Numerical solution of second-order stochastic di erential ...

1 Journal ofLinear and Topological , ,2013,229-241 Numerical solution of second - order stochastic di erentialequations with Gaussian random parametersR. Farnoosha, H. Rezazadeha; , A. Sobhaniaand D. EbrahimibaghabaSchool of Mathematics, Iran University of Science and Technology, 16844, Tehran, of Mathematics, Center Branch, Islamic Azad university, Tehran, 16 September 2013; revised 4 December 2013; accepted 31 December this paper, we present the Numerical solution of ordinary di erential equations (or SDEs), from each order especially second - order with time-varying and Gaussian randomcoe cients. We indicate a complete analysis for second - order equations in special case ofscalar linear second - order equations (damped harmonic oscillators with additive or multi-plicative noises).

2 Making stochastic di erential equations system from this equation, it couldbe approximated or solved numerically by di erent Numerical methods. In the case of linearstochastic di erential equations system by Computing fundamental matrix of this system, itcould be calculated based on the exact solution of this system. Finally, this stochastic equa-tion is solved by numerically method like Euler-Maruyama and Milstein. Also its Asymptoticstability and statistical concepts like expectation and variance of solutions are 2013 IAUCTB. All rights : stochastic di erential equation, linear equations system, Gaussian randomvariables, damped harmonic oscillators with noise, multiplicative AMS Subject Classi cation: 65C30, 65U05, 60H10, IntroductionOne of the most important and applicable concepts in various sciences is Newtons sec-ond law of motion which relates force and acceleration together.

3 Therefore, second -orderdi erential equations are most common in various scienti c applications, As we can seein some articles, the famous and well-known di erential equations of second - order suchas OrnsteinUhlenbeck process and Random harmonic oscillator have been solved by dif-ferent methods like as Monte Carlo and other Numerical methods [1, 6, 14]. The study Corresponding address: ( H. Rezazadeh).Print ISSN: 2252-0201c 2013 IAUCTB. All rights ISSN: 2345-5934 Rezazadeh et al. / J. Linear. Topological. (04) (2013) Numerical methods of second - order ordinary di erential equations is one of the mostapplicable branches in Numerical analysis issues[9]. With attention to stochastic essenceof almost all physical phenomena, the most interesting advances in recent years are thedevelopment and extension of previous methods to stochastic systems [12].

4 Numericalmethods have to replace discrete-time dynamics in place of continuous-time with generat-ing values at timest0; t1; :::; tn. In this process,t0andtn, are xed points. Some criterionsfor speci cation a good Numerical method are order of convergency, comparison with ex-act solution and produced error of a used method and determining its discrete-timedynamics which has an appropriate stationary density as close as possible to that ofthe corresponding continuous-time system that Wiener chaos expansion (WCE), is oneof these methods[4],[17]. The di erential equation which describe second - order systemscontains some parameters known as damping. The stationary density is completely in-dependent of damping, but dynamical quantities, and the specially the nal numericalalgorithms, are strongly dependent on it.

5 With some change of variables, the systembecomes rst order this paper, we intend to extend our previous work about solving second - order linearstochastic di erential equation [16], to solve second - order stochastic di erential equationBy rst- order stochastic linear system equation which has been mentioned in variousbooks like [10], [11], [13] and [15].That is, we considerXt2 RnandXt2L2n(0; T), as a stochastic process and uniquesolution of following :{dXt= (A(t):Xt+B(t))dt+ (C(t):Xt+D(t))dWX(0) =X0:(0 t T)(1)such thatW(:), is a m-dimensional Brownian motion. Afterwards, By constructionfundamental or Hamiltonian matrix for this system, we solve it with Numerical methodsas EulerMaruyama( ), Milstein and Rung-Kutta method. nally, in the end of ourconclusions, we investigate some statistical properties like expectation and variance bynumerical simulation like predictor-corrector [8], and will have a comparison betweenexact solution and its solution and nd least square error for this paper is organized as follow.}

6 In section 2, we consider the stochastic linear equa-tion system of stochastic linear second - order equation which has been mentioned invarious books like [13] and [11] and latest articles[3] and [5]. Afterwards, by construc-tion fundamental matrix for this system we solve it numerically by Rung-Kutta this section we consider second order examples solve them by this mentionedmethod and stochastic Numerical simulation like predictor-corrector EulerMaruyama andMilstein'method [8]. Also, we take a discussion about expectation, variance of these equa-tions solutions . In nal section, the conclusion of this paper has been said Making stochastic Di erential Equation SystemLet the general form of a second - order stochastic di erential equation is de ned suchthis equation:{ Xt= (f(Xt; t) +^f(Xt; t) (t))_Xt+ (g(Xt; t) + ^g(Xt; t) (t))Xt+h(Xt; t);Xt(0) =X0;_X0=X1:(2)H.}

7 Rezazadeh et al. / J. Linear. Topological. (04) (2013) that we de neh(Xt; t) = (h(Xt; t) +^h(Xt; t) (t)) andXtis a 1-dimensionalstochastic process de ned on closed time interval [0; T] and the real functionsf(Xt; t); g(Xt; t); h(Xt; t) and also^f(Xt; t);^g(Xt; t);^h(Xt; t), are stochastic integrablefunctions. In special case, It shall be considered equation of the following form whichhas been discussed in[3]:{ Xt=f(Xt) ^f2(Xt; t)_Xt+"^f(Xt; t) (t);Xt(0) =X0;_X0=X1:(3)The damping parameter and The amplitude of the random forcing are denoted by and"respectively such that related to the temperatureTand damping coe cient by the uctuation-dissipation relation[7], We have"2= 2 KT. (t), is de ned as White noise that has derivative relation with Wiener process_Wt=dWdt= (t);De nition them-dimensional vectorW(t) of real stochastic processesWi(t);(i= 1;2; :::; m).}

8 It is named Wiener process or Brownian motion if;(a)W(0) = 0 (almost sure with probability one),(b)W(t) W(s) is normal distribution (i:e:W N(0; t s));for all 0 s t(c) The random variablesW(t1); W(t2) W(t1); :::; W(tn) W(tn 1), for times0 t1 t2 ::: tn, are independent this de nition, about expectation and variance related toWi(t), it could beconcluded thatE[Wi(t)] = 0; E[W2i(t)] =t f or i= 1;2; :::; m:(4)The expectation integral for these functions has these propertiesE[ (t); (s)] = 0(s t); E[W(s); W(t)] =min(s; t):We can write (2), as a pair of rst- order equations forXtandVt, the position and velocitystochastic processes:{dXt=Vtdt;dVt=(f(Xt; t)Vt+g(Xt; t)Xt+h(Xt; t))dt+(^f(Xt; t)Vt+ ^g(Xt; t)Xt+^h(Xt; t))dW(t):(5)With initial conditionsXt(0) =X0; V0=X1:Now the second - order SDE(2), could bewritten again as a rst- order system by matrix (XtVt)=( (Xt; t)(XtVt)+(0h(Xt; t)))dt+( (Xt; t)(XtVt)+(0^h(Xt; t)))dWt:(6)232H.}

9 Rezazadeh et al. / J. Linear. Topological. (04) (2013) we have (Xt; t) =(01g(Xt; t)f(Xt; t)); (Xt; t) =(00^g(Xt; t)^f(Xt; t)):If (Xt; t) = (t) and (Xt; t) = (t), we reach to a stochastic linear equation system:d(XtVt)=( (t)(XtVt)+(0h(t)))dt+( (t)(XtVt)+(0^h(t)))dWt:(7)If in special case, we haved(XtVt)= (t)(XtVt)dt+(0^h(t))dWt;(8)This special case is a better model ofBrownian movementwhich is provided by theOrn-steinUhlenbeckequation. For solving this matrix form (8), it could be done by di erentnumerical methods like Euler, Milstein or even Rung-kutta that in [16], it was done thatis named linear second - order SDEs in narrow sense case. As another special cases, if wehaved(XtVt)= (t)(XtVt)dt+ (t)(XtVt)dWt:(9)This stochastic di erential equation system is well-known toGeometric Brownian MotionBut as we will observe later, this system just in special case has an exact solution likeBlack-Scholes model in 1-dimentional we perform the same work in extended case on interval[0; T], to solve the systemof equations (27) explicitly and numerically.

10 Suppose time interval has been separated toequal subintervals [ti; ti+1];(i= 1;2; :::; n) and we de ne the following one step recursiveequation even for general case(6).(Xi+1Vi+1)=(10)(XiVi)+( (Xi; ti)(XiVi)+(0h(Xi; ti))) ti+( (Xi; ti)(XiVi)+(0^h(Xi; ti))) Wi:(11)such that ti=ti+1 ti=Tn;and according to Wiener process de nition: W i=Wi+1 Wi N(0; ti):With the next theorem, we want to express the existence and uniqueness of second -orderstochastic di erential equations based on some thatA(t):Xt+B(t) :Rn [0; T]!RnandC(t):Xt+D(t) :Rn [0; T]!Mm nare continuous and satisfy the following properties:H. Rezazadeh et al. / J. Linear. Topological. (04) (2013) (1) A(t):(X(t) ^Xt(t)) L: Xt(t) ^Xt(t) ;(12) C(t):(Xt(t) ^Xt(t) L: Xt(t) ^Xt(t) :for all 0 t T; Xt;^Xt2Rn(2) A(t):Xt+B(t) Lj1 + Xt j; C(t):Xt+D(t) Lj1 + Xt j:for all 0 t T; Xt2Rn;for some suitableL2R:LetX02 Rnis a random variable such that:E[X20]<1so, There exist a uniquesolutionXt2L2n(0; T) of following :{dXt= (A(t):Xt+B(t))dt+ (C(t):Xt+D(t))dWX(0) =X0:(0 t T)(13)whereW(:), is a m-dimensional Brownian motion[6].)}