Moment estimation based on the composite Heun scheme for parameters in uncertain differential equations

1 Introduction

In 2007, the uncertain theory was proposed by Liu [1] in order to model the human uncertainty. After that, based on the normality, duality, subadditivity and product axioms, Liu [2] perfected the uncertain theory in 2009. Similar to Wiener process in probability theory, Liu process was introduced based on the uncertain measure. The uncertain differential equations involving Liu processes describe dynamic uncertain systems with uncertain noises. In recent ten years, there has been an increasing interest in uncertain differential equations about the analytical and numerical solutions. If the coefficients in uncertain differential equations satisfy the linear growth and Lipschitz continuous conditions, the existence and uniqueness of solution were obtained by Chen and Liu in [3]. The analytical methods for solving the linear uncertain differential equations were also proposed. Moreover, Liu [4] and Yao [5] studied analytical methods for several special types of nonlinear uncertain differential equations. In [6], Yao and Chen pioneered a numerical method for solving the uncertain differential equations. Then several numerical schemes based on the method were designed by Gao [7], Zhang et al. [8], Yang and Ralescu [9], etc.

In recent years, the uncertain differential equations have been developed in many applications such as chemical reaction [10], population model [11], optimal control [12], pharmacokinetics [13], finance [14], and so on. For uncertain differential equations, there always exist unknown parameters in practical problems. Consequently, it is an essential problem to estimate the parameters based on the observations. By means of Euler scheme which is used to approximate the uncertain differential equations, the method of moments was given by Yao and Liu [15]. Motivated by this method, Liu [16] presented the generalized moment estimation method. Sheng et al. [17] explored the least squares estimation method. Following that, Zhang et al. [18] used the method of least squares to estimate parameters in the high-order uncertain differential equations. Liu and Yang [19] proposed the method of moments based on the Euler scheme for the high-order uncertain differential equations. Gao et al. [20] estimated the parameters in uncertain delay differential equations by the method of moments. For uncertain differential equations with time-varying parameters, Zhang and Sheng [21] used the least squares estimation method to obtain estimates of a set of time-varying parameters and presented the time-varying parameter function by regression fitting. These methods were all based on the Euler scheme. Tang and Yang [22] derived the Milstein scheme for uncertain differential equations and proved that it was superior to the Euler scheme.

In order to increase the accuracy of the difference scheme, we derive the composite Heun scheme to approximate uncertain differential equations and use the method of moments to estimate the parameters in uncertain differential equations. The rest of the paper is organized as follows. The composite Hean scheme and the trapezoidal scheme for uncertain differential equations are introduced in Section 2. In Section 3, the two schemes are used to estimate parameters by the method of moments. After that, several examples are given to illustrate the viability of the composite Heun scheme in Section 4. Finally, conclusions are presented in the last section.

2 The composite Heun scheme

Consider the general uncertain differential equation dX t = f ( t , X t ) dt + g ( t , X t ) dC t ,(1) where C _t is a Liu process, f (t, x) and g (t, x) are the drift and diffusion terms which are known functions. In this work, we assume that f (t, x) and g (t, x) satisfy the Lipschitz continuous condition and the linear growth condition [3], then the uncertain differential equation (1) has a unique solution with an initial value X₀.

Note that if the solution X _t satisfies the uncertain differential equation (1), it must be the solution of the following uncertain integral equation X t = X 0 + ∫ 0 t f ( s , X s ) ds + ∫ 0 t g ( s , X s ) dC s .(2)

Given a partition of the interval [0, T] with 0 = t 0 < … < t n < … < t N = T , we have [ c ] rl X t n + 1 = X t n + ∫ t n t n + 1 f ( s , X s ) ds + ∫ t n t n + 1 g ( s , X s ) dC s ,(3) for the subinterval [t _n , t_n+1]. Assume that f (t, x) is twice continuously differentiable, then based on the linear interpolation, there exits η _n  ∈ (t _n , t_n+1) which is dependent on t such that [ c ] rl f ( t , X t ) = f ( t n , X t n ) t - t n + 1 t n - t n + 1 + f ( t n + 1 , X t n + 1 ) t - t n t n + 1 - t n + p ( η n , X η n ) ( t - t n ) ( t - t n + 1 ) ,(4) where [ c ] rl p ( t , x ) = ∂ 2 f ∂ t 2 ( t , x ) + 2 ∂ 2 f ∂ x ∂ t ( t , x ) f ( t , x ) + ∂ 2 f ∂ x 2 ( t , x ) f 2 ( t , x ) + ∂ f ∂ x ( t , x ) ( ∂ f ∂ t ( t , x ) + ∂ f ∂ x ( t , x ) f ( t , x ) ) . For the second term in the RHS of (3), we have [ c ] rl ∫ t n t n + 1 f ( s , X s ) ds = f ( t n , X t n ) t n - t n + 1 ∫ t n t n + 1 ( s - t n + 1 ) ds + f ( t n + 1 , X t n + 1 ) t n + 1 - t n ∫ t n t n + 1 ( s - t n ) ds + R n 1 [ 6 pt ] = 1 2 [ f ( t n , X t n ) + f ( t n + 1 , X t n + 1 ) ] Δ t n [ 6 pt ] + R n 1 [ 6 pt ](5) where Δt _n  = t_n+1 - t _n and R n 1 = 1 2 ∫ t n t n + 1 p ( η n , X η n ) ( s - t n ) ( s - t n + 1 ) ds .

Because the solution X _t depends on time t and Liu process C _t , we suppose that g (t, X _t ) = p (t, C _t ) and g (t, X _t ) is continuously differentiable with respect to t and X _t . From the fundamental theorem of uncertain calculus, we obtain [ c ] rl dg ( t , X t ) = ∂ p ∂ t ( t , C t ) dt + ∂ p ∂ C ( t , C t ) dC t = ( ∂ g ∂ t ( t , X t ) + ∂ g ∂ X ( t , X t ) f ( t , X t ) ) dt + ∂ g ∂ X ( t , X t ) g ( t , X t ) dC t . Therefore, it follows that [ c ] rl g ( t , X t ) = g ( t n , X t n ) + ∫ t n t ∂ g ∂ X ( s , X s ) g ( s , X s ) dC s + ∫ t n t ( ∂ g ∂ t ( s , X s ) + ∂ g ∂ X ( s , X s ) f ( s , X s ) ) ds . For the third term in the RHS of (3), by integrating g (t, X _t ) with respect to C _t on the interval [t _n , t_n+1], we have [ c ] rl ∫ t n t n + 1 g ( s , X s ) dC s = ∫ t n t n + 1 g ( t n , X t n ) dC s + R n 2 = g ( t n , X t n ) ( C t n + 1 - C t n ) + R n 2 ,(6) where [ c ] rl R n 2 = ∫ t n t n + 1 ∫ t n s ( ∂ g ∂ t ( u , X u ) + ∂ g ∂ X ( u , X u ) f ( u , X u ) ) dudC s + ∫ t n t n + 1 ∫ t n s ∂ g ∂ X ( u , X u ) g ( u , X u ) dC u dC s . From (5) and (6), for the increment of X _t during the time interval [t _n , t_n+1] in (3), we obtain the following trapezoidal scheme by ignoring the residual terms R_n1 and R_n2 [ c ] rl X t n + 1 = X t n + g ( t n , X t n ) ( C t n + 1 - C t n ) + 1 2 [ f ( t n , X t n ) + f ( t n + 1 , X t n + 1 ) ] Δ t n .(7) Compare with the Euler scheme [ c ] rl X t n + 1 = X t n + f ( t n , X t n ) Δ t n + g ( t n , X t n ) ( C t n + 1 - C t n ) , the implicit trapezoidal scheme (7) offers a substantial improvement. Considering that the expected value of increment C _{t n+1}  - C _{t n} is zero, we use X ¯ t n + 1 = X t n + f ( t n , X t n ) Δ t n , as the predicted value of X _{t n+1} . Then, combining the trapezoidal scheme, we obtain the Heun scheme [ c ] rl X t n + 1 = X t n + g ( t n , X t n ) ( C t n + 1 - C t n ) + 1 2 [ f ( t n , X t n ) + f ( t n + 1 , X t n + f ( t n , X t n ) Δ t n ) ] Δ t n .(8)

More generally, we introduce a relaxation parameter λ in the process and obtain the following composite Heun scheme [ c ] rl X t n + 1 = X t n + g ( t n , X t n ) ( C t n + 1 - C t n ) + Δ t n [ ( 1 - λ ) f ( t n , X t n ) + λ f ( t n + 1 , X t n + f ( t n , X t n ) Δ t n ) ] .(9) Obviously the composite Heun scheme is a predictor-corrector method, its approximation is more accurate than the Euler approximation. When λ = 0, the composite Heun scheme degenerates to the Euler scheme.

The problem of parameter estimation is very important in applications of uncertain differential equations. By means of the composite Heun scheme and the trapezoidal scheme, we propose two parameter estimation methods.

Theorem 1. Consider the uncertain differential equation dX t = f ( t , X t ; θ ) dt + g ( t , X t ; θ ) dC t ,(10) where θ is the unknown vector which contains K parameters to be estimated. Suppose that f (t, X_t ;  θ) is twice continuously differentiable and g (t, X_t ; θ) is continuously differentiable. Then for the observations x_t0, x_t1, ⋯ , x_tN of X_t, the moment estimate θ ˆ of θ based on the composite Heun scheme satisfies [ c ] rl 1 N ∑ n = 0 N - 1 ( h n ( θ ) ) k = ∫ 0 1 ( Φ - 1 ( α ) ) k d α , k = 1 , ⋯ , K(11) where [ c ] rl h n ( θ ) = x t n + 1 - x t n g ( t n , x t n ; θ ) Δ t n - ( 1 - λ ) f ( t n , x t n ; θ ) g ( t n , x t n ; θ ) - λ f ( t n + 1 , x t n + f ( t n , x t n ) Δ t n ; θ ) g ( t n , x t n ; θ ) , and [ c ] rl ∫ 0 1 ( Φ - 1 ( α ) ) k d α = ∫ 0 1 ( 3 π ln α 1 - α ) k d α = ( 3 ) k ( 2 k - 2 ) | B k | , with B_k is the k-th Bernoulli number.

Proof. From the composite Heun scheme (9), the following difference form of the uncertain differential equation (10) is given by [ c ] rl X t n + 1 = X t n + g ( t n , X t n ; θ ) ( C t n + 1 - C t n ) + Δ t n [ ( 1 - λ ) f ( t n , X t n ; θ ) + λ f ( t n + 1 , X t n + f ( t n , X t n ) Δ t n ; θ ) ] . Rewriting the above equation, we have [ c ] rl X t n + 1 - X t n g ( t n , X t n ; θ ) Δ t n - ( 1 - λ ) f ( t n , X t n ; θ ) g ( t n , X t n ; θ ) - λ f ( t n + 1 , X t n + f ( t n , X t n ) Δ t n ; θ ) g ( t n , X t n ; θ ) = C t n + 1 - C t n Δ t n .(12) Note that the term in right hand side is a standard normal uncertain variable. Thus the terms in the left hand side of (12) satisfy [ c ] rl X t n + 1 - X t n g ( t n , X t n ; θ ) Δ t n - ( 1 - λ ) f ( t n , X t n ; θ ) g ( t n , X t n ; θ ) - λ f ( t n + 1 , X t n + f ( t n , X t n ) Δ t n ; θ ) g ( t n , X t n ; θ ) ~ N ( 0 , 1 ) . We replace X _{t n} and X _{t n+1} with the observations x _{t n} and x _{t n+1} respectively, and obtain [ c ] rl h n ( θ ) = x t n + 1 - x t n g ( t n , x t n ; θ ) Δ t n - ( 1 - λ ) f ( t n , x t n ; θ ) g ( t n , x t n ; θ ) - λ f ( t n + 1 , x t n + f ( t n , x t n ) Δ t n ; θ ) g ( t n , x t n ; θ ) ,(13) with n = 0, 1, ⋯ , N - 1, which can be regarded as N samples of N ( 0 , 1 ) . Note that the k-th population moment is ∫ 0 1 ( Φ - 1 ( α ) ) k d α = ∫ 0 1 ( 3 π ln α 1 - α ) k d α . From the method of moments, we obtain that the moment estimate θ ˆ satisfies the following equations [ c ] rl 1 N ∑ n = 0 N - 1 ( h n ( θ ) ) k = ∫ 0 1 ( Φ - 1 ( α ) ) k d α , where k = 1, 2, ⋯ , K.□

Similarly, it is easy to obtain the following parameter estimation method based on the trapezoidal scheme.

Theorem 2. Consider the uncertain differential equation dX t = f ( t , X t ; θ ) dt + g ( t , X t ; θ ) dC t , t ∈ [ 0 , T ] where θ is the unknown vector which contains K parameters to be estimated. Suppose that f (t, X _t  ;  θ) is twice continuously differentiable and g (t, X _t  ;  θ) is continuously differentiable. Then for the observations x _{t 0} , x _{t 1} , ⋯ , x _{t N} of X _t , the moment estimate θ ˆ of θ based on the trapezoidal scheme satisfies [ c ] rl 1 N ∑ n = 0 N - 1 ( m n ( θ ) ) k = ∫ 0 1 ( Φ - 1 ( α ) ) k d α , k = 1 , ⋯ , K(14)where [ c ] rl m n ( θ ) = x t n + 1 - x t n g ( t n , x t n ; θ ) Δ t n - f ( t n , x t n ; θ ) + f ( t n + 1 , x t n + 1 ; θ ) 2 g ( t n , x t n ; θ ) and [ c ] rl ∫ 0 1 ( Φ - 1 ( α ) ) k d α = ∫ 0 1 ( 3 π ln α 1 - α ) k d α = ( 3 ) k ( 2 k - 2 ) | B k | , with B_k is the k-th Bernoulli number.

In this section, we present some numerical results obtained by the above two parameter estimation methods as shown in Section 3. In the following numerical experiments, a number of observations of an uncertain process are needed in computations. Thus an algorithm is designed to obtain the samples of N ( 0 , 1 ) . With the true values of unknown parameters, the observations of uncertain process can be calculated.

Step 1 Generate samples η _n of L ( 0 , 1 ) , n = 1, ⋯ , N, whose uncertainty distribution is Ψ ( x ) = { [ c ] cl 0 , if x ≤ 0 x , if 0 < x < 1 1 , if x ≥ 1 .

Step 2 Compute the corresponding values ξ _n  = Φ^-1 (η _n ), n = 1, ⋯ , N, which are regarded as the samples of N ( 0 , 1 ) . The inverse uncertainty distribution Φ^-1 (x) of the standard normal uncertain variable is Φ - 1 ( x ) = 3 π ln x 1 - x .

Step 3 Generate corresponding x _{t 1} , ⋯ , x _{t N} with the initial value x _{t 0}  = X₀.

Step 4 Based on the composite Heun scheme, calculate the moment estimate θ ˆ of θ with the observations obtained in Step 3.

Example 4.1 Suppose the uncertain process X _t follows dX t = μ X t dt + σ X t dC t ,(15) with initial value X₀ = 1. The two parameters μ and σ are unknown and need to be estimated. Based on the composite Heun scheme, we have [ c ] rl h n ( μ , σ ) = x t n + 1 - x t n - Δ t n ( ( 1 - λ ) μ x t n + λ μ ( x t n + μ x t n Δ t n ) ) σ x t n Δ t n = 1 σ ( x t n + 1 - x t n x t n Δ t n - μ - λ μ 2 Δ t n ) , where n = 0, 1, ⋯ , N - 1. Because the first two moments of N ( 0 , 1 ) are 0 and 1, the system of equations in (11) becomes { [ c ] r 1 N ∑ n = 0 N - 1 1 σ ( x t n + 1 - x t n x t n Δ t n - μ - λ μ 2 Δ t n ) = 0 , 1 N ∑ n = 0 N - 1 1 σ 2 ( x t n + 1 - x t n x t n Δ t n - μ - λ μ 2 Δ t n ) 2 = 1 .(16) By solving the above equations, we have the moment estimate (σ^*, μ^*) for (σ, μ).

Analogously, according to the trapezoidal scheme, we have [ c ] rl m n ( μ , σ ) = x t n + 1 - x t n - Δ t n ( μ x t n + μ x t n + 1 ) / 2 σ x t n Δ t n = 1 σ ( x t n + 1 - x t n x t n Δ t n - μ 2 - μ x t n + 1 2 x t n ) , where n = 0, 1, ⋯ , N - 1. From (14), the moment estimate (μ^*, σ^*) satisfies { [ c ] r 1 N ∑ n = 0 N - 1 1 σ ( x t n + 1 - x t n x t n Δ t n - μ 2 - μ x t n + 1 2 x t n ) = 0 , 1 N ∑ n = 0 N - 1 1 σ 2 ( x t n + 1 - x t n x t n Δ t n - μ 2 - μ x t n + 1 2 x t n ) 2 = 1 .(17)

Obviously the above system of equations has a solution { [ c ] r μ * = 2 ∑ n = 0 N - 1 x t n + 1 - x t n x t n Δ t n / ( N + ∑ n = 0 N - 1 x t n + 1 x t n ) , σ * = 1 N ∑ n = 0 N - 1 ( x t n + 1 - x t n x t n Δ t n - μ * 2 - μ * x t n + 1 2 x t n ) 2 .

Some observations of the solution X _t are needed to obtain the estimated values. Note that the uncertain differential equation \eqref N1 is equivalent to dX t X t = μ dt + σ dC t . Integrating from t _n to t_n+1, we get [ c ] rl lnX t n + 1 = lnX t n + μ Δ t n + σ ( C t n + 1 - C t n ) = lnX t n + ( μ + σ C t n + 1 - C t n Δ t n ) Δ t n , which can be rewritten as X t n + 1 = X t n exp ( ( μ + σ C t n + 1 - C t n Δ t n ) Δ t n ) . Then the samples of X _t can be obtained by x t n + 1 = x t n exp ( ( μ + σ z n ) Δ t n ) ,(18) where n = 0, 1, ⋯ , N - 1, z _n are the samples of N ( 0 , 1 ) . The parameters in the uncertain differential equation \eqref N1 are set as μ = 0.2 and σ = 0.5. Then the samples of X _t generated by \eqref N13 are obtained and shown in Table 1.

By solving the system of equations \eqref N11 with λ = 0.9, we get the estimated values μ * = 0.2187 , σ * = 0.5417 . Hence the estimated equation is dX t = 0.2187 X t dt + 0.5417 X t dC t . Based on the trapezoidal scheme, we have dX t = 0.2223 X t dt + 0.5257 X t dC t .

The estimated values based on the Euler scheme and the Milstein scheme are also shown in Table 2. In this example, the composite Heun scheme performs best compared to the Euler scheme and the Milstein scheme, and as good as the trapezoidal scheme. From Fig. 1, all the observations of X _t fall into the area between the 0.05-path and the 0.90-path, so these four estimates are plausible.

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Fig. 1

Observations and α-paths of X _t in Example 4.1.

Example 4.2 Suppose the uncertain process X _t follows dX t = ( μ 1 - μ 2 X t ) dt + σ dC t ,(19) with initial value X₀ = 1, the parameters μ₁, μ₂ and σ are unknown numbers. Using the composite Heun scheme, we have h n ( μ 1 , μ 2 , σ ) = 1 σ ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 x t n + λ μ 1 μ 2 Δ t n - λ μ 2 2 Δ t n x t n ) , where n = 0, 1, ⋯ , N - 1. Because the first three moments of N ( 0 , 1 ) are 0, 1 and 0, the system of equations in (11) becomes { [ c ] rl 1 N ∑ n = 0 N - 1 1 σ ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 x t n + λ μ 1 μ 2 Δ t n - λ μ 2 2 Δ t n x t n ) = 0 , 1 N ∑ n = 0 N - 1 1 σ 2 ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 x t n + λ μ 1 μ 2 Δ t n - λ μ 2 2 Δ t n x t n ) 2 = 1 , 1 N ∑ n = 0 N - 1 1 σ 3 ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 x t n + λ μ 1 μ 2 Δ t n - λ μ 2 2 Δ t n x t n ) 3 = 0 .(20) By solving the above nonlinear system of equations, we can get the moment estimate ( μ 1 * , μ 2 * , σ * ) for (μ₁, μ₂, σ).

Analogously, according to the trapezoidal scheme, we have [ c ] rl m n ( μ 1 , μ 2 , σ ) = 1 σ ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 2 ( x t n + x t n + 1 ) ) , for n = 0, 1, ⋯ , N - 1. From (14), the moment estimate ( μ 1 * , μ 2 * , σ * ) satisfies { [ c ] rl 1 N ∑ n = 0 N - 1 1 σ ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 2 ( x t n + x t n + 1 ) ) = 0 , 1 N ∑ n = 0 N - 1 1 σ 2 ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 2 ( x t n + x t n + 1 ) ) 2 = 1 , 1 N ∑ n = 0 N - 1 1 σ 3 ( x t n + 1 - x t n Δ t n - μ 1 + μ 2 2 ( x t n + x t n + 1 ) ) 3 = 0 .(21)

From Corollary 3.1 in [3], the analytic solution of (19) satisfies [ c ] rl X t = exp ( - μ 2 t ) ( X 0 + μ 1 μ 2 ( exp ( μ 2 t ) - 1 ) + σ ∫ 0 t exp ( μ 2 s ) dC s ) . Then we have [ c ] rl exp ( μ 2 t ) X t = X 0 + μ 1 μ 2 ( exp ( μ 2 t ) - 1 ) + σ ∫ 0 t exp ( μ 2 s ) dC s . For t _n and t_n+1, it follows that [ c ] rl exp ( μ 2 t n ) X t n = X 0 + μ 1 μ 2 ( exp ( μ 2 t n ) - 1 ) + σ ∫ 0 t n exp ( μ 2 s ) dC s ,(22) and [ c ] rl exp ( μ 2 t n + 1 ) X t n + 1 = X 0 + μ 1 μ 2 ( exp ( μ 2 t n + 1 ) - 1 ) + σ ∫ 0 t n + 1 exp ( μ 2 s ) dC s .(23) Subtracting \eqref N24 from \eqref N23 implies that [ c ] rl exp ( μ 2 t n + 1 ) X t n + 1 = exp ( μ 2 t n ) X t n + μ 1 μ 2 ( exp ( μ 2 t n + 1 ) - exp ( μ 2 t n ) ) + σ ∫ t n t n + 1 exp ( μ 2 s ) dC s , which means [ c ] rl X t n + 1 = exp ( - μ 2 Δ t n ) X t n + μ 1 μ 2 ( 1 - exp ( - μ 2 Δ t n ) ) + σ ∫ t n t n + 1 exp ( μ 2 ( s - t n + 1 ) ) dC s . Using Theorem 13.6 in [1], one has [ c ] rl σ ∫ t n t n + 1 exp ( μ 2 ( s - t n + 1 ) ) dC s ~ N ( 0 , σ μ 2 ( 1 - exp ( - μ 2 Δ t n ) ) ) . Thus the samples of X _t can be calculated by [ c ] rl x t n + 1 = exp ( - μ 2 Δ t n ) x t n + μ 1 μ 2 ( 1 - exp ( - μ 2 Δ t n ) ) + σ μ 2 ( 1 - exp ( - μ 2 Δ t n ) ) ) z n ,(24) where n = 0, 1, ⋯ , N - 1, z _n are the samples of N ( 0 , 1 ) . The parameters in the differential equation \eqref N2 are set as μ₁ = 15, μ₂ = 2.4 and σ = 1.5. Then the samples of X _t generated by \eqref N25 are shown in Table 3.

By solving the equations \eqref N21 with λ = 0.6, we get the estimated values μ 1 * = 15.2088 , μ 2 * = 2.3550 , σ * = 1.3263 . Hence the estimated equation is dX t = ( 15.2088 - 2.3550 X t ) dt + 1.3263 dC t . Based on the trapezoidal scheme, we have dX t = ( 11.8253 - 1.8958 X t ) dt + 0.9243 dC t .

The estimated values of μ₁, μ₂ and σ based on the Euler scheme and the Milstein scheme are also shown in Table 4. Table 5 shows that the composite Heun scheme performs best compared to the Euler scheme, the trapezoidal scheme and the Milstein scheme. However, the observations fall into the area between the 0.05-path and the 0.90-path of the uncertain differential equations based on the composite Heun scheme and the Milstein scheme as shown in Fig. 2. That means only these two estimates are plausible for this case.

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Fig. 2

Observations and α-paths of X _t in Example 4.2.

In the following two numerical examples, the observations of X _t are given to estimate the unknown parameters in uncertain differential equations.

Example 4.3 Suppose the uncertain process X _t follows dX t = μ ( 1 - cln X t ) X t dt + σ X t dt ,(25) where the three parameters μ, c and σ are unknown numbers. Assume that there are twenty groups of observed data of X _t as shown in Table 6.

By using the composite Heun scheme, we have h n ( μ , c , σ ) = x t n + 1 - x t n σ Δ t n x t n - ( 1 - λ ) F n σ x t n - λ μ ( 1 - cln ( x t n + Δ t n F n ) ) ( x t n + Δ t n F n ) σ x t n , where F _n  = μ (1 - clnx _{t n} ) x _{t n} . Thus the system of equations (11) becomes { [ r ] rl 1 19 ∑ n = 1 19 1 σ ( x t n + 1 - x t n Δ t n x t n - ( 1 - λ ) F n x t n - λ μ ( 1 - cln ( x t n + Δ t n F n ) ) ( x t n + Δ t n F n ) x t n ) = 0 , ∥ 1 19 ∑ n = 1 19 1 σ 2 ( x t n + 1 - x t n Δ t n x t n - ( 1 - λ ) F n x t n - λ μ ( 1 - cln ( x t n + Δ t n F n ) ) ( x t n + Δ t n F n ) x t n ) 2 = 1 , ∥ 1 19 ∑ n = 1 19 1 σ 3 ( x t n + 1 - x t n Δ t n x t n - ( 1 - λ ) F n x t n - λ μ ( 1 - cln ( x t n + Δ t n F n ) ) ( x t n + Δ t n F n ) x t n ) 3 = 0 .(26)

By solving the system of equations \eqref N31 with λ = 0.4, we get the estimated values μ * = 0.0307 , c * = 1.0145 , σ * = 0.0758 . Hence the estimated equation is dX t = 0.0307 ( 1 - 1.0145 ln X t ) X t dt + 0.0758 X t dt .

The estimated values based on the trapezoidal scheme, the Euler scheme and the Milstein scheme are also shown in Table 7. In order to compare the numerical results of the four schemes, the up (down) errors [23] which are the sum of absolute errors between the sample data and the 0.90-path (0.05-path) at each time t _n are computed as shown in Table 8. In this example, the composite Heun scheme has the least up error and down error. From Fig. 3, all the observations of X _t fall into the area between the 0.05-path and the 0.90-path, so these four estimates are plausible.

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Fig. 3

Observations and α-paths of X _t in Example 4.3.

Example 4.4 Suppose the uncertain process X _t follows dX t = - μ X t 2 dt - σ X t 2 dt ,(27) where the parameters μ and σ are unknown numbers. Assume that there are twelve groups of observed data of X _t as shown in Table 9.

By using the composite Heun scheme, we have h n ( μ , σ ) = 1 σ ( - x t n + 1 - x t n Δ t n x t n 2 - μ + 2 μ 2 λ Δ t n x t n - λ μ 3 Δ t n 2 x t n 2 ) . Thus the system of equations (11) becomes { [ r ] rl 1 11 ∑ n = 1 11 1 σ ( - x t n + 1 - x t n Δ t n x t n 2 - μ + 2 μ 2 λ Δ t n x t n - λ μ 3 Δ t n 2 x t n 2 ) = 0 , ∥ 1 11 ∑ n = 1 11 1 σ 2 ( - x t n + 1 - x t n Δ t n x t n 2 - μ + 2 μ 2 λ Δ t n x t n - λ μ 3 Δ t n 2 x t n 2 ) 2 = 1 , ∥ 1 11 ∑ n = 1 11 1 σ 3 ( - x t n + 1 - x t n Δ t n x t n 2 - μ + 2 μ 2 λ Δ t n x t n - λ μ 3 Δ t n 2 x t n 2 ) 3 = 0 .(28)