STOCHASTIC Volterra integral equations (SVIEs) is a

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1 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 The Petrov-Gaerkin Method for Numerica oution of tochastic Voterra Integra Equations F. Hosseini hekarabi 1, M. Khodabin 2 and K. Maeknejad 3 Abstract In this paper, we introduce the Petrov-Gaerkin method for soution of stochastic Voterra integra equations. Here, we use continues Lagrange-type k- eements, since these eements have simpe structure and via them, the soution of stochastic Voterra integra equation is reduced to agebraic equations. Aso the error anaysis of this method is done. In Comparison with other methods, this method has ess computation. Index Terms Petrov-Gaerkin method; Continuous Lagrange-type k- eements; tochastic Voterra integra equations; Itô integra; Brownian motion process. I. INTRODUCTION TOCHATIC Voterra integra equations (VIEs) is a fast deveoping fied, with appications in economics, socioogy, bioogy, medica modes and anthropoogy. Background materia and countess references can be found in [1-9]. tochastic Voterra integra equations arise when a random noise is introduced into Voterra integra equations. These systems are dependent on a noise source, which is a Gaussian white one. The Brownian motion process B(t) serves as a basic mode for the cumuative effect of pure noise. Generay, we are not abe to find expicit formuae for the soutions of VIEs and thus need to use a numerica method to approximate the soutions. The Petrov-Gaerkin method is a numerica method based on Gaerkin method but with different tria and test spaces. This method has been used for approximation of the numerica soution of Fredhom second kind integra equations [1-11], Hammerstein integra equations [12-13], and integrodifferentia equations [14-15]. We consider the one-dimensiona inear stochastic Voterra integra equation u(t) = f(t) + k 1 (s, t)u(s)ds + k 2 (s, t)u(s)db(s) (1) t, s [, T ] = I, where, u(t), f(t), k 1 (s, t) and k 2 (s, t), for s, t [, T ), are the stochastic processes defined on the same probabiity space (Ω, F, P ), and u(t) is unknown random function, B(t) is a Brownian motion process and k 2(s, t)u(s)db(s) is the Itô integra. We rewrite this equation in operator form as (I K 1 K 2 )u(t) = f(t), (2) Manuscript received May 16, 214; revised June 12, 214. F. Hosseini hekarabi, M. Khodabin and K. Maeknejad is with the Department of Mathematics, Coege of Basic ciences, Karaj Branch, Isamic Azad University, Aborz, Iran. ( 1 E-mai: f hosseini@srttu.edu 2 Corresponding Author. E-mai Address: m-khodabin@kiau.ac.ir 3 E- mai: maeknejad@kiau.ac.ir). where (K 1 u)(t) := (K 2 u)(t) := k 1 (s, t)u(s)ds, k 2 (s, t)u(s)db(s) t I. In this artice, we are going to use continuous Lagrangetype k- eements of Petrov-Gaerkin system to approximate the numerica soution of stochastic Voterra integra equation. The content of this paper is arranged in five sections. In ection II, we introduce some genera concepts concerning the Petrov-Gaerkin method and stochastic concepts. ection III, presents error anaysis. In ection IV, we show numerica resuts. Finay, ection V provides the concusion. II. PRELIMINARIE The Petrov-Gaerkin method uses reguar pairs X n, Y n } of piecewise poynomia spaces that are caed Petrov- Gaerkin eements. In this section, we summarize the key concepts and resuts of Petrov-Gaerkin method and stochastic cacuus. A. Petrov-Gaerkin method Let X be a Banach space and X be its dua space of continuous inear functionas, for each positive integer n, we assume that X n X,Y n X, and X n and Y n are finite dimensiona vector spaces with dim X n =dim Y n. In addition, we assume the foowing property: (H) If x X and y Y, then there are sequences x n and y n, with x n X, y n Y for a n such that x n x, y n y. Definition 1. For x X, an eement P n x X n is caed the generaized best approximation from X n to x with respect to Y n by the equation < x P n x, y n >=, y n Y n. (3) Lemma 1. [1] For each x X, the generaized best approximation from X n to x with respect to Y n exists uniquey if and ony if X n Y n = }, (4) where, X n = x X; < x, x >=, x X n }. When condition (4) satisfied, P n defines a projection; that P 2 n = P n. In addition, we must add some properties to (4) in order to bring a situation that P n converges point-wise to the identity operator. Definition 2. X n, Y n } is caed reguar pair if a inear operator n : X n Y n with n X n = Y n exists and (Advance onine pubication: 28 November 214)

2 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 satisfy the foowing two conditions (H-1) x n C 1 < x n, x n > 1/2, x n X n. (H-2) n x n C 2 x n, x n X n. C 1 and C 2 are constants independent of n and. is L 2 norm. Theorem 1. Let pairx n.y n } satisfies dimx n = dimy n and condition (H) then the corresponding generaized projection P n satisfies: (P1) P n x x as n. (P2) P n C, n = 1, 2, 3,... for some constants C. (P3) P n x x C Q n x x, n = 1, 2, 3,... for some constants C, where Q n, is the best approximation from X n to x. Proof. ee [1]. B. tochastic concepts of Itô integra Definition 3.[16] (Brownian motion process). A reavaued stochastic process B(t), t [, T ] is caed Brownian motion, if it satisfies the foowing properties: (i) (Independence of increments) B(t) B(s), t > s, is independent of the past, that is, of B(u), u s, or of F s, the σ-fied generated by B(u), u s. (ii) (Norma increments) B(t) B(s) has Norma distribution with mean and variance t s. (iii) (Continuity of paths) B(t), t are continuous functions of t. Definition 4. [16] Let N(t)} t be an increasing famiy of σ-agebras of sub-sets of Ω. A process g(t, ω) from [, ) Ω to R n is caed N(t)-adapted if for each t the function ω g(t, ω) is N(t)-measurabe. Definition 5. [16] Let ν = ν(, T ) be the cass of functions f(t, ω) : [, ) Ω R such that, (i) (t, ω) f(t, ω), is B F-measurabe, where B denotes the Bore σ-agebra on [, ) and F is the σ-agebra on Ω. (ii) f(t, ω) is F t -adapted, where F t is the σ-agebra generated by the random variabes B(s); s t. (iii) E [ T f 2 (t, ω)dt] <. Definition 6. [16] (The Itô integra). Let f ν(, T ), then the Itô integra of f (from to T) is defined by T f(t, ω)db(t)(ω) = im n T (imit in L 2 (P )) ϕ n (t, ω)db(t)(ω), where, ϕ n is a sequence of eementary functions such that E [ T (f(t, ω) ϕ n (t, ω)) 2 dt ], as n. Theorem 2. E [ ( T (The Itô isometry), Let f ν(, T ), then (f(t, ω)db(t)(w)) 2] = E [ T f 2 (t, ω)d(t) ]. Proof. ee [16]. Definition 7. (1-dimensiona Itô processes), [16]. Let B(t) be 1-dimensiona Brownian motion on (Ω, F, P ). A 1- dimensiona Itô process (stochastic integra) is a stochastic process X(t) on (Ω, F, P ) of the form X(t) = X() + u(s, ω)ds + v(s, ω)db(s), or where dx(t) = udt + vdb(t), (5) P [ v 2 (s, ω)ds <, for a t ] = 1, P [ u(s, ω) ds <, for a t ] = 1. Theorem 3. (The 1-dimensiona Itô formua). Let X(t) be an Itô process given by (1) and g(t, x) C 2 ([, ) R), then is again an Itô process, and Y (t) = g ( t, X(t) ), dy (t) = g ( ) g ( ) t, X(t) dt + t, X(t) dx(t)+ t x 1 2 g ( )( ) 2, t, X(t) dx(t) (6) 2 x 2 where (dx(t)) 2 = (dx(t))(dx(t)) is computed according to the rues dt.dt = dt.db(t) = db(t).dt =, Proof. ee [16]. db(t).db(t) = dt. (7) C. Appying the Petrov-Gaerkin method to sove VIEs Consider u(t) k 1 (s, t)u(s)ds + L(t, ω, u(t)), with random input parameters L(t, ω, u(t)) = f(t). k 2 (s, t)u(s)db(s) := In the first step, Petrov-Gaerkin method for Eq.(1) is a numerica method for finding u n = n i= c iϕ i (t) X n, such that c i is unknown and must be determined. o L(t, ω, n c i ϕ i (t)) = f(t). i= Using inner product, we take a projection of the equation onto each basis poynomia ψ n Y n, i.e. < L(t, ω, n c i ϕ i (t)), ψ n >=< f(t), ψ n > ψ n Y n. i= (8) Eq. (8) can be derive from P n x = for any x X if and ony if < x, ψ n >= for a ψ n Y n. In this case our probem reduced to a inear system and we can use any appropriate method for finding coefficients. (Advance onine pubication: 28 November 214)

3 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 D. Continuous Lagrange-type k- eements Here we give a brief review of construction of continuous Lagrange-type k eements [1]. We subdivide the interva [, 1] into n subintervas. uppose that X n be the space of continuous piecewise poynomias of degree not exceeding k, and with knots at t i, i = 1, 2,..., n 1. It is cear that dimx n = nk + 1. Let = t < t 1 <... < t n = 1 and I i = [t i 1, t i ]; h i = t i t i 1 for i = 1,..., n. The basis of X n is constructed by concept of Lagrange poynomias, for this purpose, et τ j = j k, j =, 1,..., k and t(i) j = t i 1 + τ j h i, j =, 1,..., k, i = k = t i. 1,..., n.then, it is cear that t i 1 = t (i) <... < t (i) Then nk + 1 functions ϕ (i) ϕ (i) j (t) = j k = j (t) as basis of X n is given by t t (i) t (i) j t(i) t I i, t / I i. for i = 1, 2,..., n, j = 1, 2,..., k 1; i = 1, j = ; i = n, j = k. k 1 t t (i) = t I ϕ (i) t (i) i, k (t) = k t(i) k t t (i+1) =1 t I t (i+1) t (i+1) i+1, (1) t / I i+1 I i. i = 1, 2,..., n 1 To construct the test space Y n, we define ψ (1) 1 t < h 1 = 2k, (11) otherwise. ψ (i) 1 ti 1 + 2j 1 j = 2k h i t < t i 1 + 2j+1 2k h i, otherwise. (12) i = 1,..., n, j = 1, 2,..., k 1, ψ (i) 1 k = ti 1 + 2k 1 2k h i t < t i + 1 otherwise. ψ (n) 1 ti 1 + k = 2k 1 2k h n t < 1, otherwise. 2k h i+1, (9) (13) (14) Furthermore, for any x n X n, we have k x n = x n (t (i) j )ϕ(i) j (t), t I i, i = 1,..., n, (15) j= and it is proved in [1] that with inear operator X n Y n ; k x n (t) = x n (t (i) j )ψ(i) j, (16) n n j= t I i, i = 1,..., n, X n, Y n } is a reguar pair, therefore it can be used in Petrov- Gaerkin system. III. ERROR ANALYI In this section, we investigate error anaysis for our method. By Theorem 1 (P1) im P nx x =. n Furthermore, we assume that the speed of convergence is specified by (P n I)x Ch m x. (17) We are interested to find u n (t) X n such that for t [, T ] u n (t) = P n f + P n K 1 u n (t) + P n K 2 u n (t). Let e(t) = u n (t) u(t) be the error of this method where, u n is a PG soution and u is the exact soution of the stochastic Voterra integra Eq.(1). Theorem 4. Assume that 1. f(t) <, 2.P (w Ω : u(ω, t) < ) = k i (s, t) < i = 1, 2, then sup t T (E( (u(t) u n (t)) ) 2 ) 1/2 = O(h m ). Proof: From u n (t) u(t) = P n f f +P n K 1 u n K 1 u+p n K 2 u n K 2 u, we get e = (P n I)f +(P n I)K 1 u n +K 1 e+(p n I)K 2 u n +K 2 e, so E( u n u 2 ) 3[E( (p n I)f) 2 )+ (18) E( (P n I)K 1 u n +K 1 e 2 )+E( (P n I)K 2 u n +K 2 e 2 )]. Furthermore, from (17) we have (I 1) E( (p n I)f) 2 ) E(C 2 h 2m f 2 ) = Ch 2m E( f 2 ). (I 2) E( (P n I)K 1 u n + K 1 e 2 ) 3[E( (P n I)K 1 u 2 ) + E( (P n I)K 1 (u n u) 2 )+ E( K 1 (u n u) 2 )]. (I 3) E( (P n I)K 1 u 2 ) C.h 2m E( u 2 ). (I 4) E( (P n I)K 1 (u n u) 2 ) < Ch 2m E( (u n u) 2 ). (I 5) E( K 1 (u n u) 2 ) C. E( (u n u) 2 )ds. Then E( (P n I)K 1 u n + K 1 e 2 ) C.h 2m E( u 2 ) + Ch 2m E( (u n u) 2 )+ C. E( (u n u) 2 )ds. For third term on the right-hand side of Eq. (18) we can write E( (P n I)K 2 u n + K 2 e 2 ) 3[E( E( E( (P n I)k 2 udb(s) 2 )+ (P n I)k 2 (u n u)db(s) 2 )+ k 2 (u n u)db(s) 2 )] (Advance onine pubication: 28 November 214)

4 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 3[ E( (P n I)k 2 u 2 )ds+ E( (P n I)k 2 (u n u) 2 )ds+ E( k 2 (u n u) 2 )ds] Ch 2m E( u 2 )+Ch 2m E( u n u 2 )+C Finay, E( (u u n ) 2 ) Ch 2m + C 2 E( u u n 2 )ds, and by using Gronwa s inequaity E( (u u n ) 2 ) Ch 2m. Proposition 1. uppose that X n, Y n } is a reguar pair that satisfies dimx n = dimy n and condition (H). Then, for any given u X, there exists a positive integer N such that, for a n N, the Petrov-Gaerkin equation, < (I K 1 u n K 2 u n, y n >=< f, y n >, y n Y n, has a unique soution u n X n that satisfies E( u n u 2 ) C inf xn X n u x n, n N. Proof: By Theorem 1 (P1), P n converges pointwise to the identity operator I in X. Hence, it foows from Theorem 4 that there exists an integer N > for which E( u n u 2 ) C P n u u ; n > N. Using this estimate and Theorem 1 (P3), the proof is competed. IV. NUMERICAL EXAMPLE In this section, we present four different exampes for performance of the Petrov-Gaerkin method. In these exampes the Error is defined as Exampe 1. [5] integra equation, u(t) = 1 + e = max x n (t (i) j ) x(t(i) j ). Consider the inear stochastic Voterra s 2 u(s)ds + su(s)db(s) s, t [, 1), (19) with the exact soution u(t) = e t3 6 + sdb(s), for t < 1. The numerica resuts are shown in Tabes I and II. In theses tabes, n is the number of iterations, x E is error mean, and s E is standard deviation of error. Exampe 2. [5] Consider the foowing inear stochastic Voterra integra equation, u(t) = cos(s)u(s)ds + sin(s)u(s)db(s) s, t [, 1), (2) with the exact soution u(t) = 1 12 e t sin(2t) 4 +sin(t)+ 8 + sin(s)db(s), for t < 1. For this exampe, the numerica resuts are shown in Tabes III and IV. Exampe 3. [1] Consider the foowing inear stochastic Voterra integra equation, u(t) = n(s+1)u(s)ds+ with the exact soution E( u n u 2 )ds. u(t) = 1 3 e 1 2 t+ 1 2 tn(t+1)+ 1 2 n(t+1)+ n(s + 1)u(s)dB(s), s, t [,.5), (21) n(s+1)db(s), for t <.5. The numerica resuts are shown in Tabe V and Tabe VI. As can be seen, by ess computation, we get good accuracy. Exampe 4. The Langevin mode (Pau Langevin, 198) has been quite successfuy used to study rotationa motion of moecues in gases, iquids, and soids. uppose that a sma macroscopic partice of mass m (such as a poen grain) is immersed in a iquid at a temperature T. In addition to any macroscopic motion that the partice may have, its veocity fuctuates due to the random coisions of the partice with the moecues of the iquid. For simpicity, we confine ourseves to one-dimensiona motion aong the x-axis. Then the equation of motion of the partice may be written in the form of stochastic differentia equation m d2 x dt 2 = αdx dt dv dx + F(t). (22) The first term on the right-hand side is due to the viscosity of the fuid and α is the friction constant. The second term, where V (x) is a potentia, represents the interaction of the partice with any externa forces, such as gravity. The fina term is the random force due to coisions with the moecues of the iquid. Ceary to compete the specification of the dynamics of the partice we need to give (i) the initia position and veocity of the partice, and (ii) the statistics of the random force F(t) (the noise term). Note that since F(t) is a random variabe, soving the Langevin equation wi give x(t) as a random variabe [17-18]. Eq. (21) may be equivaent written as m dv dt dx dt = v, = αv dv dx + F(t). (23) A particuary we-known case is when the Brownian partice is moving in the harmonic potentia V (x) = ax2 2 or doube-we potentia V (x) = (x 2 1) 2. Aso, F (t) = σdb(t), where, σ = 2kT τm is diffusion coefficient, and B(t) is Brownian motion. Here we appy this method for soving the foowing system x(t) = x (t) + v(s)ds, mv(t) = v (t) αv(s)ds dv dx (s)ds + σdb(s). (24) In order to conform the resuts above, initia vaue x() =, v() = 5 are chosen and parameters σ =.1, m = 1, α = 1. (Advance onine pubication: 28 November 214)

5 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 TABLE I MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 1 WITH k = TABLE II MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 1 WITH k = , TABLE III MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 2 WITH k = TABLE IV MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 2 WITH k = , TABLE V MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 3 WITH k = V. CONCLUION We introduced the Petrov-Gaerkin method for numerica soution of stochastic Voterra integra equations. The advantages of this method is that it can be constructed simpy, and by choosing the degree of test space ower than tria space s, we can achieve the same order of Gaerkin method convergence with ess computationa cost. This method specific properties ead to numerica resuts with considerabe accuracy and ess computationa effort. REFERENCE [1] M. Khodabin, K. Maeknejad, F. Hosseini hekarabi, Appication of Trianguar Functions to Numerica oution of tochastic Voterra Integra Equations, IAENG Internationa Journa of Appied Mathematics, 43:1, (213), pp (Advance onine pubication: 28 November 214)

6 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 TABLE VI MEAN, TANDARD DEVIATION AND CONFIDENCE INTERVAL FOR ERROR MEAN IN EXAMPLE 3 WITH k = Fig. 1. Curve1: The exact soution x(t) of exampe 4 with V (x) =, σ =. Curve2: The trajectory of the approximate soution x(t) of exampe 4 with V (x) =, σ =.1. Fig. 2. Curve1: The exact soution x(t) of exampe 4 with V (x) = x2, σ =. Curve2: The trajectory of the approximate soution x(t) of exampe 4 2 with V (x) = x2 2, σ =.1. Fig. 3. Curve1: The exact soution x(t) of exampe 4 with V (x) = (x 2 1) 2, σ =. Curve2: The trajectory of the approximate soution x(t) of exampe 4 with V (x) = (x 2 1) 2, σ =.1. [2] M. Khodabin, K. Maeknejad, M. Rostami, M. Nouri, Numerica soution of stochastic differentia equations by second order Runge- Kutta methods, Mathematica and Computer Modeing, 53, (211) pp [3] Fima C Kebaner, Intoduction To tochastic Cacuus With Appications, Monash University, Austraia, econd edition 25. [4] P.E. Koeden, E. Paten, Numerica oution of tochastic Differentia Equations, Appications of Mathematics, pringer-verag, Berin, [5] K. Maeknejad, M. Khodabin, M. Rostami, Numerica oution of tochastic Voterra Integra Equations By tochastic Operationa Matrix (Advance onine pubication: 28 November 214)

7 IAENG Internationa Journa of Appied Mathematics, 44:4, IJAM_44_4_2 Based on Bock Puse Functions, Mathematica and Computer Modeing, (211) pp [6] C.H. Wena, T.. Zhangc, Improved rectanguar method on stochastic Voterra equations, Journa of Computationa and Appied Mathematics 235 (211) pp [7] Kun Du, Guo Liu, and Guiding Gu, A Cass of Contro Variates for Pricing Asian Options under tochastic Voatiity Modes, IAENG Internationa Journa of Appied Mathematics, 43:2, (213), pp [8] Kun Du, Guo Liu, and Guiding Gu, Acceerating Monte Caro Method for Pricing Muti-asset Options under tochastic Voatiity Modes, IAENG Internationa Journa of Appied Mathematics, 44:2, (214), pp [9] Chares I. Nkeki, Continuous Time Mean-Variance Portfoio eection Probem with tochastic aary and trategic Consumption Panning for a Defined Contribution Pension cheme, IAENG Internationa Journa of Appied Mathematics, 44:2, (214), pp [1] Z. Chen, Y. Xu, The Petrov-Gaerkin and iterated Petrov-Gaerkin methods for second-kind integra equations, IAM J. Number. Ana. 35(1) (1998) pp [11] K. Maeknejad, M. Karami, Using the WPG method for soving integra equations of the second kind, Appied Mathematics and Computation 166 (25) pp [12] H. Kaneko, R.D. Noren, B. Novaprateep, Waveet appication to the Petrov-Gaerkin method for Hammerstein equations, Appied Numrica Mathematics 45 (23) pp [13] K. Maeknejad, M. Karami, M. Rabbani, Using the Petrov-Gaerkin eements for soving Hammerstein integra equations, Appied Mathematics and Computation 172 (26) pp [14] K. Maeknejad; M. Rabbani; N. Aghazadeh; M. Karami, A waveet Petrov-Gaerkin method for soving integro-differentia equations, Internationa Journa of Computer Mathematics, 86: 9,(29) pp [15] Tao Lin, Yanping Lin, Ming Rao, huhua Zhang, Petrov-Gaerkin methods for inear Voterra integro-differentia equations, IAM J. Numer. Ana. Vo. 38(3), (2) pp [16] B. Oksenda, tochastic Differentia Equations, An Introduction with Appications, Fifth Edition, pringer-verag, New York, [17] G.N. Mistein and M.V Tretyakov: Computing ergodic imits for Langevin equations, Physica D, 229 (27) pp [18] J. Carsson, K.. Moon, A. zepessy, R. Tempone, G. Zouraris, tochastic Differentia Equations: Modes and Numerics, February 2, 21. (Advance onine pubication: 28 November 214)