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In this paper, the parameter estimation problem is investigated for the continuous time stochastic logistic diffusion system. A new continuous process is built based on the likelihood ratio scheme, the Radon-Nikodym derivative and the explicit expressions of the error of estimation are given under this new continuous process. By using the random time transformations, law of large numbers for martingales, law of iterated logarithm and stationary distribution of solution, the consistency property are proved for the estimation error. Finally, a numerical simulation is presented to demonstrate the effectiveness of the proposed method in this paper.

In the past few decades, parameter estimation problem for the stochastic differential equation have been studied by many scholars whose results mostly base on the discrete observation. In order to get more accurate estimators, we should observe through continuous time. Deterministic model, which parameters are deterministic irrespective of environmental fluctuations, are usually used to describe the overall impact of changes between different factors. these models obviously impose limitations in mathematical modeling of whole real systems. However, in the real world, many random factors (i.e. earthquakes, typhoons, car accidents and other unforeseen factors) may make the parameters into random variables. Therefore, it is more reasonable to use the stochastic differential equation with to describe the real systems disturbed by random noises. For example, the stochastic logistic diffusion model has been widely used in the field of social life, application of stochastic logistic diffusion model has been used in the field of applied economics [

The stochastic logistic diffusion model can be described by the following stochastic differential equation:

d X t = ( α X t − β X t 2 ) d t + ε X t d W t , X 0 = x (1.1)

where X t represents the population capacity at time t, α > 0 represents the

natural birth rate, β > 0 represents mortality, α β represents load capacity,

usually also represents the largest population that environmental resources can support. ε > 0 represents the dynamic effect of noise on X t . W t is a Wiener process modelling the random factor. [

In this paper, the continuous observations shall be used to obtain more accurate results than discrete observations, and the likelihood ratio will be employed to get Radon-Nikodym derivative which can be used to solve the parameter estimation problem for logistic diffusion model. As we all know that logistic diffusion model is a diffusion process, for a general diffusion model d X t = μ ( X t | θ ) + σ ( X t | θ ) , the parameter θ enter into the description of X t through μ or σ or both. However, the nature of diffusions allows us to evaluate σ exactly under a given continuous record, from the formula ∑ j = 1 2 n ( X j s 2 − n − X ( j − 1 ) s 2 − n ) 2 → ∫ 0 s σ 2 ( X u ) d s a.s. as n → + ∞ (a.s. means almost surely) as pointed out in [

This paper is organized as follows. In Section 2, a new method of estimating parameters is given and estimators are obtained. In Section 3, the strong consistency properties of estimators and estimation of asymptotic normality of error are proved. In Section 4, a numerical example for the estimators and error of estimation between estimators and trues is given to demonstrate the effectiveness of the proposed results. The conclusion is given in Section 5.

In this paper, the parameter estimation problem shall be studied for the logistic diffusion model described by a stochastic differential equation as given in (1.1). In this model α , β are unknown parameters. We can calculate ε use the method in [

Assumption 1: α , β and ε are positive, X 0 is positive and independent with W t .

Assumption 2: 2 α > ε 2 , which implies that X t cannot reach zero.

Assumption 3: X 0 is a positive random variable, and there is a Q > 2 such that E [ X 0 Q ] < ∞ hold.

Next, the specific steps with respect to derivations of the likelihood function and parameter estimators are given below.

Assume that ε 2 is known for all X t . X t can be observed continuously throughout the time interval 0 ≤ s ≤ t . For observations in this detail enable the true diffusion parameter ε 2 to be determinate through following result

∑ j = 1 2 n ( X j s 2 − n − X ( j − 1 ) s 2 − n ) 2 → ∫ 0 s σ 2 ( X u ) d s a . s . as n → + ∞ .

For all s ∈ [ 0, t ] , above equation can be rewritten descriptively as follows:

∫ 0 t ( d X s ) 2 = ∫ 0 t ( ε X s ) 2 d s a . s ..

Then, ε 2 may there be assumed know as:

ε 2 = ∫ 0 t ( d X s ) 2 ∫ 0 t X s 2 d s . (2.1)

The parameter α and β enter into the description of X t of (1.1), we can get ε exactly by (2.1). We begin with a class of probability ( Ω , F , P α , β ) , where the real stochastic process X = X s ; s ≥ 0 on ( Ω , F ) evolves according to one of probability laws P α , β . For each t ≥ 0 define

F t = σ ( X s : 0 ≤ s ≤ t , N ) , (2.2)

with σ-field generated by sets w ∈ Ω : X s = X s ( w ) ∈ B , and B is a Borel set on R and the class N is P α , β -null set of F .

Define

ρ ( t ) = ∫ 0 t ( ε X s ) 2 d s , t ≥ 0 (2.3)

and

Y ρ ( t ) = X t , t ≥ 0. (2.4)

From the theory of random time transformations, we have

d Y t = α Y t − β Y t 2 ( ε Y t ) 2 d t + d W ′ t (2.5)

where W ′ t is a standard Brownian motion(or its measure). (1.1) with the initial distribution μ . W t is a Brownian motion with respect to the filtration F t and X 0 = x is a F 0 -measurable random variable with distribution μ . Then, for the process of option times (2.3), we can create the inverse process that is X t = Y ρ ( t ) with filtration G t = F ρ ( t ) , and we can find a Brownian motion W ′ t with respect to G . For more details please see reference [

d P α , β t d W ′ t = exp { ∫ 0 ρ ( t ) ( α Y s − β Y s 2 ) ε 2 Y s 2 d Y s − 1 2 ∫ 0 ρ ( t ) ( α Y s − β Y s 2 ) 2 ε 2 Y s 4 d s } = exp { ∫ 0 t ( α X s − β X s 2 ) ε 2 X s 2 d X s − 1 2 ∫ 0 t ( α X s − β X s 2 ) 2 ε 2 X s 4 d ρ ( s ) } (2.6)

by substituting ρ ( s ) for s,

= exp { ∫ 0 t ( α X s − β X s 2 ) ε 2 X s 2 d X s − 1 2 ∫ 0 t ( α X s − β X s 2 ) 2 ε 2 X s 2 d s } . (2.7)

Similarly, we can get,

d P α ^ , β ^ t d W ′ t = exp { ∫ 0 t ( α ^ X s − β ^ X s 2 ) ε 2 X s 2 d X s − 1 2 ∫ 0 t ( α ^ X s − β ^ X s 2 ) 2 ε 2 X s 2 d s } . (2.8)

Thus,

d P α ^ , β ^ t d P α , β t = [ d P α ^ , β ^ t d W ′ t ] / [ d P α , β t d W ′ t ] . (2.9)

Writing P α , β t for the restriction of P α , β to F t , and we can now define following likelihood function as a Radon-Nikodym derivative:

L t ( α ^ , β ^ ) = d p α ^ , β ^ t d p α , β t = exp { ∫ 0 t ( α ^ X s − β ^ X s 2 ) − ( α X s − β X s 2 ) ε 2 X s 2 d X s − 1 2 ∫ 0 t ( α ^ X s − β ^ X s 2 ) 2 − ( α X s − β X s 2 ) 2 ε 2 X s 2 d s } , ∀ t > 0. (2.10)

Let l t ( α ^ , β ^ ) = log L t ( α ^ , β ^ ) , (the “log()” function has the basement “e”) solving following equation

{ ∂ l t ( α ^ , β ^ ) ∂ α ^ = 0 ∂ l t ( α ^ , β ^ ) ∂ β ^ = 0 , (2.11)

we can obtain the estimators as follows:

{ α ^ = ∫ 0 t d X s X s ∫ 0 t X s 2 d s − ( X t − X 0 ) ∫ 0 t X s d s t ∫ 0 t X s 2 d s − ( ∫ 0 t X s d s ) 2 , β ^ = ∫ 0 t d X s X s ∫ 0 t X s d s − t ( X t − X 0 ) t ∫ 0 t X s 2 d s − ( ∫ 0 t X s d s ) 2 . (2.12)

In this section, we shall give the asymptotic distribution of the estimated errors and the corresponding proof. It’s easy to konow the solution of Equation (1.1) has following expression:

X t = exp { ( α − ε 2 2 ) t + ε W t } x + β ∫ 0 t exp { ( α − ε 2 2 ) s + ε W s } d s . (3.1)

Firstly, let us give following four lemmas.

Lemma 1 (The law of iterated logarithm) [

lim t → + ∞ sup W t 2 t log log t = 1 a . s .. (3.2)

Lemma 2 [

− λ : = λ max + ( − 2 C β ) = sup X t ∈ R + , | X t | = 1 ( − 2 C β ) X t 2 < 0. (3.3)

Then, for any given initial value x ∈ ℝ + , the solution X t of equation d X t = ( α X t − β X t 2 ) d t + ε X t d W t has following properties

lim t → + ∞ sup ∫ t t + 1 E | X ( s ) | 2 d s ≤ 4 C 2 | α | λ 2 ( 1 + | C α | C ) , (3.4)

lim t → + ∞ sup E ( sup t ≤ s ≤ t + 1 | X s | ≤ 2 C 2 | α | C ( 1 + | C α | C ) + 6 | C ε | C C 2 | α | λ 2 ( 1 + | C α | C ) ) , (3.5)

and

lim t → + ∞ sup log ( | X ( t ) | ) log t ≤ 1 a . s .. (3.6)

Lemma 3 [

lim t → + ∞ inf log ( | X ( t ) | ) log ( t ) ≥ − ε 2 2 α − ε 2 a . s .. (3.7)

Lemma 4 Assume that X t is a solution to the stochastic differential Equation (1.1) and Assumptions 1 - 3 hold. Then, we have

lim t → + ∞ 1 t ∫ 0 t X s d s = α − ε 2 2 β . (3.8)

proof: It is known from lemma 2 that the solution (3.1) obeys

lim t → + ∞ sup log ( X t ) log t ≤ t a . s ..

While, from Assumption 2 we have 2 α > ε 2 . Then, by the properties in lemma 2, the solution (3.1) satisfies

lim t → + ∞ inf log ( X t ) log t ≥ − ε 2 2 α − ε 2 a . s ..

Consequently,

lim t → + ∞ 1 t log ( X t ) = 0 a . s ..

By the Itô formula, it is easy to know that

log ( X t ) = log ( x ) + ( α − ε 2 2 ) t − β ∫ 0 t X s d s + ε W t .

Dividing both side by t and then letting t → + ∞ , we obtain

lim t → + ∞ 1 t ∫ 0 t X s d s = α − ε 2 2 β a . s .. (3.9)

The proof is complete.,

Remark: (The one-dimensional Itô formula) Let x ( t ) be an Itô process on t ≥ 0 with the stochastic differential

d x ( t ) = f ( t ) d t + g ( t ) d B ( t ) , (3.10)

where f ∈ L 1 ( R + ; R ) and g ∈ L 2 ( R + ; R ) . Let V ∈ C 2,1 ( R × R + ; R ) .Then V ( x ( t ) , t ) is again an Itô process with the stochastic differential given by

d V ( x ( t ) , t ) = [ V t ( x ( t ) , t ) + V x ( x ( t ) , t ) f ( t ) + 1 2 V x x ( x ( t ) , t ) g 2 ( t ) ] d t + V x ( x ( t ) , t ) g ( t ) d B t a . s .. (3.11)

Theorem 1 Let X t be a solution to the stochastic differential Equation (1.1) and Assumptions 1-3 hold. Then, we have

l i m t → + ∞ 1 t ∫ 0 t X s 2 d s = α 2 − α ε 2 2 β 2 . (3.12)

proof: It follows from (1.1) that X t − x = α ∫ 0 t X s d s − β ∫ 0 t X s 2 d s + ε ∫ 0 t X s d W s , dividing both sides by t and then letting t → + ∞ , one has

lim t → + ∞ X t − x t = lim t → + ∞ α t ∫ 0 t X s d s − l i m t → + ∞ β t ∫ 0 t X s 2 d s + l i m t → + ∞ ε t ∫ 0 t X s d W s .

Since

E [ ∫ 0 t X s d W s ] = ∫ 0 t E [ X s d W s ] = ∫ 0 t E [ E [ X s d W s | F s ] ] = ∫ 0 t E [ X s E [ d W s ] ] = 0,

and

E [ ∫ 0 t X s d W s | F t − ] = ∫ 0 t − X s d W s + E [ ∫ t − t X s d W s | F t − ] = ∫ 0 t − X s d W s + ∫ t − t X s [ d W s ] = ∫ 0 t − X s d W s .

∫ 0 t − X s d W s is a matingale with zero mean with respect to the σ-algebra F t − . Moreover, according to (1.1), X t i − X t i − 1 = ( α X t i − 1 − β X t i − 1 2 ) Δ + ε X t i − 1 Δ ϵ t i . The equation f ( x ) = x + ( α x − β x 2 ) Δ (among them Δ = max | t i − t i − 1 | , 0 ≤ t i ≤ t ,

ϵ t i ~ N ( 0 , 1 ) ) gets maximum when x = α Δ + 1 2 β , thus,

X t ≤ ( α Δ + 1 ) 2 4 β < ∞ .

Therefore, E [ ( X s d W s ) 2 ] = E [ X s 2 ] is bounded. It then follows that

lim t → + ∞ sup ∫ 0 t X s 2 d s t < ∞ a . s .. (3.13)

By the strong law of large numbers of martingales, we have

lim t → + ∞ ∫ 0 t X s d W s t = 0 a . s ..

Together with (3.1) and Lemma 1, we obatain

0 < lim t → + ∞ X t = lim t → + ∞ exp { ( α − ε 2 2 ) t + 2 t log log t } x − 1 + β ∫ 0 t exp { ( α − ε 2 2 ) s + 2 s log log s } d s . (3.14)

By LHospital rule, we get

lim t → + ∞ exp { ( α − ε 2 2 ) t + 2 t log log t } x − 1 + β ∫ 0 t exp { ( α − ε 2 2 ) s + 2 s log log s } d s = α − ε 2 2 β .

Then,

lim t → + ∞ X t t = 0 a . s ..

According to (3.10), we get

lim t → + ∞ 1 t ∫ 0 t X s 2 d s = α 2 − α ε 2 2 β 2 a . s .. (3.15)

The proof is now complete.,

Remark: (LHospital rule) The general form of LHospital rule covers many cases. Let c and L be extended real numbers.The real valued function f and g are assumed to be differentiable on an open interval with endpoint c, and additionally

g ′ ( x ) ≠ 0 on the interval. It is also assumed that lim x → c f ′ ( x ) g ′ ( x ) = L . Thus the

rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, and not to situations in which that ratio fluctuates permanenty as x gets closer and closer to c.

If either

lim x → c f ( x ) = lim x → c g ( x ) = 0 (3.16)

or

lim x → c | f ( x ) | = lim x → c | g ( x ) | = ∞ , (3.17)

then

lim x → c f ( x ) g ( x ) = L . (3.18)

Theorem 2 Under Assumptions 1-3, α ^ and β ^ are strongly consistent.

Proof: Substituting (1.1) into the expression of α ^ yields

α ^ − α = ε W t ∫ 0 t X s 2 d s − ε ∫ 0 t X s d W s ∫ 0 t X s d s t ∫ 0 t X s 2 d s − ( ∫ 0 t X s d s ) 2 . (3.19)

Letting t → + ∞ , and according to Lemma 4 and Theorem 1, we have

α ^ − α = ε W t t ∫ 0 t X s 2 d s t − ε t ∫ 0 t X s d W s 1 t ∫ 0 t X s d s 1 t ∫ 0 t X s 2 d s − ( 1 t ∫ 0 t X s d s ) 2 = lim t → + ∞ ε W t t α 2 − α ε 2 2 β 2 − ε t α − ε 2 2 β α 2 − α ε 2 2 β 2 − ( α − ε 2 2 ) 2 β 2 = 2 α ε lim t → + ∞ W t t − 2 β ε lim t → + ∞ ∫ 0 t X s d W s t .

It follows from (3.12) that

α ^ − α → 0 a . s .. (3.20)

Substituting (1.1) into the expression of β ^ yields

β ^ − β = ε W t ∫ 0 t X s d s − t ε ∫ 0 t X s d W s t ∫ 0 t X s 2 d s − ( ∫ 0 t X s d s ) 2 . (3.21)

Similarly, we have

β ^ − β = ε W t t ∫ 0 t X s d s t − ε t ∫ 0 t X s d W s 1 t ∫ 0 t X s 2 d s − ( 1 t ∫ 0 t X s d s ) 2 = lim t → + ∞ ε W t t α − ε 2 2 β − ε t ∫ 0 t X s d W s α 2 − α ε 2 2 β 2 − ( α − ε 2 2 ) 2 β 2 = 2 α ε lim t → + ∞ W t t − 4 β 2 2 α ε − ε 3 lim t → + ∞ ∫ 0 t X s d W s t

It is easy to get from (3.12) that

β ^ − β = 0 a . s .. (3.22)

Thus, α ^ and β are strongly consistent. The proof is complete.,

Theorem 3 Under Assumptions 1-3, we have

ε t 2 α ( α ^ − α ) → L N ( 0,1 ) , ε t 2 β ( β ^ − β ) → L N ( 0,1 ) . (3.23)

Proof: It follows from (3.15) that

α ^ − α = ε W t t ∫ 0 t X s 2 d s t − X t − x 0 t ∫ 0 t X s d s t + α ( ∫ 0 t X s d s t ) 2 − β ∫ 0 t X s 2 t ∫ 0 t X s d s t ∫ 0 t X s 2 d s t − ( ∫ 0 t X s d s t ) 2 .

Substituting (3.9) and (3.11) into the above expression and then letting t → + ∞ , we have

α ^ − α = l i m t → + ∞ ε W t t α 2 − α ε 2 2 β 2 − X t − x t α − ε 2 2 β 2 α ε 2 − ε 2 4 β 2 .

According to (3.13), one has

l i m t → + ∞ t X t − x t α − ε 2 2 β = 0 a . s ..

Then,

l i m t → + ∞ ε t 2 α ε W t t α 2 − α ε 2 2 β 2 − X t − x t α − ε 2 2 β 2 α ε 2 − ε 2 4 β 2 → L N ( 0,1 ) . (3.24)

Similarly, it follows easily from (3.17) that

β ^ − β = ε W t t ∫ 0 t X s d s t − X t − x 0 t + α ∫ 0 t X s d s t − β ∫ 0 t X s 2 d s t ∫ 0 t X s 2 d s t − ( ∫ 0 t X s d s t ) 2 .

Substituting (3.9) and (3.11) into the above expression and then letting t → + ∞ yields

β ^ − β = lim t → + ∞ ε W t t α − ε 2 2 β − X t − x t 2 α ε 2 − ε 4 4 β 2 (3.25)

Therefore,

l i m t → + ∞ ε t 2 β ε W t t α − ε 2 2 β − X t − x t 2 α ε 2 − ε 4 4 β 2 → L N ( 0,1 ) . (3.26)

The proof is complete.,

In this section, a numerical simulation example shall be presented to demonstrate the effectiveness of the approach results.

The simulation is based on (2.6) (2.7) and (2.12). First according to (2.6) and (2.7), for given values of α , β and t such as α = 0.3 , β = 0.6 and t = 500 , we can get the sample values based on the likelihood ratio estimation and MATLAB. Then, for substituting the sample values into (2.12), the values of ( α ^ , β ^ ) can be obtained. Subsequently, we calculate the average values of the estimators. Finally, the average errors between estimators can also be calculated. Simulation results are shown in

True value ( α , β ) | Average value | Absolute value | |||
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t | α ^ − L R | β ^ − L R | α ^ | β ^ | |

500 | 0.2989 | 0.5992 | 0.0037 | 0.0013 | |

(0.3, 0.6) | 1000 | 0.2992 | 0.5997 | 0.0027 | 0.0005 |

1500 | 0.2998 | 0.5999 | 0.0006 | 0.0001 | |

500 | 0.3991 | 0.6989 | 0.0023 | 0.0016 | |

(0.4, 0.7) | 1000 | 0.3995 | 0.6992 | 0.0013 | 0.0011 |

1500 | 0.3999 | 0.6998 | 0.0003 | 0.0003 | |

500 | 0.4991 | 0.7992 | 0.0018 | 0.0011 | |

(0.5, 0.8) | 1000 | 0.4994 | 0.7995 | 0.0012 | 0.0006 |

1500 | 0.4998 | 0.7998 | 0.0004 | 0.0003 | |

500 | 0.5991 | 0.8991 | 0.0015 | 0.0012 | |

(0.6, 0.9) | 1000 | 0.5995 | 0.8994 | 0.0008 | 0.0006 |

1500 | 0.5999 | 0.8998 | 0.0001 | 0.0002 |

In this paper, parameter estimation problem has been studied for the continuous time stochastic logistic diffusion model by using likelihood ratio. The explicit expressions for the estimation errors have been given and the according asymptotic properties have been proved by applying the he law of iterated logarithm, random time transformations, stationary distribution of solutions of stochastic differential equations and the law of large numbers for martingales. To get more accurate results, we use continuous observation method, and the proposed estimators are closer to the true value that be demonstrated by a simulation example. In the future research, we will consider the state estimation problem for non-linear systems with incomplete observation [

Zheng, Z.W., Shu, H.S., Kan, X., Fang, Y.Y. and Zhang, X. (2017) Parameter Estimation for the Continuous Time Stochastic Logistic Diffusion Model. Open Journal of Statistics, 7, 1039-1052. https://doi.org/10.4236/ojs.2017.76072