Strong Convergence of the Split-Step θ-Method for Stochastic Age-Dependent Capital System with Random Jump Magnitudes

and Applied Analysis 3 measurable (from [0, T] × Ω into V), and satisfying E∫ T 0 ‖K t ‖ dt < ∞. Here C([0, T]; V) denotes the space of all continuous functions from [0, T] to V; (D3) it satisfies the following equation:


Introduction
Stochastic partial differential equations are becoming increasingly used to model real-world phenomena in different fields, such as economics, biology, and physics.Recently, the study of the stochastic age-dependent (vintage) capital system has received a great deal of attention.For example, Wang studied stability of solutions for stochastic investment system [1].Zhang et al. studied the convergence and exponential stability of numerical solutions to the stochastic age-dependent capital system [2,3].
In the stochastic age-dependent capital system, due to the effects of external environment for capital system, such as innovations in technique, introduction of new products, natural disasters, and changes in laws and government policies, the size of the capital systems increases or decreases drastically.So Poisson jumps with deterministic jump magnitude have been used in stochastic age-dependent population equations.For example, Li et al. [4] studied the Euler numerical method for stochastic age-dependent population equations with Poisson jumps.L. Wang and X. Wang [5] analysed the convergence of the semi-implicit Euler method for stochastic age-dependent population equations with Poisson jumps.Rathinasamy et al. [6] developed the numerical method for stochastic age-dependent population equations with Poisson jump and phase semi-Markovian switching.However, the random jump magnitude is now commonly seen in financial models [7][8][9].In this paper, we will consider the following stochastic age-dependent capital system with random jump magnitudes as shown in [10] where (, ) denotes the stock of capital goods of age  at time , d(, ) = ((, )/ + (, )/)d, (0) =  0 (0), and  = [0, ] × [0, ].N() is defined as total 2 Abstract and Applied Analysis output produced in year ; also  is the age of the capital; the investment () in the new capital.The  is the appreciation (when  ≥ 0) or depreciation (when  ≤ 0) of the production capacity, and  represents the volatility of the capital stock.The value of ℎ is the actual jump and   is the underlying random variables of the magnitudes, and often it is called "mark" of the jump.
Also () is a standard Wiener process.() is a scalar Poisson process with intensity  1 .It is assumed that for some  ≥ 1 there is a constant  such that E[|  | 2 ] ≤ ; that is, the 2th moment of the jump magnitude is bounded.The maximum physical lifetime of capital , the planning interval of calendar time [0, ), the depreciation rate (, ) of capital, and the capital density  0 () (the initial distribution of capital over age)are given.The () denotes the accumulative rate of capital at the moment of , 0 < () < 1, and () is the technical progress at the moment of .This makes that total output produced in year  be defined as N() = ∫  0 (, )d.
In each sector all the firms have an identical neoclassical technology and produce output using labor and capital.The production function ((), ∫  0 (, )d) is neoclassical, where ∫  0 (, )d is the total sum of capital at time  and () is the labor force.
The integral version of ( 1) is given by the equation where   = (, ) for fixed .
Since the system (1) does not have closed form solutions, it is necessary to develop numerical methods for (1).Recently, Zhang and Rathinasamy [10] first derived the numerical solutions for stochastic age-dependent capital system with random jump magnitudes.However, their method belongs to the classic explicit Euler method and has a lower accuracy in [10] if we do not consider the appropriate step sizes.
Higham and Kloeden [11] first constructed the splitstep backward Euler (SSBE) method for nonlinear stochastic differential equations with Poisson jumps.Tan and Wang [12] studied the convergence and stability of the SSBE method for linear stochastic delay integrodifferential equations.Ding et al. [13] developed the split-step  method for solving the stochastic differential equations.Rathinasamy [14] investigated the split-step  methods for stochastic age-dependent population equations with Markovian switching.Thus, we can construct the SS method for stochastic age-dependent population equations with random Poisson jumps.
In this paper, we will investigate the convergence of the SS method for system (1).The outline of the paper is as follows.In Section 2, we will introduce some preliminary results which are essential for our analysis.Section 3 will show us the SS method for solving stochastic age-dependent population equations with random Poisson jumps.In Section 4, several lemmas which are useful for our main result are proved.We give the main result that the numerical solutions converge to the true solutions with strong order 1/2 in Section 5.At last, a numerical example is given to verify the results obtained from the theory.

Preliminaries
Throughout this paper, it will be denoted by  2 ([0, ]) the space of functions that are square-integrable over the domain where / is generalized partial derivative with respect to age  and  is a Sobolev space. =  2 ([0, ]) such that  →  ≡   →   .  =  −1 ([0, ]) is the dual space of .We denote by ‖ ⋅ ‖, | ⋅ | and ‖ ⋅ ‖ * the norms in , , and   , respectively, by (⋅, ⋅) the scalar product in .⟨⋅, ⋅⟩, the duality product between  and   , is defined by Let () be a Wiener process defined on (Ω, F, P) and taking its values in the separable Hilbert space , where {  } ≥0 is an orthonormal set of eigenvectors of ,   () are mutually independent real Wiener processes with incremental covariance   > 0,   =     , and tr  = ∑ ∞ =1   (tr denotes the trace of an operator).For an operator  ∈ L(, ) to be the space of all bounded linear operators from  into , it is denoted by ‖‖ 2 the Hilbert-Schmidt norm; that is, Let  = ([0, ]; ) be the space of all continuous function from [0, ] into  with sup-norm ‖‖  =sup 0≤≤ ||(),    =   ([0, ]; ), and    =   ([0, ]; ).Let (Ω, F, P) be a complete probability space with a filtration {F  } ≥0 , satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets).Definition 1.Let (Ω, F, {F  }, P) be the stochastic basis and   a Wiener process.Suppose that  0 is a random variable such that E| 0 | 2 < ∞.A stochastic process   ≡ (, ) for fixed  is said to be a solution on Ω to the stochastic age-structured capital system for  ∈ [0, ] if the following conditions are satisfied: for all V ∈ ,  ∈ [0, ], a.e. ∈ Ω, where the stochastic integrals are understood in the Itô sense.
In an analogous way to the corresponding proof presented in [15], the following existence and uniqueness of solutions is established: under the conditions (H1)-(H4), (1) has a unique continuous solution (, ) on (, ) ∈ .
To answer the question of the existence of numerical solution, we will give the following lemma.Lemma 3. Let conditions (H2) and (H3) hold, and let 0 <  < 1 and 0 < Δ < 1/( + ); then the implicit equation (12a) can be solved uniquely for  *  , with probability 1.
Proof.Writing (12a) as and using condition (H2) and (H3), we have Then the result follows from the classical Banach contraction mapping theorem [18].
When Lemma 3 followed, we find it is convenient to use continuous-time approximation solution in our strong convergence analysis; hence for ∈ [  ,  +1 ), we can define the following step functions: where  is the largest number such that Δ ≤ .
It is easy to verify that  1 (  , ) =   = (  , ); that is,  1 (, ) and (, ) coincide with the discrete solutions at the grid points.Hence we refer to (, ) as a continuous-time extension of the discrete approximation {  }.So our plan is to prove a strong convergence result for (, ).

Several Lemmas
In this section, we will give several lemmas which are useful for the following main result.Lemma 4.Under the conditions (H1)-(H4), there are constants  ≥ 2 and  1 > 0 such that The proof is similar to that in [3].
Lemma 8.Under conditions (H1)-(H4), for any  ≥ 2, there exists a positive constant  7 such that Proof .The proof is similar to that of Lemma 4.

Main Result
Now we use the above lemmas to prove a strong convergent result.
It is easy to verify that the conditions (H1)-(H4) are satisfied.Then the approximate solution will converge to the true solution of (1) for any (, ) ∈ (0, 1) × (0, 1) in the sense of Theorem 11.Obviously, (, ) in (72) cannot be solved explicitly.It is necessary to know numerical approximation (, ) of (, ).Now we fix step sizes Δ = 0.005 and Δ = 0.05 and change the parameter  in all figures.Then we compare the expectation of the numerical solution, where E[(, )] = (1/1000) ∑ 1000 =1   (, ).In Figure 1, we show the expectation of numerical solution of the system (72) by Euler method in [10] and splitstep -method with  = 0.2, respectively.From the figure, we can see that the split-step -method also reveals the agedependent capital system tendency.
In Figures 2 and 3, the four pictures are simulation for numerical solution   (, ) of the system (72) by split-step method with  = 0.3, 0.4, 0.5, and 0.6, respectively.

Conclusion
Due to the complexity of this model, the comparison between the Euler approximation and split-step methods is not possible in this paper.Similarly, error analysis is also not obtained in this paper due to unavailability of closed form solutions.They will be discussed in the future researches.But the visualization of Figures 1, 2, and 3 provides the information about the split-step methods which coincide with the Euler approximation.The comparison between Euler approximation and split-step methods in the stochastic differential equations in [11][12][13] shows that the split-step methods are better than the Euler approximation.Similarly, we believe that this new approximation of the system (1) in this paper is a good approximation when compared with the Euler approximation.