Stationary distribution and extinction of a stochastic

Applied Mathematics Letters 73 (2017) 8–15

Contents lists available at ScienceDirect

Applied Mathematics Letters www.elsevier.com/locate/aml

Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation Qun Liua,b , Daqing Jianga,c,d, * a

School of Mathematics and Statistics, Key Laboratory of Applied Statistics of MOE, Northeast Normal University, Changchun 130024, Jilin Province, PR China b School of Mathematics and Statistics, Guangxi Colleges and Universities Key Laboratory of Complex System, Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, Guangxi, PR China c Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia d College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, PR China

article

info

Article history: Received 17 March 2017 Received in revised form 19 April 2017 Accepted 19 April 2017 Available online 26 April 2017

abstract In this paper, we analyze a stochastic SIR model with nonlinear perturbation. By the Lyapunov function method, we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model. Moreover, sufficient conditions for extinction of the disease are also obtained. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Stochastic SIR model Stationary distribution Ergodicity Extinction Nonlinear perturbation

1. Introduction The SIR model is one of the most important models in epidemiological patterns and disease control which was initially proposed and studied by Kermack and McKendrick [1]. From then on, various of SIR models have been investigated by many researchers due to their theoretical and practical significance and the study of the model will continue to be one of the most interesting hot topics in both epidemiology and mathematical ecology [2]. The classical SIR model can be expressed as follows ⎧ dS(t) ⎪ ⎪ = Λ − µ1 S(t) − βS(t)I(t), ⎨ dt (1.1) ⎪ ⎪ ⎩ dI(t) = βS(t)I(t) − (µ2 + α + γ)I(t), dt

*

Corresponding author at: College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, PR China. E-mail addresses: [email protected], [email protected] (D. Jiang).

http://dx.doi.org/10.1016/j.aml.2017.04.021 0893-9659/© 2017 Elsevier Ltd. All rights reserved.

Q. Liu, D. Jiang / Applied Mathematics Letters 73 (2017) 8–15

9

where S(t) denotes the number of the individuals susceptible to the disease, I(t) denotes the number of the individuals who are infectious. The parameter Λ is the influx of individuals into the susceptible, µi (i = 1, 2) represent the natural death rates of S and I compartments, respectively, β is the contact rate between compartments S and I, α is the disease-caused death rate and γ is the recovery rate of the infective individuals. All the parameters are assumed to be positive constants. Since system (1.1) always has a disease-free equilibrium E0 = (S0 , I0 ) = ( µΛ1 , 0), the basic reproduction number for system (1.1) can be defined as R0 = µ1 (µ2βΛ +γ+α) which means the average new infections caused by a single infected individual in a whole susceptible population. When R0 < 1, the disease-free equilibrium E0 is globally asymptotically stable. When R0 > 1, then E0 is unstable and there is a globally asymptotically stable endemic equilibrium E ∗ = (S ∗ , I ∗ ), where S ∗ = µ1ΛR0 , I ∗ = µβ1 (R0 − 1). However, it is well known that epidemic models are inevitably affected by the environmental noise which is an important component in an ecosystem [3–6]. May [7] has revealed that due to the continuous fluctuation in the environment, the birth rates, death rates, transmission coefficient and other parameters involved in the system should exhibit random fluctuation to a greater or lesser extent. Hence the deterministic SIR epidemic model has some limitations in predicting the future dynamics accurately and the stochastic one can make it [8]. As an extension of system (1.1), we adopt a different approach to introduce random perturbation into it by replacing the parameters −µ1 and −µ2 by −µ1 → −µ1 + (σ11 + σ12 S(t))B˙ 1 (t), − µ2 → −µ2 + (σ21 + σ22 I(t))B˙ 2 (t), 2 where B˙ i (t) are the white noise, namely, Bi (t) are mutually independent standard Brownian motions, σij >0 denote the intensities of the white noise, i, j = 1, 2. Here we introduce the nonlinear perturbation because the random perturbation may be dependent on square of the state variables S and I in system (1.1), respectively. Based on this viewpoint, the stochastic version corresponding to system (1.1) takes the following form { dS(t) = [Λ − µ1 S(t) − βS(t)I(t)]dt + S(t)[σ11 + σ12 S(t)]dB1 (t), (1.2) dI(t) = [βS(t)I(t) − (µ2 + γ + α)I(t)]dt + I(t)[σ21 + σ22 I(t)]dB2 (t).

Throughout this paper, unless otherwise specified, let (Ω , F, {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets). Moreover, let R+ = (0, +∞), Rn+ = {(x1 , . . . , xn ) ∈ Rn : xi > 0, i = 1, . . . , n}. If f is a bounded function on [0, +∞), define f u = supt∈[0,+∞) f (t). The organization of this paper is as follows. In Section 2, we establish sufficient conditions for the existence of an ergodic stationary distribution. In Section 3, we establish sufficient conditions for extinction of the disease. The following lemma concerns the existence and uniqueness of the global positive solution. Since the proof is standard [9], we omit it here. Lemma 1.1. For any initial value (S(0), I(0)) ∈ R2+ , there is a unique solution (S(t), I(t)) ∈ R2+ of system (1.2) on t ≥ 0 and the solution will remain in R2+ with probability one.

2. Existence of ergodic stationary distribution of system (1.2) In this section, we shall establish sufficient conditions for the existence of a unique ergodic stationary distribution. First of all, we present a lemma which will be used later.


10

Let X(t) be a homogeneous Markov process in Ed (Ed denotes d-dimensional Euclidean space) and be described by the following stochastic differential equation dX(t) = b(X)dt +

k ∑

gr (X)dBr (t).

r=1

The diffusion matrix is defined as follows A(x) = (aij (x)), aij (x) =

k ∑

gri (x)grj (x).

r=1

Lemma 2.1 ([10]). The Markov process X(t) has a unique ergodic stationary distribution µ(·) if there exists a bounded domain D ⊂ Ed with regular boundary Γ and ∑d 2 (B.1): there is a positive number M such that i,j=1 aij (x)ξi ξj ≥ M |ξ| , x ∈ D, ξ ∈ Rd . (B.2): there exists a nonnegative C 2 -function V such that LV is negative for any Ed \ D. Then } { ∫ ∫ 1 T f (X(t))dt = f (x)µ(dx) = 1 Px lim T →+∞ T 0 Ed for all x ∈ Ed , where f (·) is a function integrable with respect to the measure µ. Define a parameter R0s =

βΛ 2 + (µ1 + σ11

2Λσ12 σ11 )(µ2

+γ+α+

2 σ21 2 )

.

Theorem 2.1. Assume that R0s > 1, then for any initial value (S(0), I(0)) ∈ R2+ , system (1.2) admits a unique stationary distribution µ(·) and it has the ergodic property. Proof . By Lemma 1.1, we have obtained that for any initial value (S(0), I(0)) ∈ R2+ , system (1.2) has a unique global solution (S(t), I(t)) ∈ R2+ on t ≥ 0. In order to prove Theorem 2.1, we only need to validate conditions (B.1) and (B.2) in Lemma 2.1. Now we verify the condition (B.1). The diffusion matrix of system (1.2) is given by 2 ∑ i,j=1

( ) ) (σ11 S + σ12 S 2 )ξ1 aij (S, I)ξi ξj = (σ11 S + σ12 S )ξ1 (σ21 I + σ22 I )ξ2 (σ21 I + σ22 I 2 )ξ2 2

(

2

= (σ11 S + σ12 S 2 )2 ξ12 + (σ21 I + σ22 I 2 )2 ξ22 ¯ k ⊂ R2+ , ξ = (ξ1 , ξ2 ) ∈ R2+ , ≥ M ∥ξ∥2 f or any (S, I) ∈ D ¯ k = [ 1 , k] × [ 1 , k], then the condition where M = min(S,I)∈D¯ k {(σ11 S + σ12 S 2 )2 ξ12 + (σ21 I + σ22 I 2 )2 ξ22 } and D k k (B.1) holds. To prove the condition (B.2), we define a C 2 -function V˜ : R2+ → R as follows V˜ (S, I) = M [−c1 ln S − c2 ln I +

2c1 (σ11 + σ12 S)θ ] + (1 + S)p + I p , θ θ(1 − θ)σ11

where ci (i = 1, 2) are positive constants to be determined later, θ, p ∈ (0, 1) and M > 0 is a sufficiently large number such that −M λ + f1u + f2u ≤ −2, f1 (·) and f2 (·) will be determined later. It is easy to check that lim inf

k→+∞,(S,I)∈R2 \Dk +

V˜ (S, I) = +∞,


11

where Dk = ( k1 , k) × ( k1 , k). Moreover, V˜ (S, I) is a continuous function. Therefore V˜ (S, I) must have a minimum point (S0 , I0 ) in the interior of R2+ . Then we define a nonnegative C 2 -function V : R2+ → R+ in the following form V (S, I) = V˜ (S, I) − V˜ (S0 , I0 ). Denote V1 (S, I) = −c1 ln S − c2 ln I +

2c1 θ (σ11 θ(1−θ)σ11

+ σ12 S)θ , V2 (S, I) = (1 + S)p + I p . Making use of the

Itˆ o’s formula [9] to V1 (S, I), we get ( ) ( ) σ2 σ2 2 LV1 = − c1SΛ + c1 µ1 + 211 + c1 βI + c1 σ11 σ12 S + c21 σ12 S 2 − c2 βS + c2 µ2 + γ + α + 221 [ 2 2 I + σ12 Λθ(σ11 + σ12 S)θ−1 − σ12 µ1 θS(σ11 + σ12 S)θ−1 + c2 σ21 σ22 I + c22 σ22 ] θ(θ − 1) 2 2 2c1 − σ12 βθSI(σ11 + σ12 S)θ−1 + σ12 S (σ11 + σ12 S)θ θ 2 θ(1 − θ)σ11 ) ) ( ( c1 2 2 c1 Λ σ2 σ2 − c2 βS + c1 µ1 + 11 + c2 µ2 + γ + α + 21 + c1 βI + c1 σ11 σ12 S + σ12 S ≤− S 2 2 2 [ ] θ(1 − θ) θ 2 2 2c1 c2 2 2 θ−1 (2.1) I + σ12 Λθσ11 − σ11 σ12 S + c2 σ21 σ22 I + σ22 θ 2 2 θ(1 − θ)σ11 ( ) ) ( √ σ2 c1 2 2 c1 2 2 c1 2 σ2 S + σ12 S + σ11 ≤ −2 βΛc1 c2 + c1 µ1 + 11 + c2 µ2 + γ + α + 21 + σ12 2 2 2 2 2 2c1 Λσ12 c2 2 2 2 + σ22 I + (c1 β + c2 σ21 σ22 )I + − c1 σ12 S2 2 (1 − θ)σ11 ( ) ) ( √ 2Λσ12 σ2 2 + = −2 βΛc1 c2 + c1 µ1 + σ11 + c2 µ2 + γ + α + 21 (1 − θ)σ11 2 c2 2 2 + σ22 I + (c1 β + c2 σ21 σ22 )I. 2 Let ) ( ) ( 2 2Λσ12 σ21 2 c1 µ1 + σ11 + = Λ, = c2 µ2 + γ + α + (1 − θ)σ11 2 then we have c1 =

Λ 2 + µ1 + σ11

2Λσ12 (1−θ)σ11

Λ

, c2 =

Substituting these expressions into (2.1) leads to   βΛ  LV1 ≤ −2Λ√ 2 + 2Λσ12 )(µ + γ + α + (µ1 + σ11 2 (1−θ)σ11

µ2 + γ + α +

2 σ21 2 )

+ 2Λ +

2 σ21 2

.

c2 2 2 σ I + (c1 β + c2 σ21 σ22 )I 2 22

1 c2 2 2 = −2Λ[(R0s (θ)) 2 − 1] + σ22 I + (c1 β + c2 σ21 σ22 )I 2 c2 2 2 := −λ(θ) + σ22 I + (c1 β + c2 σ21 σ22 )I, 2

where R0s (θ) =

βΛ σ2 2 + 2Λσ12 )(µ +γ+α+ 21 ) (µ1 +σ11 2 2 (1−θ)σ11

1

and λ(θ) = 2Λ[(R0s (θ)) 2 − 1].

Due to the continuity of the function R0s (θ) and R0s > 1, we can choose θ sufficiently small such that R0s (θ) > 1. Therefore, we have λ(θ) > 0 and c2 2 2 c2 2 2 LV1 ≤ −λ(θ) + σ22 I + (c1 β + c2 σ21 σ22 )I → −λ + σ22 I + (c1 β + c2 σ21 σ22 )I as θ → 0+ , 2 2 1

where λ = 2Λ[(R0s ) 2 − 1] > 0.


12

On the other hand, we obtain p(p − 1) (1 + S)p−2 S 2 (σ11 + σ12 S)2 + pI p−1 (βSI − (µ2 + γ + α)I) LV2 = p(1 + S)p−1 (Λ − µ1 S − βSI) + 2 p(p − 1) p + I (σ21 + σ22 I)2 2 p(1 − p) 2 p+2 p(1 − p) 2 p+2 ≤− σ12 S σ22 I + pΛ − + pβSI p . 2 2 Hence ] p(1 − p) [ p(1 − p) 2 p+2 c2 2 2 2 I + (c1 β + c2 σ21 σ22 )I − σ12 S p+2 + pΛ − σ22 I + pβSI p LV ≤ M −λ + σ22 2 2 2 [ ] c2 2 2 := f1 (S) + f2 (I) + M −λ + σ22 I + (c1 β + c2 σ21 σ22 )I + pβSI p 2 := G(S, I), where f1 (S) = −

p(1 − p) 2 p+2 p(1 − p) 2 p+2 σ12 S + pΛ, f2 (I) = − σ22 I 2 2

and [ ] c2 2 2 G(S, I) = f1 (S) + f2 (I) + M −λ + σ22 I + (c1 β + c2 σ21 σ22 )I + pβSI p . 2 Then we obtain ⎧ ⎨G(+∞, I) → −∞, as S → +∞, G(S, I) ≤ G(S, +∞) → −∞, as I → +∞, ⎩ u f1 + f2u − M λ ≤ −2, as S → 0+ or I → 0+ . Thus there exists a sufficiently large k such that LV (S, I) ≤ −1 f or any (S, I) ∈ R2+ \ Dk , where Dk = ( k1 , k) × ( k1 , k). Therefore the condition (B.2) is verified. It follows from Lemma 2.1 that system (1.2) is ergodic and has a unique stationary distribution. This completes the proof.

3. Extinction of the disease In this section, we shall give a sharp result for extinction of the disease in the stochastic model (1.2). Define a parameter ∫∞ β xπ(x)dx 0 ˆ0 = R , σ2 µ2 + γ + α + 221 where −2−

π(x) = Qx

2(2Λσ12 +µ1 σ11 ) σ3 11

−2+

(σ11 + σ12 x)

2(2Λσ12 +µ1 σ11 ) σ3 11

e

−σ

2Λσ12 +µ1 σ11 2 (Λ+ ) σ11 11 (σ11 +σ12 x) x

, x ∈ (0, +∞).

Theorem 3.1. Let (S(t), I(t)) be the solution of system (1.2) with any initial value (S(0), I(0)) ∈ R2+ . If ˆ0 < 1, then the solution (S(t), I(t)) of system (1.2) satisfies R lim I(t) = 0 a.s.

t→+∞

and the distribution of S(t) converges weakly to the measure which has the density −2−

π(x) = Qx

2(2Λσ12 +µ1 σ11 ) σ3 11

−2+

2(2Λσ12 +µ1 σ11 ) σ3 11

(σ11 + σ12 x) ∫∞ where Q is a constant such that 0 π(x)dx = 1.

e

−σ

2Λσ12 +µ1 σ11 2 (Λ+ ) σ11 11 (σ11 +σ12 x) x

, x ∈ (0, +∞),


13

Proof . Consider the following auxiliary logistic equation with random perturbation dX(t) = [Λ − µ1 X(t)]dt + X(t)(σ11 + σ12 X(t))dB1 (t),

(3.1)

with the initial value X(0) = S(0) > 0. Setting b(x) = Λ − µ1 x, σ(x) = x(σ11 + σ12 x), x ∈ (0, +∞), we compute that ∫ b(u) du = σ 2 (u) =

=

) µ1 Λ − du u2 (σ11 + σ12 u)2 u(σ11 + σ12 u)2 2Λσ12 σ11 + σ12 u Λ 2Λσ12 ln − − 2 3 (σ11 + σ12 u) σ11 u σ11 u(σ11 + σ12 u) σ11 µ1 µ1 σ11 + σ12 u − + 2 ln +Q σ11 (σ11 + σ12 u) σ11 u Λ 2Λσ12 + µ1 σ11 2Λσ12 + µ1 σ11 σ11 + σ12 u ln − − 2 + Q. 3 σ11 u σ11 u(σ11 + σ12 u) σ11 (σ11 + σ12 u)

∫ (

Hence ∫ e

b(u) du σ 2 (u)

= eQ (

Obviously, we obtain ∫ ∞ 0

∫x 1 1 e σ 2 (x)

σ11 + σ12 u ) u

2b(τ ) dτ σ 2 (τ )

2Λσ12 +µ1 σ11 σ3 11

−σ

e

2Λσ12 +µ1 σ11 1 ( Λ +. ) σ11 11 (σ11 +σ12 u) u

∞

∫

x−2 (σ11 + σ12 x)−2 (

= 0

−σ

×e

σ11 + σ12 x ) x

2Λσ12 +µ1 σ11 2 ( Λ. + ) σ11 11 (σ11 +σ12 x) x

.

2(2Λσ12 +µ1 σ11 ) σ3 11

(3.2)

dx < ∞.

Therefore, the condition of Theorem 1.16 in [11] follows clearly from (3.2). So system (3.1) has the ergodic property and the invariant density is given by −2−

2(2Λσ12 +µ1 σ11 ) σ3 11

−2+

2(2Λσ12 +µ1 σ11 ) 3

−

2

(Λ+

2Λσ12 +µ1 σ11

)

σ σ11 11 , x ∈ (0, +∞), (σ11 + σ12 x) e σ11 (σ11 +σ12 .x) x ∫∞ where Q is a constant such that 0 π(x)dx = 1. From the ergodic theorem it follows that ∫ ∫ ∞ 1 t (3.3) lim x(s)ds = xπ(x)dx a.s. t→+∞ t 0 0

π(x) = Qx

Let X(t) be the solution of SDE (3.1) with the initial value X(0) = S(0) > 0, then using the comparison theorem for stochastic differential equation [12], we have S(t) ≤ X(t) a.s. On the other hand, applying the Itˆ o’s formula to ln I, we get ] [ 2 2 σ22 σ21 2 − σ21 σ22 I − I dt + σ21 dB2 (t) + σ22 IdB2 (t). d ln I(t) = βS(t) − (µ2 + γ + α) − 2 2

(3.4)

(3.5)

Integrating the both sides of (3.5) from 0 to t, we obtain ( ) ∫ t ∫ t 2 2 ∫ t σ21 σ22 ln I(t) − ln I(0) = β S(s)ds − µ2 + γ + α + t − σ21 σ22 I(s)ds − I 2 (s)ds + σ21 B2 (t) 2 2 0 0 0 ∫ t (3.6) + σ22 I(s)dB2 (s) ( ) ∫ t 0 ∫ t σ2 σ2 ≤β S(s)ds − µ2 + γ + α + 21 t − 22 I 2 (s)ds + σ21 B2 (t) + M (t), 2 2 0 0


14

where M (t) = σ22

∫t 0

I(s)dB2 (s) whose quadratic variation is ⟨M, M ⟩(t) =

2 σ22

∫

t

I 2 (s)ds.

0

By the exponential martingales inequality [9], for any positive constants T , α and β, we have } { ] [ α P sup M (t) − ⟨M, M ⟩(t) > β ≤ e−αβ . 2 0≤t≤T Choose T = k, α = 1, β = 2 ln k, then it follows that { ] } [ 1 1 P sup M (t) − ⟨M, M ⟩(t) > 2 ln k ≤ 2 . 2 k 0≤t≤k Using the Borel–Cantelli Lemma [9] leads to that for almost all ω ∈ Ω , there exists a random integer k0 = k0 (ω) such that for k ≥ k0 , we get ] [ 1 sup M (t) − ⟨M, M ⟩(t) ≤ 2 ln k. 2 0≤t≤k That is 1 1 2 M (t) ≤ 2 ln k + ⟨M, M ⟩(t) = 2 ln k + σ22 2 2

∫

t

I 2 (s)ds

0

for all 0 ≤ t ≤ k, k ≥ k0 a.s. Substituting this inequality into (3.6) yields ( ) ∫ t 2 σ21 ln I(t) − ln I(0) ≤ β S(s)ds − µ2 + γ + α + t + σ21 B2 (t) + 2 ln k, 2 0 for all 0 ≤ t ≤ k, k ≥ k0 a.s. In other words, we have shown that for 0 ≤ k − 1 ≤ t ≤ k, ( ) ∫ t 2 σ21 σ21 B2 (t) 2 ln k −1 −1 t [ln I(t) − ln I(0)] ≤ βt S(s)ds − µ2 + γ + α + + + 2 t t ) ( ∫0 t 2 σ B (t) 2 ln k σ 21 2 ≤ βt−1 + , X(s)ds − µ2 + γ + α + 21 + 2 t k −1 0

(3.7)

where in the second inequality, we have used (3.4). Taking the superior limit on the both sides of (3.7), ˆ0 < 1 leads to together with (3.3), and note that limt→+∞ B2t(t) = 0 a.s. and R ( ) ∫ ∞ ln I(t) σ2 lim sup ≤β xπ(x)dx − µ2 + γ + α + 21 < 0 a.s., t 2 t→+∞ 0 which shows that lim I(t) = 0 a.s.

t→+∞

Hence for any small ϵ > 0 there exist t0 and a set Ωϵ ⊂ Ω such that P(Ωϵ ) > 1 − ϵ and βSI ≤ ϵS for t ≥ t0 and ω ∈ Ωϵ . Now from [Λ − µ1 S(t) − ϵS(t)]dt + S(t)(σ11 + σ12 S(t))dB1 (t) ≤ dS(t) ≤ [Λ − µ1 S(t)]dt + S(t)(σ11 + σ12 S(t))dB1 (t), it follows that the distribution of the process S(t) converges weakly to the measure with the density π. This completes the proof.


15

Acknowledgments This work was supported by NSFC of China Grant No. 11371085, the Fundamental Research Funds for the Central Universities (No. 15CX08011A), 2016GXNSFBA380006 and KY2016YB370. References

[1] W. Kermack, A. McKendrick, Contributions to the mathematical theory of epidemics (Part I), Proc. Soc. Lond. Ser. A 115 (1927) 700–721. [2] Y. Lin, D. Jiang, Long-time behavior of perturbed SIR model by white noise, Discrete Contin. Dyn. Syst. Ser. B 18 (2013) 1873–1887. [3] T.C. Gard, Persistence in stochastic food web models, Bull. Math. Biol. 46 (1984) 357–370. [4] X. Meng, S. Zhao, T. Feng, T. Zhang, Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl. 433 (2016) 227–242. [5] Y. Zhao, S. Yuan, T. Zhang, The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching, Commun. Nonlinear Sci. Numer. Simul. 37 (2016) 131–142. [6] W. Zhao, J. Li, T. Zhang, X. Meng, T. Zhang, Persistence and ergodicity of plant disease model with Markov conversion and impulsive toxicant input, Commun. Nonlinear Sci. Numer. Simul. 48 (2017) 70–84. [7] R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, NJ, 2001. [8] Y. Zhou, W. Zhang, S. Yuan, Survival and stationary distribution of a SIR epidemic model with stochastic perturbations, Appl. Math. Comput. 244 (2014) 118–131. [9] X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997. [10] R.Z. Has’minskii, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. [11] N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977) 619–633. [12] A.Y. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2003.