Almost sure stability of the weak backward euler maruyama method for the stochastic lotka volterra m

Page 1

Mathematical Computation June 2014, Volume 3, Issue 2, PP.19-25

Almost Sure Stability of the Weak Backward Euler-Maruyama Method for the Stochastic Lotka-Volterra Model in One Dimension Wei Liu Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K. Email: w.liu@lboro.ac.uk

Abstract The almost sure exponential stability of the weak backward Euler-Maruyama (EM) method is discussed for the stochastic version of the Lotka-Volterra model in one dimension. As the nonlinear term exists in the drift coefficient, the explicit Euler-Maruyama method is not a good candidate [6]. The backward EM method is normally a naturally replacement in the nonlinear case, but we show that in this model the backward EM method may not be well defined. Then we turn to the weak backward EM method, in which the normal distribution is replaced by a two-point distribution, and we prove that this method can reproduce the almost sure exponential stability of the underlying model. Keywords: Almost Sure Exponential Stability; Weak Backward Euler-Maruyama Method; Nonlinear SDEs

1 INTRODUCTION Stochastic differential equations (SDEs) have been employed to model uncertain scenarios in many areas for decades [5, 7] . Due to the difficulties to find explicit solutions to most nonlinear SDEs [5], the research on numerical approximations to SDEs has been blooming in recent years [4, 6]. Among different aspects of numerical analysis for SDEs, the almost sure exponential stability is one of the most popular topics. This paper is devoted to the study on reproducing the almost sure exponential stability of the underlying SDE by using numerical solutions. There already exist many literatures on this direction, and we just mention a few of them here [1, 2, 8] and the references therein. However, few works have been contributed to the weak method, in which the two-point distribution replaces the normally distributed Brownian increment. We refer the readers to [1] for the weak methods in the linear case. In this paper, we are investigating one type of nonlinear SDE, the one dimension stochastic Lotka-Volterra model, which is a stochastic population model [5]. We first show that there is positive probability that the explicit EulerMaruyama (EM) method may blow up while the underlying SDE is stable. In many papers, the backward EM method (also called the semi-implicit EM method) is a naturally good substitute to the explicit method in dealing with stability problems [2]. But this is not the case in this paper, as we will see the backward EM method may even not be well defined for the stochastic Lotka-Volterra model. Then we employ the weak backward EM method, which has been proved converge weakly to the underlying solution [4]. We first show that by properly choosing the stepsize, the weak backward EM can preserve the positivity. This is a desirable property, as the underling equation is a population model and its solution can only be nonnegative. More importantly, the positivity property is essential to the proof of the almost surely exponential stability. This paper is organised in the following way. In Section 2, the mathematical preliminary is briefed as well as the properties of the underlying equation. Section 3, firstly, sees the discussions of the failures of the explicit EM method and the backward EM method then the main result is presented. Some numerical simulations are used to illustrate the theoretical result in Section 4. We conclude this paper by some further discussion in Section 5. - 19 www.ivypub.org/mc

2 MATHEMATICAL PRELIMINARY AND THE UNDERLYING EQUATION Throughout this paper, let (Ί, Ć‘, P) be a complete probability space with a filtration {Ć‘đ?‘Ą }đ?‘Ąâ‰Ľ0 satisfying the usual conditions (that is, it is right continuous and increasing while Ć‘0 contains all P-null sets). Let B (t) be a scalar Brownian motion defined on the probability space. Our aim of this paper is to study the almost sure stability of different numerical approaches to the one-dimensional Lotka-Volterra model dx(t) = (bx(t) - ax 2 (t))dt + Ďƒx(t)dB(t), x(0) = x0 ,

(2.1)

Where a, b and Ďƒ are positive parameters, and the initial value, x0 >0, is a constant. It can be seen that the drift coefficient of this SDE does not satisfy either linear growth condition or one-sided Lipschitz condition. We first present the following theorem about the existence, uniqueness and positivity of the solution. We refer the reader to [5] for the proof. Theorem 2.1 For any given initial value x (0) > 0, and positive parameters a, b and Ďƒ, there exists a unique positive global solution x (t) to the equation (2.1) on t ≼ 0. As we are investigating reproduction of the stability of the underlying equation by some numerical methods, we quote here the result of the almost sure stability of the underlying equation as follows. Theorem 2.2 (see for example [5]) Assume a, b, Ďƒ > 0 and 1

b - đ?œŽ 2 < 0,

(2.2)

2

Then, for any initial value x (0) >0, the underlying SDE (2.1) is almost surely exponential stable, i.e. đ?‘™đ?‘–đ?‘šđ?‘ đ?‘˘đ?‘? đ?‘Ąâ†’∞

đ?‘™đ?‘œđ?‘”â Ą|đ?‘Ľ(đ?‘Ą)| đ?‘Ą

1

≤ b - đ?œŽ2. 2

However, little knowledge about what kind of numerical scheme could reproduce this stability for this model is known.

3

NUMERICAL SOLUTION

3.1 The Euler-Maruyama Method Due to the simple algebraic structure and cheap computational cost, the Euler-Maruyama (EM) method has been one of the most popular methods [9]. Applying the EM method to the equation (2.1), we see that Xđ?‘˜+1 = Xđ?‘˜ + (bXđ?‘˜ - aXđ?‘˜2 ) Δt +ĎƒXđ?‘˜ ΔBđ?‘˜ ,

X0 = x (0),

(3.1)

Where Δđ??ľđ?‘˜ = B (đ?‘Ąđ?‘˜+1 ) - B (đ?‘Ąđ?‘˜ ) is a Brownian motion increment and đ?‘Ąđ?‘˜ =kΔt. In the next lemma we will see that no matter how small the step size is there is always a positive probability that the numerical solution will blow up as time advances. And this contradicts to the behaviour of the underlying solution stated in Theorem 2.2. Lemma 3.1 For any positive parameters a, b and Ďƒ, assume 0 < Δt < 1/đ?œŽ 2 . If |đ?‘‹1 | ≼ 24+đ?‘? / (aΔt) in (3.1), then P (|đ?‘‹đ?‘˜ | ≼

2đ?‘˜+3+đ?‘? đ?‘Žđ?›Ľđ?‘Ą

for any k 1)

exp (-4đ?‘’ −2/(đ?œŽâˆšđ?›Ľđ?‘Ą) ).

Proof. Firstly, we show that |Xk | ≼

2đ?‘˜+3+đ?‘? aΔt

and

2đ?‘˜

|ΔBđ?‘˜ | ≤

Ďƒ

indicates

|Xk+1 | ≼

To see this, suppose |Xk | ≼ 2đ?‘˜+3+đ?‘? / (aΔt). Then |Xk+1 | ≼ |Xk |(aΔt|Xk | - 1 - bΔt - |ΔBđ?‘˜ |) ≼⠥

2đ?‘˜+3+đ?‘? aΔt

(2đ?‘˜+3+đ?‘? – 1 - bΔt - |ΔBđ?‘˜ |).

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2đ?‘˜+4+đ?‘? aΔt

(3.2)

Hence, |Xk+1 | ≼ 2đ?‘˜+4+đ?‘? /(â ĄaΔt) if 2đ?‘˜+3+đ?‘? – 1 - bΔt - |ΔBđ?‘˜ | ≼ 2, that is |ΔBđ?‘˜ | ≤ 2đ?‘˜+3+đ?‘? - 3 – bΔt. It is clear that 2đ?‘˜+3+đ?‘? - 3 – bΔt ≼2đ?‘˜+3+đ?‘? - 3 – b ≼ 2đ?‘˜ for any k ≼ 0 and b > 0, then (3.2) follows. Given that |X1 | ≼ 24+đ?‘? / (aΔt), the event that {2đ?‘˜+3+đ?‘? /(aΔt)}, for any 1 ≤ k ≤ H} contains the event that {|ΔBđ?‘˜ | ≤ 2đ?‘˜ / for any 1 ≤ k ≤ H }. As the {ΔBđ?‘˜ } are independent, we see that P (|Xk | ≼

2đ?‘˜+3+đ?‘? aΔt

for any 1 ≤ k ≤ H)≼ â Ą âˆ?đ??ť đ?‘˜=1 P(|ΔBđ?‘˜ | â Ą ≤

2đ?‘˜ Ďƒ

).

(3.3)

Because ΔBđ?‘˜ ~ N (0, Δt), we have that P (|ΔBđ?‘˜ | ≼ â Ą =

2

2đ?‘˜

đ?‘˜

∞

đ?‘’ −đ?‘Ľ âˆŤ √2Ď€ 2đ?‘˜ /(ĎƒâˆšÎ”t) ≤

đ?‘˜

B | 2 ) = P (|Δ√Δt ≼ ) Ďƒ ĎƒâˆšÎ”t 2 /2

â Ą đ?‘‘đ?‘Ľ

∞ 2 đ?‘’ −x â Ą đ?‘‘đ?‘Ľ âˆŤ √2Ď€ 2đ?‘˜ /(ĎƒâˆšÎ”t)

=

2 √2Ď€

exp (-

2đ?‘˜ ĎƒâˆšÎ”t

).

Thus, we see from (3.3) that P (|Xk | ≼

2đ?‘˜+3+đ?‘? aΔt

for any 1 ≤ k ≤ H) ≼ âˆ?đ??ť đ?‘˜=1(1 − â Ąexp(âˆ’â Ą

2đ?‘˜ ĎƒâˆšÎ”t

)).

Since Log (1-u)

-2u

for 0 < u < 0.5,

we then have that log (P (|Xk | ≼

2đ?‘˜+3+đ?‘? aΔt

for any 1 ≤ k ≤ H)) -2 ∑đ??ť đ?‘˜=1 exp(âˆ’â Ą

∑đ??ť đ?‘˜=1 log(1 − â Ąexp(âˆ’â Ą 2đ?‘˜ ĎƒâˆšÎ”t

).

2đ?‘˜ ĎƒâˆšÎ”t

â Ą)) (3.4)

Using 2đ?‘˜ ≼ 2k, ∑đ??ť đ?‘˜=1 exp(âˆ’â Ą

2đ?‘˜ ĎƒâˆšÎ”t

) ≤ ∑đ??ť đ?‘˜=1 exp(âˆ’â Ą

2đ?‘˜ ĎƒâˆšÎ”t

).

The right hand side is a geometric series that converges monotonically from below to đ?‘’ −2/(ĎƒâˆšÎ”t) / (1- đ?‘’ −2/(ĎƒâˆšÎ”t) ) 2đ?‘’ −2/(ĎƒâˆšÎ”t) . Hence (3.4) indicates that log (P (|Xk | ≼

2đ?‘˜+3+đ?‘? aΔt

for any 1 ≤ k ≤ H))

-4 đ?‘’ −2/(ĎƒâˆšÎ”t) ,

and the assertion holds. The results stated in Lemma 3.1 contrasts to the initial-data-independent exponential stability of the underlying SDE, shown in Theorem 2.2. Hence the EM method is not a candidate any more.

3.2 The Backward Euler-Maruyama Method The backward Euler-Maruyama method is often a good substitute to the classic Euler-Maruyama method, as it is better at dealing with nonlinearity and stability. The backward EM method for the equation (2.1) is 2 Xđ?‘˜+1 = Xđ?‘˜ + (bX đ?‘˜+1 - aXđ?‘˜+1 ) Δt +ĎƒXđ?‘˜ ΔBđ?‘˜ ,

By solving the quadratic equation (3.5), we see the iteration that - 21 www.ivypub.org/mc

X0 = x (0).

(3.5)

Xđ?‘˜+1 =

(bΔt−1)Âąâˆš(bΔt−1)2 +4đ?‘ŽÎ”t(1+ĎƒÎ”Bđ?‘˜ )Xđ?‘˜ 2đ?‘ŽÎ”t

.

However, due to the unboundedness of the Brownian motion increment under the square root, the backward EM may not be well defined.

3.3 The Weak Backward Euler-Maruyama Method In this section, the weak backward EM method is investigated. There already exist some works on the advantages of the weak methods [1, 4]. The weak backward EM method, in which the normal distributed Brownian motion increment is replaced by a two-point distribution, is defined by 2 Xđ?‘˜+1 = Xđ?‘˜ + (bXđ?‘˜+1 - aXđ?‘˜+1 ) Δt +ĎƒXđ?‘˜ √Δt⠥ΔVđ?‘˜ ,

X0 = x (0),

(3.6)

where Δ Vđ?‘˜ k=0,1,2,‌ are independently identically distributed random variables following the two-point distribution with P(Δ Vđ?‘˜ = 1) = P(Δ Vđ?‘˜ = -1) = 0.5. The replacement allows cheaper computation, and more importantly the two-point distribution makes the numerical solution well-defined. The expression of Xđ?‘˜+1 is now given by Xk+1 =

(bΔt−1)+√(bΔt−1)2 +4aΔt(1+ĎƒâˆšÎ”t⠥ΔVđ?‘˜ )Xk 2aΔt

.

Since Ďƒ > 0, we have that 1 + ĎƒâˆšÎ”t⠥ΔVđ?‘˜ ≼ 1 -â ĄĎƒâˆšÎ”tâ Ą > 0, if Δt < 1/Ďƒ2 . Then for any X0 > 0, the weak backward EM method can guarantee that Xk > 0 for all k > 0, which is in line with the positivity of the underlying solution discussed in Theorem 2.1. The next theorem discusses the stability of the weak backward EM method (3.6). Theorem 3.2 Assume b - 0.5đ?œŽ 2 < 0, for any Îľâˆˆ(0,|b-0.5đ?œŽ 2 |) there exists a Δđ?‘Ą ∗ ∈(0,1) such that for all Δt < Δđ?‘Ą ∗ the weak backward EM solution (3.6) is almost surely exponential stable, i.e. đ?‘™đ?‘–đ?‘šđ?‘ đ?‘˘đ?‘?đ?‘˜â†’∞

đ?‘™đ?‘œđ?‘”â Ą|đ?‘‹đ?‘˜ |

≤ b - 0.5đ?œŽ2 + Îľ < 0.

đ?‘˜đ?›Ľđ?‘Ą

Proof. Taking square on both sides of (3.6), we have that 2 |Xk+1 |2 = Xk+1 (X k +ĎƒXk ΔVđ?‘˜ √Δtâ Ą) + Xk+1 (bXk+1 - aXk+1 )⠥Δt.

Since X k >0 for all k >0, we see that 3 2 2 2 ΔtXk+1 (bXk+1 - aXk+1 ) = (bXk+1 – a Xk+1 )⠥Δt ≤ bΔtXk+1 ,

and 2 Xk+1 (Xk +ĎƒXk ΔVđ?‘˜ √Δtâ Ą) ≤ 0.5â ĄXk+1 + 0.5â Ą(Xk â Ą + ĎƒXk ΔVđ?‘˜ √Δtâ Ą)2 .

Given Δt < 1/(2b), we obtained that |Xk+1 |2 ≤ =

1 1−2đ?‘?Δt

|Xk |2

(Xk â Ą + ĎƒXk ΔVđ?‘˜ √Δtâ Ą)2

(1+2ĎƒÎ”Vđ?‘˜ √Δtâ Ą+Ďƒ2 ΔVđ?‘˜2 Δt)

1−2đ?‘?Δt

=

|Xk |2 1−2đ?‘?Δt

(1 + Îś),

(3.7)

where Îś = 2ĎƒÎ”Vđ?‘˜ √Δtâ Ą+ Ďƒ2 ΔVđ?‘˜2 Δt. For any p∈(0,1) we have an inequality that đ?‘?

đ?‘?(đ?‘?−2)

2

8

(1 + đ?‘˘)đ?‘?/2 ≤ 1 + u +

u2 +

đ?‘?(đ?‘?−2)(đ?‘?−4) 23 Ă—3!

u3 ,

u ≼ -1.

So we have from (3.7) that E (|Xk+1 |đ?‘? â Ą|â ĄĆ‘đ?‘˜Î”t ) ≤

|Xk |đ?‘?

đ?‘?

(1−2đ?‘?Δt)đ?‘?/2

E (1 + Îś + 2

đ?‘?(đ?‘?−2) 2 Îś 8

- 22 www.ivypub.org/mc

+

đ?‘?(đ?‘?−2)(đ?‘?−4) 3 Îś â Ą|â ĄĆ‘đ?‘˜Î”t), 23 Ă—3!

(3.8)

where Ć‘đ?‘˜Î”t is the sigma algebra generated by {ΔVđ?‘– : i = 0, ‌, k}. It is not hard to derive that E (Îśâ Ą|â ĄĆ‘đ?‘˜Î”t ) =Ďƒ2 Δt, E (â ĄÎś2 â Ą|â ĄĆ‘đ?‘˜Î”t ) = 4Ďƒ2 Δt + Ďƒ4 Δt 2 > 4Ďƒ2 Δt - Ďƒ4 Δt 2 , E (â ĄÎś3 â Ą|â ĄĆ‘đ?‘˜Î”t ) < (12Ďƒ4 +Ďƒ6 )Δt 2 . Substituting those into (3.8), we have that E (|Xk+1 |đ?‘? â Ą|â ĄĆ‘đ?‘˜Î”t ) ≤

|Xk |đ?‘? (1−2đ?‘?Δt)đ?‘?/2

≤

đ?‘?

đ?‘?(đ?‘?−2)

2

8

(1 + Ďƒ2 Δt + |Xk |đ?‘? (1−2đ?‘?Δt)đ?‘?/2

(1 +

p2 2

đ?‘?(đ?‘?−2)(đ?‘?−4)

(4Ďƒ2 Δt - Ďƒ4 Δt 2 ) +

Ďƒ2 Δt -

đ?‘? 2

23 Ă—3!

(12Ďƒ4 +Ďƒ6 ) Δt 2 )

Ďƒ2 Δt + CΔt 2 ),

(3.9)

where C is a positive constant depending on p and . Taking expectation on both sides of (3.9) yields E (|Xk+1 |đ?‘? ) ≤

1â Ą+â Ą

p2 2 đ?‘? Ďƒ Δtâ Ąâˆ’â Ąâ Ą â ĄĎƒ2 Δtâ Ą+â ĄCΔt 2 2 (1−2đ?‘?Δt)đ?‘?/2

E (|Xk |đ?‘? ).

Now for any Îľâˆˆ(0,|b-0.5Ďƒ2 |), we can choose p sufficiently small such that pĎƒ2 ≤ Îľ/4. Then, for sufficiently smallΔt, we have that (1 − 2đ?‘?Δt)đ?‘?/2 ≼ 1 – pbΔt - C1 Δt 2 > 0, Where C1 is a positive constant depending on p and b. By further reducingΔt, we ensure that 1

1

C1 Δt < ξ,

CΔt < pξ, 8

1

1

4

2

|p (b+ ξ)⠥Δt| ≤ .

4

Hence E (|Xk+1 |đ?‘? ) ≤

đ?‘? 2

1 2 1 1−đ?‘?(đ?‘?+ Îľâ Ą)⠥Δt 4

1â Ą+â Ąâ Ą (âˆ’â ĄĎƒ2 â Ą+â Ąâ Ą Îľ)Δt

E (|Xk |đ?‘? ).

Note that for any u∈[-0.5,0.5], 1

∞

1−đ?‘˘

∞

= 1 + u + đ?‘˘2 ∑đ?‘–=0 uđ?‘– ≤1 + u + đ?‘˘2 ∑đ?‘–=0 0.5đ?‘– = 1 + u + 2đ?‘˘2 .

By further reducing Δt so that 1

1

1

1

4

2

4

4

4p(bâ Ą + â Ą Îľ)2 Δt + (âˆ’â ĄĎƒ2 â Ą + â Ąâ Ą Îľ)(p(bâ Ą + â Ą Îľ)Δt + 2(p(bâ Ą + â Ą Îľ)Δt)2 ) ≤ Îľ, We get that 1

E (|Xk+1 |đ?‘? ) ≤ (1 + p (b - â Ą Ďƒ2 + )⠥Δt) E (|Xk |đ?‘? ). 2

By iteration, we see that 1

E (|Xk |đ?‘? ) ≤ |X0 |đ?‘? exp (p (b - â Ą Ďƒ2 + ) kΔt). 2

By chebyshev's inequality, 1

|X0 |đ?‘?

2

k2

P (|Xk |đ?‘? ≼k 2 exp (p (b - â Ą Ďƒ2 + )kΔt)) ≤

.

Then by the Borel-Cantelli lemma, we see for almost all Ď‰âˆˆΊ that 1

|Xk |đ?‘? ≤ k 2 exp (p(b - â Ą Ďƒ2 + )kΔt) 2

Holds for all but finitely many k. Hence there exists a k 0 (ω) such that for all Ď‰âˆˆΊ excluding a P-null set, the inequality above holds whenever k ≼ k 0 . Hence 1 đ?‘˜Î”t

1

2logk

2

pkΔt

â Ąlog |Xk | ≤ (b - â Ą Ďƒ2 + ) +

Letting k → ∞, the assertion holds. - 23 www.ivypub.org/mc

.

4 NUMERICAL EXAMPLE To illustrate the theorem, we use present a numerical example. Set a = 1, b = 1.5 and = 2. It is easy to check that (2.2) holds. Thus, by Theorem 2.2, we know that the underlying solution (2.1) is almost sure stability. Now we choose Δt = 0.2 such that Δt < 1/σ2 and Δt < 1/ (2b), and simulate one path with 100 iterations starting from the initial value x0 = 10.

FIGURE 1 ONE PATH OF THE MODEL (2.1) USING THE WEAK BACKWARD EULER-MARUYAMA METHOD

It can be seen that after a few oscillations the solution stays at zero as time advances, that is to say the solution is almost surely stable. This is in line with the Theorem 3.2.

5 CONCLUSIONS In this short paper, we discuss the reproduction of the almost stability of the underlying equation by the weak backward Euler-Maruyama method and demonstrate the theoretical result by some computational example. We see the advantages of the weak scheme over the classical schemes, as the unbounded noise term is replaced by the bounded one. It is interesting to see if this kind of weak scheme would be applied to other nonlinear SDEs, to which the classical methods are not proper to apply.

REFERENCES [1]

Desmond J. Higham. “Mean-square and asymptotic stability of the stochastic theta method.” SIAM journal on numerical analysis, 2000, 38(3): 753-769.

[2]

Desmond J. Higham, Xuerong Mao, Chenggui Yuan. “Almost sure and moment exponential stability in the numerical simulations of stochastic differential equations.” SIAM journal on numerical analysis, 2007, 45(2): 592-609.

[3]

Desmond J. Higham, Xuerong Mao, Andrew Stuart. “Exponential mean-square stability of numerical solutions to stochastic differential equations.” LMS Journal of Computation and Mathematics, 2003, 6: 297-313.

[4]

Peter E. Kloeden, Eckhard Platen. “Numerical Solution of Stochastic Differential Equations.” Berlin, Springer-Verlog, 1992.

[5]

Xuerong Mao. “Stochastic Differential Equations and Applications.” 2nd Edition, Chichester, U.K, Horwood, 2007.

[6]

Hutzenthaler, Martin, Arnulf Jentzen, Peter E. Kloeden. "Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2011, 467(2130): 1563-1576.

[7]

Bernt Øksendal. “Stochastic differential equations.” Springer Berlin Heidelberg, 2003.

[8]

Xuerong Mao, Yi Shen, Alison Gray. “Almost sure exponential stability of backward Euler–Maruyama discretizations for hybrid stochastic differential equations.” Journal of Computational and Applied Mathematics, 2011, 235(5): 1213-1226.

[9]

Desmond J. Higham. “Stochastic ordinary differential equations in applied and computational mathematics.” IMA journal of applied mathematics, 2011, 76(3): 449-474. - 24 www.ivypub.org/mc

AUTHORS 1

Wei Liu received his first degree from the University of Strathclyde in 2010, then he obtained his PhD degree

from Strathclyde in 2013. He is currently a research associate working at Loughborough University

- 25 www.ivypub.org/mc

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