ADOMIAN DECOMPOSITION METHOD APPLIED TO LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

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1 International Journal of Pure and Applied Mathematics Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i3.1 PAijpam.eu ADOMIAN DECOMPOSITION METHOD APPLIED TO LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS Abdelouahab Bibi 1, Fateh Merahi 2 Département de Mathématiques Université Constantine UMC1 ALGERIA Abstract: The paper deals with an application of Adomian Decomposition Method for solving linear stochastic differential equations. We derive new formulas such as the analytical approximate solution which convergces rapidely to the exact solution. The numerical experiments which are obtained show the efficiency of this method in the field of stochastic differential equations. AMS Subject Classification: 4A5, 4A25, 45G5 Key Words: Adomian Decomposition Method, Brownian motion, Linear stochastic differential equations, Approximate solution. 1. Introduction Adomian Decomposition Method (ADM) is widely used to solve linear and nonlinearequations ofvariouskind[3, 2,17]. Thesolution isfoundasaninfinite series wich converges quickly towards an accurate solution, and its terms can be easily determined. The convergence of the ADM has been discussed by many authors, in particular Cherruault [8] is the first who proved the convergence, after that Abbaoui and Cherruault [1, 2], Himoun, et al. [11, 12], Hosseini and Nasabzadeh [13], Lesnic [14, 15], and Rach [17, 18]. El-Kalla [9] gave the Received: Revised: Published: April 1, 218 c 218 Academic Publications, Ltd. url: Correspondence author

2 52 A. Bibi, F. Merahi error analysis of the method, Babolian and Biazar [5], Boumenir and Gordon [6] discussed the order and rate of convergence for ADM respectively. Theaim of this work is to extend theadm for solving stochastic differential (SDE) which play an important role in various field such as finance, automatic control and physics, chemistry, astronomy, engineering, biology and others, the SDE are the fundamental tool to solve some problems in mathematics and finance stochastic modeling. The continuous-time models can be interpreted as a solution of SDE, these models are often assumed to be linear and may be Gaussian and continue to gain a growing interest of researchers (see for instance Brockwell [7] and the references therein). On the other hand, SDE are the more general framework of study and analysis of continuous-time random process. The rest of paper is organized as follows. Having defined the linear SDE and their properties. ADM is given in Section 3. The numerical experiments are provided in Section 4 and conclusion is in Section Linear Stochastic Differential Equations We consider the process (X(t)) t R generated by the following linear SDE dx(t) = (α 1 X(t)+α )dt+σdw(t) (1) where (w(t)) t is the standard Brownian motion (Bm) in R defined on some basic probability space (Ω,A,P), α 1,α and σ are constants, with σ >. The initial state X() = X is a random variable, defined on (Ω,A,P), independent of w such that E{X()} = m() and Var{X()} = V(). The existence and uniqueness of the solution process (X(t)) t of equation (1) is ensured by the general results on stochastic differential equations [4] and under the above conditions. So, the solution of equation (1) is interpreted as satisfying the following integral equation X(t) X() = (α 1 X(s)+α )ds+σw(t) (2) The term (α 1 X(s) + α ) is referred to as instantaneous mean of the process, and σ the instantaneous standard deviation.the solution of (1) can be written as X(t) = e α1t X()+α e α1(t s) ds+σ e α1(t s) dw(s) (3) Let the mean of X(t) be denoted by m(t). It satisfies the equation m(t) = α α 1 (e α 1t 1)+e α 1t m(). (4)

3 ADOMIAN DECOMPOSITION METHOD APPLIED R(t,s) is the covariance of X(t) given by R(t,s) = e α 1(t s) V(s), for all t s (5) where V(s) denotes the varians of X(s) given by We can approach by s V(s) = e 2α1s V()+σ 2 e 2α1(s u) du, for all s. (6) X(t+h) = X(t) Euler Approximation +h t (α 1 X(t)+α )ds+σ +h t dw(s), (7) X(t j+1 ) X(t j )+(α 1 X(t j )+α )h+σ(w(t j+1 ) w(t j )), (8) where h = t j+1 t j. The Euler approximation suppose that the integrals are constant over theinterval ofintegration. Sincethevariables(w(t j+1 ) w(t j ))are independent and normal with variance h, we obtain the scheme X (j+1)h = X jh +(α 1 X jh +α )h+σ hz j (9) where(z j ) j N isagaussianwhitenoiseprocesswithmeanandfinitevariance. Phillips [16] showed that the exact discrete model corresponding to (1) is given by X (j+1)h = e α 1h X jh + α α 1 (e α 1h 1)+σ 1 e 2α 1h 2α 1 Z j. (1) 3. First Application of Adomian Decomposition Method Adomian s method consists in calculating the solution X(t) of (1) in a series form X(t) = X j (t) (11) j= j= Putting equation (11) into (1) gives ( X j (t) = X()+α t+l 1 α 1 X j (t) +L 1 σ dw(t) ), (12) dt j=

4 54 A. Bibi, F. Merahi where L = d dt is the derivative of 1-order, then the corresponding L 1 operator can be written in the form L 1 [.] = (.)ds. The iterates are determined by following recursive way and n, we get X 1 (t) = α 1 tx()+ α 1α 2 t2 +σα 1 X 2 (t) = α 2 t X()+ α2 1 α 6 t3 +σα 2 1 X 3 (t) = α 3 t X()+ α3 1 α 24 t4 +σα 3 1. X n (t) = α n 1 X (t) = X()+α t+σw(t) (13) X n+1 (t) = α 1 L 1 [X n (t)], (14) w(t)dt w(t)dtdt w(t)dtdtdt t n n! X()+ αn 1 α (n+1)! tn+1 +σα n 1 }{{} n fold Finally, we approximate the solution by the truncated series N 1,Φ N (t) = N 1 n= w(t)dt }{{} dt n fold X n (t). (15) In order to show the convergence almost sure (a.s) of the series to the exact solution we need the following Theorem which is given and proved by I. El-Kalla [1]. Theorem 3.1 (Theorem 1, I. El-Kalla [1] ). If q(t) is integrable and at least the derivative of 1-order of r(t) exists then we can prove that e r(t) e r(t) q(t)dt = dr(t) dr(t) ( 1) k dt }{{} dt k= q(t)dtdt }{{} dt. (k+1) fold In particular, if dq dt exists then we have e r(t) e r(t) dq(t) = dt k= ( 1) k dr(t) dr(t) }{{} dt q(t)dtdt }{{} dt.

5 ADOMIAN DECOMPOSITION METHOD APPLIED Proof. See the proof of Theorem 1 in [1]. Then the convergence of the approximate solution Φ N (t) to the exact solution process is given by the folowing theorem : Theorem 3.2. Let Φ N (t) is the approximate solution given by Adomian decomposition method of the linear SDE (1), then we have with probability 1 Proof. First, Φ N (t) can be written as follow lim Φ N(t) = X(t) (16) N + N 1 Φ N (t) = X() k= (α 1 t) k k! + α α 1 N (α 1 t) k k! N 1 +σ α k 1 k= }{{} w(t)dt }{{} dt (17) and apply Theorem 3.1 on the third term of (17) we get k= Then we obtain α k 1 }{{} lim Φ N(t) = X() N + (α 1 t) k k= w(t)dt }{{} dt = k! + α α 1 = X()e α 1t + α α 1 (e α 1t 1)+σ and the proof is complete. (α 1 t) k +σ k! e α 1(t u) dw(u) (18) e α 1(t u) dw(u) e α 1(t u) dw(u) = X(t). 4. Second Application of Adomian Decomposition Method In this section we give another approximate solution of the linear SDE (1). This approach follows from the fact that a normalized Brownian motion w(t) can be written in the form w(t) = Z 2π t+ 2 π Z k sin(kt) for t [,π] k

6 56 A. Bibi, F. Merahi where Z,Z 1,... be mutually independent random variables with identical normal distributions N (, 1). Then The iterates are determined by following recursive way χ (t) = X()+α t+σ Z 2π t+σ 2 π Z k sin(kt) (19) k and n, χ n+1 (t) = α 1 L 1 [χ n (t)], (2) we obtain 2 π Z k t 3 2 2π 6 +α2 1σ π Z k χ 1 (t) = α 1 tx()+ α 1α 2 t2 +α 1 σ Z 2π t 2 2 +α 1σ χ 2 (t) = α 2 t X()+ α2 1 α 6 t3 +α 2 1σ Z χ 3 (t) = α 3 t X()+ α3 1 α 24 t4 +α 3 1σ Z t 4 2π 24 +α3 1σ. χ n (t) = (α 1t) n X()+ α (α 1 t) n+1 n! α 1 (n+1)! + σ Z (α 1 t) n+1 α 1 2π (n+1)! ( ) 2 +α n 1σ π Z ( 1) [n 2 ] k sin(kt) if n is even kn+1 χ n (t) = (α 1t) n X()+ α (α 1 t) n+1 n! α 1 (n+1)! + σ Z (α 1 t) n+1 α 1 2π (n+1)! ( ) 2 +α n 1σ π Z ( 1) [n 2 ]+1 k k n+1 cos(kt) if n is odd Finally, we approximate the solution by the truncated series N 1, Ψ N (t) = N 1 n= ( 1) k 2 cos(kt) ( 1) k 3 sin(kt) 2 π Z ( 1) 2 k k 4 cos(kt) χ n (t). (21)

7 ADOMIAN DECOMPOSITION METHOD APPLIED Numerical Experiments The obtained results for various values of the parameters α,α 1 and σ where compared with the exact solution X and where shown graphicaly. Assume the process X(t) is observed in the interval [,T] at points (t 1,t 2,...,t m ), where h = t i+1 t i is the fixed space scale. Then we have the sequence of m observations X h,x 2h,...,X mh. In our experiment we choose T = 1. We use two kinds of approximated solution Φ 1 (t) = X which is an approximation with one term, and Φ 2 (t) = X +X 1 using a two terms of approximation. We can prove that the discrete models corresponding to Φ 1 (t) and Φ 2 (t) are given respectively by and Φ 1 (t j+1 ) = Φ 1 (t j )+α h+σ hz j, Φ 2 (t j+1 ) = Φ 2 (t j )+(α +α 1 X())h+ α α 1 2 (2j +1)h2 +σ(α 1 j +1) hz j, for all j with t = and (Z j ) j N is a Gaussian white noise process with mean and finite variance equal to 1. The table 1 shows the results from t =.1 to t = 1 with an increment h =.1 where we compute the absolute error of the both approximate solutions Φ 1 (t) and Φ 2 (t) with the parameters α =.2, α 1 =.5 and σ = 1. t X(t) Φ 1 (t) Φ 2 (t) Φ 1 (t) X(t) Φ 2 (t) X(t) Table 1: Absolut error of Φ 1 and Φ 2, with α =.2, α 1 =.5, σ = 1, h =.1.

8 58 A. Bibi, F. Merahi The table 2 shows the results from t =.1 to t = 1 but with an increment h =.1, we compute the absolute error of the approximate solutions Φ 1 (t) and Φ 2 (t) with the parameters α =, α 1 =.1 and σ = 1. Finally, in oredr to show that Adomian series is a rapidly converging to the exact solution process X(t). Figure 1, 2 present the plots of Φ 1 (t) and Φ 2 (t) with the parameters α =, α 1 =.1 and σ = 1. t X(t) Φ 1 (t) Φ 2 (t) Φ 1 (t) X(t) Φ 2 (t) X(t) Table 2: Absolut error of Φ 1 and Φ 2, with α =, α 1 =.1, σ = 1, h = Conclusion In this paper we propose an extension of the ADM for solving linear SDE. The numerical results for different values of parameters confirm the theoretical results obtained and illustrate the powerful of this method. We use Matlab for the computations associated with the section of numerical experiments in this

9 ADOMIAN DECOMPOSITION METHOD APPLIED Figure 1: Exact solution X and approximate solution Φ 1 for α =, α 1 =.1, σ = 1, h =.1. Figure 2: Exact solution X and approximate solution Φ 2 for α =, α 1 =.1, σ = 1, h =.1. work.

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