX (t ) = X (t )f (X (t ))dt + X (t )g (X (t ))dB(t ),

t ≥ 0,

(1)

with X (0) ∈ R, where f and g are bounded real-valued functions, and B is a one-dimensional Brownian motion. If f and g are required to be locally Lipschitz continuous, then unique strong solutions exist for (1) and the equation has an equilibrium solution X (t ) ≡ 0 when X (0) = 0. For details on the existence and uniqueness of solutions, see Mao [1] and Karatzas and Shreve [2]. Equations of this form (with positive initial conditions) arise as single-species population models under intrinsic stochastic perturbation (see, for example, Murray [3]), and in that context the equilibrium may be interpreted as the absence of the species. If the equilibrium solution is almost surely (a.s.) unstable, then the species will persist in the long run, with probability one. In this article, we are concerned with the nonlinear stochastic difference equation that results from applying a Euler– Milstein discretisation to (1):



Xn+1 = Xn 1 + hf (Xn ) +

√

  2 hg (Xn )ξn+1 + g (Xn ) g (Xn ) + Xn g (Xn ) (ξn+1 − 1) , h



2

′

n ∈ N,

(2)

with X0 ∈ R, h > 0 and {ξn }n∈N is a sequence of mutually independent N (0, 1) random variables. Let the coefficients f and g be bounded real-valued functions, with g continuous and differentiable in all of R, and |xg ′ (x)| bounded for all x ∈ R. Solutions of (2) are defined on a complete filtered probability space (Ω , F , {Fn }n∈N , P) where {Fn }n∈N is the filtration generated by the sequence {ξn }n∈N . We develop conditions on the relationship between f and g ensuring that, when X0 ̸= 0 and

∗

Corresponding author. Tel.: +876 957 67 87. E-mail addresses: [email protected] (C. Kelly), [email protected] (P. Palmer), [email protected], [email protected] (A. Rodkina). 0898-1221/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.camwa.2013.06.020

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

2221

for h sufficiently small,





P lim inf |Xn | > 0 = 1.

(3)

n→∞

Notice that (2) has an equilibrium solution Xn ≡ 0 when X0 = 0, and so (3) implies the a.s. instability of the equilibrium. The Euler–Milstein numerical method is a stochastic Taylor approximation with strong order of convergence 1, by contrast with the Euler–Maruyama method, which has a strong order of convergence 1/2; for details see Kloeden and Platen [4]. A linear stability analysis in mean-square of the method applied to various test systems has been carried out in Buckwar and Sickenberger [5], but we are unaware of any similar stability analysis in the a.s. sense. This article can be viewed as a first step in that direction. In the proof of our results we will use a variant of the discrete Itô formula first introduced by Appleby, Berkolaiko and Rodkina [6]. Other discrete-time analogues of the Itô formula have been developed previously; see for example Fujita [7] and Shiryaev [8]. The particular form developed in [6] has been used to examine the a.s. stability properties of the θ -Maruyama numerical method with small stepsize applied to a two-dimensional test system in Berkolaiko, Buckwar, Kelly and Rodkina [9], and to place explicit bounds on the stepsize for which the Euler–Maruyama numerical method can be guaranteed to recover the exact a.s. asymptotic stability properties of a scalar linear test equation in Palmer [10]. It turns out that separate variants must be derived for the classes of difference equations considered in this article. The paper is structured as follows. In Section 2.1 we provide background on the convergence of martingale sequences that will form an important part of our analysis, followed in Section 2.2 by an outline of the analytic strategy of the paper. There we explain the role played by the variants of the discrete form of the Itô formula developed in this paper for Eq. (2). In Section 3 we state and prove a variant of the discrete Itô formula for application in the special case where f and g are constants and determine a sufficient condition for (3) to hold. In Section 4 we generalise the analysis to include nonlinear bounded coefficients. 2. Background 2.1. On the convergence of martingale sequences Suppose that (Ω , F , P) is a probability space equipped with filtration {Fn }n∈N . The following definition, along with a detailed exposition of the theory of random sequences, may be found in Shiryaev [8]. We will require that all sequences of interest be Fn -measurable. Definition 2.1. A stochastic sequence {Mn }n∈N is an Fn -martingale, if for all n ∈ N, (i) E|Mn | < ∞; (ii) E[Mn |Fn−1 ] = Mn−1 , a.s. Lemma 2.2. Let {Yi }i∈N be a sequence of non-negative random variables defined on (Ω , F , P) and adapted to the filtration {Fn }n∈N , where each Yi satisfies the following: (i) E[Yi ] < ∞; (ii) E[Yi |Fi−1 ] = 1. Then the sequence {Mn }n∈N given by Mn =

n 

Yi ,

n ∈ N,

i =1

is an Fn -martingale. Remark 2.3. The sequence {Mn }n∈N from the statement of Lemma 2.2 is provided as an example in Shiryaev [8] and Williams [11]. However both authors assume that terms of the sequence {Yi }i∈N are mutually independent. We will be unable to make this assumption, replacing it instead with the requirement that each term be non-negative. Therefore we provide a short proof for completeness. Proof of Lemma 2.2. Property (ii) from Definition 2.1 may be shown as follows:

 E[Mn |Fn−1 ] = E

n 

 Yi |Fn−1

= E [Yn Mn−1 |Fn−1 ] = Mn−1 E [Yn |Fn−1 ] = Mn−1 .

i=1

Property (i) then follows from (4):

E|Mn | = EMn = E[E[Mn |Fn−1 ]] = EMn−1 = · · · = EM1 = EY1 < ∞. 

(4)

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C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

The following lemma, and the associated proof, can be found in Shiryaev [8, p. 509, Corollary 3]. Lemma 2.4. If {Mn }n∈N is a non-negative martingale, then limn→∞ Mn exists with probability one. Finally, for a, b ∈ [−∞, ∞] we let Lq [a, b] denote the family of Borel-measurable functions f : [a, b] → R such that

b a

|f (t )|q dt < ∞, a.s.

2.2. Analytic strategy In order to derive conditions under which the equilibrium solution of the stochastic difference equation (2) is a.s. unstable, we take the following approach. 1. Express solutions of (2) in the form Xn+1 = Xn Un+1 , where terms of the sequence {Un }n∈N are Fn -measurable with respect to the filtration generated by the sequence {Xn }n∈N . 2. For α ∈ (0, 1), write

|Xn |α = |X0 |α

n−1 

|Un+1 |α = |X0 |α Mn−1

i=0

n−1 

1

i=0

E[|Ui+1 |−α |Fi ]

,

(5)

where the non-negative sequence defined by Mn =

n −1  i =0

|Un+1 |−α , E[|Ui+1 |−α |Fi ]

is an Fn -martingale, by Lemma 2.2. 3. Use a discrete form of the Itô formula to show that, when the stepsize parameter h is sufficiently small, E[|Ui+1 |−α |Fi ] < 1, for all i ∈ N. 4. By Lemma 2.4, Mn tends to a non-negative limit a.s. and therefore, with probability one, Mn−1 will eventually be bounded away from zero. 5. For sufficiently small stepsize h, the a.s. instability of non-equilibrium solutions of (2) then follows from the fact that all factors on the right-hand-side of (5) are bounded away from zero (or will be eventually) on an event of probability one. Therefore the core of our analysis will be in the derivation of discrete forms of the Itô formula that allow us to express the conditional expectation E[|Ui+1 |−α |Fi ] explicitly in terms of the parameters of Eq. (2), and to use such formulae to determine the conditions on f and g under which each E[|Ui+1 |−α |Fi ] < 1. We look first at the linear case, since the derivation of a discrete Itô formula is considerably more complicated in the nonlinear case. 3. Case 1: Linear equation Suppose that f (x) = λ and g (x) = σ ̸= 0 for all x ∈ R. In this case Eq. (2) has the form

 Xn+1 = Xn

1 + hλ +

√

hσ ξn+1 +

h 2

σ (ξ 2

2 n +1



− 1) ,

n ∈ N.

(6)

Definition 3.1. Define the random sequence {Un }n∈N by Un+1 := 1 + hλ +

√

hσ ξn+1 +

h 2

σ 2 (ξn2+1 − 1),

n ∈ N.

(7)

3.1. A discrete form of the Itô formula for (7) Lemma 3.2. Suppose φ : R \ {0} → R is such that, for any δ > 0, φ ′′′ exists and is bounded on (−∞, −δ] ∪ [δ, ∞). If {Un }n∈N is defined as in (7), then there exists h0 > 0 such that, for all h < h0

E [φ(Un+1 )|Fn ] = φ(1) + φ ′ (1)λh +

φ ′′ (1) 2

h(σ 2 + (λ + σ 2 )O (h)) +

φ ′′′ (θ ) 6

h(λ + σ 2 )O (h),

(8)

where θ is between 1 and Un+1 , and the error terms satisfy limh→0 O (h) = 0 uniformly in n ∈ N. Remark 3.3. Since each ξi is continuously distributed on R, each Ui will take a value of zero on a set of probability zero, which will not contribute to the value of the expectation on the LHS of (8). Therefore there is no technical issue raised by the fact that φ need not be defined at the origin.

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

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Proof. We wish to identify ϵ > 0 such that Un+1 ≥ ϵ > 0 for all n ∈ N. This will ensure that φ ′′′ (θ ) is bounded, and we can apply a Taylor series expansion to φ(Un+1 ). Completing the square, write Un+1 in the form Un+1 =

h



2

1

σ ξ n +1 + √

2 +

h

1 2

  σ2 +h λ− . 2

If λ − σ /2 < 0 we choose 2

h0 <

1



4

σ2 2

 −1 −λ ,

and then, for h < h0 , Un+1 ≥

1 2

  σ2 1 − h0 λ − > = ϵ. 2

4

If λ − σ /2 ≥ 0, we have for all h > 0, 2

Un+1 ≥

1 2

>

1 4

= ϵ.

In this latter case, we take h0 := ∞. In either case, when h < h0 , Un+1 defined by (7) will never take values in an interval (−δ, δ), where δ < 1/4. Denote un+1 := hλ +

√

hσ ξn+1 +

h 2

σ 2 (ξn2+1 − 1),

so that Un+1 = 1 + un+1 . Note that E (un+1 |Fn ) = E (un+1 ), since the coefficients of ξn+1 and ξn2+1 are constants. Hence we can write

E(un+1 ) = hλ,

E(u2n+1 ) = hσ 2 + h(λ + σ 2 )O (h),

E(u3n+1 ) = h(λ + σ 2 )O (h),

where

|O (h)| ≤ C¯ h, C¯ is non-random and depends only on the constant parameters making up the coefficients of (6), and moments of ξn for n ∈ N. So C¯ is independent of n ∈ N since {ξn }n∈N is a sequence of mutually independent N (0, 1) random variables. Recall  that, by requiring that h < h0 , we are assured that Un+1 ∈ 41 , ∞ for each n ∈ N, and hence θ ∈ [1/4, ∞). A Taylor series expansion of φ(1 + u) around u = 0 therefore yields (8), where ′′′ |φ ′′′ (θ )| ≤ max {|φ (y)|} < ∞.   y∈ 14 ,∞

3.2. Almost sure instability of the equilibrium of (6) Theorem 3.4. Let {Xn }n∈N be any solution of (6) where X0 ̸= 0, and suppose that 2λ > σ 2 > 0.

(9)

¯ Then there exists h¯ > 0 such that for all h < h,   P lim inf |Xn | > 0 = 1.

(10)

n→∞

Proof. Set φ(x) = |x|−α , where α ∈ (0, 1), and choose h < h0 sufficiently that Lemma 3.2 gives

E |Un+1 |−α |Fn = 1 − αλh +





= 1−

α(α + 1) 2

h(σ 2 + (λ + σ 2 )O (h)) + h(λ + σ 2 )O (h)

αh  2

2λ − (α + 1)σ 2 + (λ + σ 2 )O (h) .



(11)

In order to prove the instability of the equilibrium solution of (6) we need to show that there is α ∈ (0, 1) and 0 < h¯ < h0 such that 2λ − (α + 1)σ 2 + (λ + σ 2 )O (h) > 0,

for all h < h¯ .

(12)

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C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

By (9), there exists γ > 0 such that 2λ/σ 2 = 1 + γ . Let α ∈ (0, 1) and ε0 > 0 be such that

α<γ

and

ε0 <

2(γ − α) 3+γ

.

(13)

¯ the error term O (h) from (11) satisfies O (h) ≤ ε0 . Then, for h < h¯ we Choose h¯ < h0 sufficiently small that for all h < h, have 2λ − (α + 1)σ 2 + (λ + σ 2 )O (h) ≥ 2λ − (α + 1)σ 2 − ε0 λ − ε0 σ 2

= (2 − ε0 )λ − (α + 1 + ε0 )σ 2 = σ 2



2λ 2 − ε0

− (α + 1 + ε0 )



σ2 2   2 − ε0 3+γ = σ 2 (1 + γ ) − (α + 1 + ε0 ) = γ − α − ε0 > 0, 2

2

(14)

so (12) holds. Finally, let {Xn }n∈N be any solution of (6) with X0 ̸= 0. Then we have

|Xn+1 |α = |Xn |α |Un+1 |α = |X0 |α

n 

|Ui+1 |α = |X0 |α

i=0

1 n 

|Ui+1 |−α

i=0

= |X0 |α

1 n

 i=0

× |−α

|Ui+1 E(|Ui+1 |−α |Fi )

n 

1

i =0

E (|Ui+1 |−α |Fi )

.

We set Mn :=

n  i=0

|Ui+1 |−α , E (|Ui+1 |−α |Fi )

  Ψn := E |Un+1 |−α |Fn .

From (11) and (14) we conclude that Ψn < 1 a.s. Since {Mn }n∈N is a non-negative martingale it converges to a finite limit c (ω) ∈ [0, ∞) with probability 1 by Lemma 2.4. So there exists an a.s. finite random variable N (ω) ∈ N, such that, for 2|X0 |α n ≥ N (ω), we have: |Xn (ω)|α > c (ω) > 0 a.s. on Ω . The result follows.  Remark 3.5. When setting up Eq. (2), we assume that {ξn }n∈N is a sequence of independent N (0, 1) random variables. This assumption allows us to make an explicit connection between Eq. (2) and the Itô differential equation (1) via the Euler– Milstein discretisation method. Indeed we see that condition (9) corresponds exactly to the necessary and sufficient condition 2λ−σ 2 > 0 for the a.s. instability of the equilibrium of (1). However, notice that the proof of Lemma 3.2 relies only upon the first six moments of each ξi and does not require any other properties of the density. So this result can also apply to random sequences solving (2) where the terms of {ξn }n∈N are non-Gaussian, but have the appropriate moment properties. 4. Case 2: Nonlinear equation Now consider the nonlinear equation



Xn+1 = Xn 1 + hf (Xn ) +

√

  2 hg (Xn )ξn+1 + g (Xn ) g (Xn ) + Xn g (Xn ) (ξn+1 − 1) , h



2

′

n ∈ N,

(15)

with nonlinear coefficients f : R → R and g : R → R satisfying

|f (x)|, |g (x)| ≤ K1 ,

for all x ∈ R,

(16)

and

|xg ′ (x)| ≤ K2 ,

for all x ∈ R,

(17)

where K1 , K2 > 0. Further denote, for each x ∈ R, g¯ (x) :=

1 2

g (x)(g (x) + xg ′ (x)).

Conditions (16) and (17) imply that

|¯g (x)| ≤ K3 ,

for all x ∈ R, K3 :=

K1 2

(K1 + K2 ).

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

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Example 4.1. The function g (x) = 2 + arctan(x),

x ∈ R,

satisfies conditions (16) and (17) with K1 = 4 and K2 = 1. Example 4.2. Let g (x) = x−1 2−2n when x ∈ (2−2n , 2−2n+1 ). We note that the intervals [2−2n , 2−2n+1 ] are non-overlapping for all n ∈ N, the maximum of g on each interval is 1 and the maximum of |xg ′ (x)| on each interval is 1 as well. Extending g on R in such a way that it is continuously differentiable yields a function satisfying

|g (x)|, |xg ′ (x)| ≤ 1,

x ∈ R.

Then conditions (16) and (17) hold true and g¯ (x) =

1 g 2

(x)(g (x) + xg ′ (x)) ≡ 0 for x ∈ [2−2n , 2−2n+1 ].

Definition 4.3. Following the template set out in Section 3, define Un+1 := 1 + hf (Xn ) +

√

hg (Xn )ξn+1 +

h 2

g (Xn )(g (Xn ) + xg ′ (Xn ))(ξn2+1 − 1),

n ∈ N.

(18)

4.1. A discrete form of the Itô formula for (18) Lemma 4.4. Consider φ : R \ {0} → R such that there exists δ > 0 and φ˜ : R → R satisfying (i) φ˜ ≡ φ on R \ Uδ , where Uδ = [−δ, δ],

(ii) φ˜ ∈ C 3 (R) and |φ˜ ′′′ (x)| ≤ M for some 0 < M < ∞ and all x ∈ R, (iii) φ − φ˜ ∈ Lq [−δ, δ] for some q > 2. Then for each n ∈ N,

E [φ(Un+1 )|Fn ] = φ(1) + φ ′ (1)f (Xn )h +

φ ′′ (1) 2

h[g 2 (Xn ) + (f (Xn ) + g 2 (Xn ))O (h)]

+ h(f (Xn ) + g 2 (Xn ))O (h),

(19)

where the error terms satisfy limh→0 O (h) = 0 uniformly in n ∈ N. Example 4.5. If we choose φ(x) = |x|−1/3 then the function

˜ x) = φ(

19 247 247 δ+6 2δ + 5 1 5δ − 12 x6 − x4 + x2 + , 19 / 3 13 / 3 7 162 (7δ − 15)δ 54 (7δ − 15)δ 54 (7δ)δ /3 162 (7δ − 15)δ 1/3  |x|−1/3 ,

  13

|x| ≤ δ; |x| > δ

satisfies properties (i)–(iii) in the statement of Lemma 4.4, with q ∈ (2, 3). Remark 4.6. Since the statement of Lemma 4.4 deals with an expectation conditioned on Fn , we can assume that ω ∈ Ω is fixed and consider the following cases: (i) (ii) (iii) (iv)

g (Xn (ω)) g (Xn (ω)) g¯ (Xn (ω)) g¯ (Xn (ω))

= 0; ̸= 0, g (Xn (ω)) + Xn (ω)g ′ (Xn (ω)) = 0; > 0; < 0.

Case (i) corresponds to the deterministic case. Case (ii) corresponds to the Euler–Maruyama discretisation and is treated in [6]. We will prove Lemma 4.4 for Case (iii); Case (iv) can be treated similarly. Remark 4.7. Note also that, if g 2 ( u) 4g¯ (u)

≤ 1 − ε0 ,

for all u ∈ R,

then it is possible to identify h0 > 0 and ε1 > 0 such that for all h ≤ h0 and all n ∈ N Un+1 ≥ ε1 .

(20)

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C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

ε

0 Suppose ε0 ∈ (0, 1) and (omitting arguments for simplicity) f − g¯ < 0 then choose h0 < 2(¯g − , which gives f)

Un+1

 g = hg¯ ξn+1 + √ =

2 +1−

2 hg¯

ε0

g2 4g¯

+ h(f − g¯ ) ≥ 1 − (1 − ε0 ) − h0 (f − g¯ ) > ε0 +

ε0 2(¯g − f )

(f − g¯ )

= ε1 .

2

Suppose f − g¯ ≥ 0 then for all h

 Un+1 = hg¯

2

g

ξn+1 + √

+1−

2 hg¯

g2 4g¯

+ h(f − g¯ ) ≥ 1 − (1 − ε0 ) + h(f − g¯ ) >

ε0 2

= ε1 .

In this instance, the proof of Lemma 4.4 proceeds in the same way as for Lemma 3.2, and Remark 3.5 applies: the proof relies only upon the first six moments of the sequence {ξn }n∈N . However, in general (20) will not hold, and the task of deriving (19) requires that we exploit the rate of decay of the tails of the density of each ξi – a much stronger constraint – and hence we must present the result separately. Proof of Lemma 4.4. As indicated in Remark 4.6, we assume that g¯ (Xn (ω)) > 0. Again, for the sake of simplicity we omit the argument Xn (ω) from the functions f , g and g¯ where possible. Note once again that, since g¯ > 0, we can write

 Un+1 = hg¯

2

g

ξn+1 + √

+1−

2 hg¯

g2 4g¯

+ h(f − g¯ ).

(21)

We begin by deriving a formula similar to (19) for φ˜ : if we set un+1 := Un+1 − 1 = hf (Xn ) +

√

hg (Xn )ξn+1 +

h 2

g (Xn )(g (Xn ) + Xn g ′ (Xn ))(ξn2+1 − 1),

we get

E (un+1 |Fn ) = hf (Xn ); E u2n+1 |Fn = hg 2 (Xn ) + h(f (Xn ) + g 2 (Xn ))O (h);





E u3n+1 |Fn = h(f (Xn ) + g 2 (Xn ))O (h).





Using a Taylor expansion of φ˜ around 1, just as in the proof of Lemma 3.2, we get





˜ Un+1 )|Fn = φ( ˜ 1) + φ˜ ′ (1)f h + E φ(

φ˜ ′′ (1) 2

h[g 2 + (f + g 2 )O (h)] + h(f + g 2 )O (h),

where, due to conditions (16) and (17), limh→0 O (h) = 0 and this convergence does not depend on n ∈ N and ω ∈ Ω . It remains to show that

  1 ˜ Un+1 )| |Fn = √ ∆n+1 := E |φ(Un+1 ) − φ(

2π



∞

˜ Un+1 (ξ ))|e−ξ |φ(Un+1 (ξ )) − φ(

2 /2

dξ

−∞

≤ Ch3/2 g 2 (Xn ),

(22)

where C does not depend on n ∈ N and ω ∈ Ω . Hence ∆n+1 will be absorbed in the last term of (19). Denote g2

Ln := 1 −

4g¯

+ h(f − g¯ ).

(23)

By the choice of φ˜ for the purpose of estimating the integral in (22) we can consider only |Un+1 | ≤ δ , i.e.



1

∆n+1 := √

2π

˜ Un+1 (ξ ))|e−ξ |φ(Un+1 (ξ )) − φ(

2 /2

dξ .

(24)

|Un+1 |≤δ

We recall here that ∆n+1 is a random variable which depends on Xn (ω) since it is an expectation conditioned on Fn . Since g¯ > 0, and using (23), we can write

 Un+1 = hg¯

g

ξn+1 + √

2

2 hg¯

+1−

g2 4g¯

+ h(f − g¯ ) ≥ 1 −

g2 4g¯

+ h(f − g¯ ) = Ln .

Since |Un+1 | ≤ δ then −δ ≤ Un+1 ≤ δ therefore δ ≥ Un+1 ≥ Ln . Hence Ln = 1 −

g2 4g¯

+ h(f − g¯ ) ≤ δ,

g2 4g¯

≥ 1 − δ + h(f − g¯ ).

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

2227

For Un+1 − Ln ≥ 0 we find ξ from Eq. (21)



g

ξ =− √

±

√ − 2√g g¯ ± Un+1 − Ln = . √ √ h g¯

Un+1 − Ln hg¯

2 hg¯

√ 2 g To estimate ∆n+1 we need a bound for e−ξ /2 which we will obtain by estimating − 2√g¯ ± Un+1 − Ln . Because we need √

g g a lower bound for ξ 2 we can consider estimating 2√g¯ − Un+1 − Ln and 2√g¯ + √ g √ − Un+1 − Ln from below we do the following transformation 2 g¯

g

√ − 2 g¯



Un+1 − Ln =



g

√ − 2 g¯ g2 4g¯

=

g √

2 g¯

Un+1 − 1 +

g2 4g¯

√

Un+1 − Ln separately. Firstly estimating

− h(f − g¯ )

2

− Un+1 + 1 − g4g¯ + h(f − g¯ ) −Un+1 + 1 + h(f − g¯ )  = . √ g 2 √ + Un+1 − Ln 2 g¯ + Un+1 − 1 + g4g¯ − h(f − g¯ )

Let

δ<

1 4

,

h<

1 4(K1 + K3 )

,

then

−δ + 1 − h(K1 + K3 ) ≥

1 2

.

Since

|Un+1 | ≤ δ,

|f − g¯ | ≤ K1 + K3 ,

we have

−Un+1 + 1 + h(f − g¯ ) ≥ −δ + 1 − h(K1 + K3 ) ≥

1 2

,

and Un+1 − Ln ≤ δ − 1 + h(K1 + K3 ) +

g2 4g¯

≤

g2 4g¯

−

1 2

≤

g2 4g¯

.

Then g

√ + 2 g¯

g Un+1 − Ln ≤ √ , g¯



so, by (25), g

√ − 2 g¯



Un+1 − Ln =

√ −Un+1 + 1 + h(f − g¯ ) 1/2 g¯ ≥ = . √ g g √ √ 2g + U n +1 − Ln g¯ 2 g¯

Also δ ≥ Un+1 ≥ Ln ≥ 0 therefore

√ √ δ ≥ Un+1 − Ln ≥ 0. This gives the bound on √

√   g g g g¯ . √ + δ ≥ √ + Un+1 − Ln ≥ √ − Un+1 − Ln ≥ 2g 2 g¯ 2 g¯ 2 g¯  g 2  √ √ 2 √ − Un+1 − Ln g¯ 1 1 2 g¯ . ≥ = √ √ √ √ 2 2g 4hg h g¯ h g¯ In (24) change the variable ξ to u by using the formula

ξ=

g √ 2 g¯

√ − u − Ln . √ √ h g¯

g √ 2 g¯

+

√

Un+1 − Ln by using

(25)

2228

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

Then estimating we arrive at  (1)

∆n+1 = √

2π

|u|≤δ, u≥Ln



1

≤ √

2π

√ g √ − u−Ln 2 g¯

2

√ √ ˜ u)| − |φ(u) − φ( h g¯ e √ √ 2 hg¯ u − Ln ˜ u)| − 1 |φ(u) − φ( e 4hg 2 du. √ √ 2 hg¯ u − Ln



1

|u|≤δ, u≥Ln

du

Let C1 , C2 , . . . , C5 be positive constants that are independent of n ∈ N and ω ∈ Ω . Applying the estimate x2 e−x ≤ 1, we can write e

x > 0,

− 12 4hg

1 4hg 2

−2



1

(1)



≤

∆n+1 ≤ √

2π

|u|≤δ, u≥Ln

= 42 h2 g 4 . Hence we get

 ˜ u)| g 4 ˜ u)| 2 2 4 |φ(u) − φ( |φ(u) − φ( 4 h g du ≤ C1 h3/2 √ √ du. √ √ g¯ 2 hg¯ u − Ln u − Ln |u|≤δ, u≥Ln g 2 (X (ω))

g 2 (X (ω))

Now fix n ∈ N and ω ∈ Ω and consider two cases: 4g¯ (Xn (ω)) ≥ 4 and 4g¯ (Xn (ω)) ≤ 4. Suppose first that n n g2 4g¯

≥ 4.

Then g2

2≤

8g¯

,

g2

u − Ln ≥ −δ − 1 − h(K2 + K3 ) +

g2

>

4g¯

4g¯

−2≥

g2 8g¯

.

So



g

u − Ln ≥

√ √

2 2 g¯

and from (26) we obtain (1)

∆n+1 ≤ C1 h3/2

 |u|≤δ

≤ C3 h

3/2 2

˜ u)| 4 |φ(u) − φ( g du ≤ C2 h3/2 √ g g¯ √ √ 2 2 g¯





˜ u)|g 3 du |φ(u) − φ( |u|≤δ

˜ u)|du ≤ C3 h3/2 g 2 ∥φ − φ∥ ˜ L1 [0,1] . |φ(u) − φ(

g

|u|≤δ

Suppose now that g2 4g¯

< 4,

then from (26) we obtain (1) ∆n+1

≤ C1 h

3/2

 |u|≤δ, u≥Ln

˜ u)| 3 g |φ(u) − φ( g √ du ≤ C4 h3/2 g 2 √ g¯ u − Ln

 |u|≤δ, u≥Ln

˜ u)| |φ(u) − φ( du. √ u − Ln

Applying the Hôlder inequality with p < 2 and q = p−1 > 2 we get p

(1) ∆n+1

≤h

3/2

C4 g

2



˜ u)|q du |φ(u) − φ(

 1q 

p

|u|≤δ, u≥Ln

≤h

3/2

C4 g

2

|u|≤δ, u≥Ln

 1q  q ˜ |φ(u) − φ(u)| du

 |u|≤δ

δ

1 p

Ln

( u − Ln ) 2

˜ Lq [−δ,δ] . ≤ h3/2 C5 g 2 ∥φ − φ∥ Estimations for  (2)

1

∆n+1 = √

2π

 |u|≤δ, u≥Ln

˜ u)| − |φ(u) − φ( e √ √ 2 hg¯ u − Ln

 1p

1

√ g √ + u−Ln 2 g¯ √ √ h g¯

2

du

( u − Ln ) 2  1p

du

du

(26)

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

2229

(1)

are analogous to, and even simpler than, those for ∆n+1 . Since (1)

(2)

∆n+1 = ∆n+1 + ∆n+1 , we have established (22), and the proof is complete.



4.2. Almost sure instability of the equilibrium of (15) Theorem 4.8. Let {Xn }n∈N be any solution of (15) where X0 ̸= 0. Suppose that conditions (16), (17), and

 lim inf

2f (u)



>1

g 2 (u)

u→0

(27)

¯ hold. Then there exists h¯ > 0 such that, for all h < h,  P



lim |Xn | = 0 = 0.

(28)

n→∞

Remark 4.9. We note that in order for condition (27) to be fulfilled, the function f must be positive in some neighbourhood of x = 0. Proof of Theorem 4.8. Condition (27) implies that for some γ > 0

 lim inf u→0

2f (u)



g 2 (u)

= 1 + 2γ .

Then there is a θ = θ (γ ) > 0 such that 2f (u) g 2 (u)

> 1 + γ,

when |u| < θ .

Assume that there is a set Ω1 ∈ Ω with P(Ω1 ) > 0, such that Xn (ω) → 0 as n → ∞ and ω ∈ Ω1 . Let N (ω, θ ) be such that

|Xn (ω)| ≤ θ ,

ω ∈ Ω1 , n ≥ N (ω, θ ).

Then for ω ∈ Ω1 and n ≥ N (ω, θ ), we have 2f (Xn (ω)) g 2 (Xn (ω))

≥ 1 + γ.

Let α ∈ (0, γ ) and

|Xn |α = |X0 |α

n 

1

|Ui+1 |−α i=0 E(|Ui+1 |−α |Fi )

×

1 N (ω,θ)



E (|Ui+1

× |−α |F

i =0

i

)

1 n  i=N (ω,θ )+1

E (|Ui+1

, |−α |F

i

)

where Mn =

n  i =0

|Ui+1 |−α E (|Ui+1 |−α |Fi )

is a non-negative martingale which converges to an a.s. finite limit. We will show that E |Ui+1 |−α |Fi ≤ 1 for all i ≥ N (ω, θ ) and ω ∈ Ω1 . Applying (19) for φ(u) = |u|−α we obtain



E |Un+1 |−α |Fn = 1 −





αh  2



2f (Xn (ω)) − (α + 1)g 2 (Xn (ω)) + (f (Xn (ω)) + g 2 (Xn (ω)))O (h) .



(29)

Recall that by Lemma 4.4, O (h) in (29) converges to zero as h → 0 uniformly in n. Therefore we can act in the same way ¯ Then for n ≥ N (ω, θ ) and as in Section 3.2 and for ε0 satisfying inequality (13), find h¯ such that O (h) < ε0 when h < h. ω ∈ Ω1 we obtain 2f (Xn (ω)) − (α + 1)g 2 (Xn (ω)) + (f (Xn (ω)) + g 2 (Xn (ω)))O (h)

≥ 2f (Xn (ω)) − (α + 1)g 2 (Xn (ω)) − ε0 f (Xn (ω)) − ε0 g 2 (Xn (ω)) = f (Xn (ω))(2 − ε0 ) − (α + 1 + ε0 )g 2 (Xn (ω))

2230

C. Kelly et al. / Computers and Mathematics with Applications 66 (2013) 2220–2230

= g 2 (Xn (ω))



2f (Xn (ω)) 2 − ε0

− (α + 1 + ε0 )



g 2 (Xn (ω)) 2   2 − ε0 ≥ g 2 (Xn (ω)) (1 + γ ) − (α + 1 + ε0 ) > 0. 2

  So, E |Ui+1 |−α |Fi ≤ 1 for all i ≥ N (ω, θ ) and ω ∈ Ω1 . Since we also have a.s. on Ω , that a non-negative martingale Mn converges and N (ω,θ)



E |Ui+1 |−α |Fi < ∞,





i=0

this guarantees that Xn (ω) does not tend to zero on Ω1 , which contradicts our assumption and therefore proves the theorem.  Remark 4.10. Notice that Eq. (28) in the statement of Theorem 4.8 may hold even if





P lim inf |Xn | = 0 > 0. n→∞

However, if we strengthen condition (27) so that the ratio 2f (u)/g 2 (u) is always bounded below by 1, then we can use a similar proof to that for Theorem 3.4 to get the following result. Theorem 4.11. Let {Xn }n∈N be any solution of (15) where X0 ̸= 0. Suppose that conditions (16), (17), and

 inf

2f (u) g 2 ( u)

u∈R



>1

(30)

¯ hold. Then there exists h¯ > 0 such that, for all h < h, 



P lim inf |Xn | > 0 = 1. n→∞

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