Stochastic Di erential Equations and Integrating Factor

1 Int. J. Nonlinear Anal. Appl. 4 (2013) No. 2, 62-67 ISSN: 2008-6822 (electronic) Differential Equations and IntegratingFactorR. Rezaeyana, E. Baloui JamkhanehbaDepartment of Statistic and Mathematics, Nour Branch, Islamic Azad University, Nour, of Statistics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, aim of this paper is the analytical solutions the family of first-order nonlinear Stochastic differ-ential Equations . We define an Integrating Factor for the large class of special nonlinear stochasticdifferential Equations . With multiply both sides with the Integrating Factor , we introduce a deter-ministic differential equation. The results showed the accuracy of the present : Stochastic Differential Equation, Analytical Solution, Integrating MSC:Primary 60H10 Secondary Introduction and PreliminariesStochastic and deterministic differential Equations are fundamentals for the modeling in science, en-gineering and mathematical finance.

2 As the computational power increases, it becomes feasible touse more accurate differential equation models and solve more demanding problems. The model canbe Stochastic by two reasons: if calibration of data implies this, as in financial mathematical, or,if fundamental microscopic laws generate Stochastic behavior when coarse-grained, as in moleculardynamics for chemistry, material science and in statistical mechanics are usually modeled by adding a Stochastic term to the de-terministic differential equation. By doing this one obtains what is called Stochastic differentialequations (SDEs), and the term Stochastic called noise [1]. Then, a SDE is a differential equationin which one or more of the terms is a Stochastic process, and resulting in a solution which is itselfa Stochastic process. Every unwanted signal that adds to the information called noise.

3 Noise indynamical system is usually considered a nuisance. Noise has the most important role in the SDE[2]. Corresponding authorEmail E. BalouiJamkhaneh)Received:August 2012 Revised:March 2013 Stochastic Differential Equations .. 4 (2013) No. 2,62-6763 When a differential Equations model for some physical phenomenon is formulated preferably theexact solution can be obtained. However, even for ordinary differential Equations , this is generallynot possible [3]. Rezaeyan and et al ([4]-[6]) discussed the application of the SDEs for the moldingelectrical circuits. In this paper, we will present an application of the Integrating Factor for analyticalsolution to the family of attention, in the next section, we describe Stochastic calculus and SDEs. In Section 3, we definethe Integrating Factor for the class of SDEs. Finally, the paper ends with an example and a Stochastic CalculusIn many physical applications one has to deal with random quantities that depend on a param-eter.

4 This phenomenon is termed Brownian motion. Each coordinate of the Brownian particle is arandom variable that depends on a parameter. A Stochastic processXt( ), is a family of randomvariables{Xt( ) :t T, }depending upon the parameter t and defined on the probabilityspace ( ,=,P) (=,is a - algebra of subsets of and P is probability measure defined on allelements of=)[3]. Stochastic calculus is a branch of mathematics that operates on Stochastic processes. It allows aconsistent theory of integration to be defined for integrals of Stochastic processes with respect tostochastic processes. The best-known Stochastic process to which Stochastic calculus is applied theWiener main part of Stochastic calculus is the Ito calculus and Stratonovich. Ito calculus extends themethods of calculus to Stochastic processes such as Brownian motion.

5 We go back to the definitionof an integral: T0f(t)dt= limn + n j=1f( j)(tj+1 tj),( )where jis in the interval [tj,tj+1]. More generally have Riemann-Stieltjes integral: T0f(t)dt= limn + n j=1f( j)(g(tj+1) g(tj)).( )For a smooth measureg(t), limit converges to a unique value regardless where j, taken in interval[tj,tj+1].The Ito and Stratonovich calculus follows the same rules as for the regular Riemann-Stieltjes calcu-lus. If our choose is lower end point, of the partition [tj,tj+1], we have the case Ito integral, but ifwe choose midpointtj+1+tj2, we got Stratonovich 0 t1 t2 tn=Tbe a partition of the interval [0,T] and = max(ti ti 1).TheIto integral T0h(t,Xt))dWtis defined as the limit in the quadratic mean T0h(t,Xt)dWt= lim n 0n i=1h( i 1,X i 1)(Wti Wti 1).( )64 Rezaeyan and BalouiIf the integrand h is jointly measurable and T0E(|h(s,Xs)|2)ds < ,( )the Stochastic integral in ( ) is defined as the limit in probability.

6 The Stratonovich integral isdefined by T0h(t,Xt)odWt= lim n 0n i=1h( i 1,X i 1+X i2)(Wti Wti 1),( )(where the symbol o, is employed).In addition to the conditions on the existence of the Ito integral, it is required for the existence ofthe Stratonovich integral in ( ) that theh(t,Xt) function be continuous in t and have continuouspartial derivatives (see [1]-[5]). Moreover T0h(t,Xt)odWt= T0h(t,Xt)dWt+12 T0g(t,Xt) h x(t,Xt)dt( )or, equivalently[1]h(t,Xt)odWt=h(t,Xt)dWt+12 g(t,Xt) h x(t,Xt)dt.( )Consider a SDE,dXt=f(t,Xt)dt+g(t,Xt)dWt,( )wherefis an n-vector valued function,gis ann pmatrix valued function,Wtis an p-dimensionalBrownian motion process or Wiener process, and the solutionXtof the Stochastic differential equa-tion ( ), is meant a processXtfor allt, in some interval [0,T].We also assume that the distribution ofX0is known and independent ofWt.

7 There is an explicitseveral-dimensional formula which expresses the Stratonovich interpretation of ( )dXt= f(t,Xt)dt+g(t,Xt)odWt,( )where: f(t,Xt) =f(t,Xt) +12p j=1n k=1 gij xjgkj,( )(see Oksendal (2000)).(I) t0 WsdWs=12[W2t W20 t],( ) Stochastic Differential Equations .. 4 (2013) No. 2,62-6765while(S) t0 WsodWs=12[W2t W20],( )Ito integral and Stratonovich integral have applications in the SDEs. In this paper, we consider SDEof Ito kind . A SDE is given byX t=f(t,Xt) +g(t,Xt) t,X0=x0,t 0,( )wherefis the deterministic part,g tis the Stochastic part, and tdenotes a generalized stochasticprocess [1]-[3].An example of generalized Stochastic processes is white noise. For a generalized Stochastic process,derivatives of any order can be defined. Suppose thatWtis a generalized version of a Wiener processwhich is used to model the motion of stock Wiener process is a time continuous process with the propertyWt N(0,t), (0 t T), usuallyit is differentiable almost nowhere.

8 White noise tis defined as t=dWtdt[1].If we replace tdtbydWtin equation (2. 13), an Ito SDE can be rewritten as:dXt=f(t,Xt)dt+g(t,Xt)dWt,( )wherefandgare drift and diffusion term, respectively, andXtis a solution which we try to findbased on the Integrating Factor [3].3. Main ResultsConsider the following nonlinear SDE,dXt=f(t,Xt)dt+C(t)XtdWt, X0=x,( )wheref:R R RandC:R Rare given continuous (deterministic) Ito processes in R. Then:d(XtYt) =XtdYt+YtdXt+dXtdYt.( )Proof .LetXtandYtbe Ito processes given bydXt=g0(t)dt+g1(t)dWt,( )dYt=h0(t)dt+h1(t)dWt.( )By Ito Lemma ( See Theorem 4. 1. 2 of [1]), we haved(XtYt) =XtdYt+YtdXt+g1(t)h1(t),( )such 0, ,( )66 Rezaeyan and BalouithendXtdYt=g0(t)h0(t)(dt)2+g1(t)h0 (t)dtdWt( )+g0(t)h1(t)dWtdt+g1(t)h1(t)(dWt)2=g1(t) h1(t)dt,sod(XtYt) =XtdYt+YtdXt+dXtdYt.( ) Remark Ito processes in R, then deduce the following general integration byparts formula t0 XsdYs=XtYt X0Y0 t0 YsdXs t0dXsdYs.

9 ( )Lemma exp( t0C(s)dWs+12 t0C2(s)ds),( )then,d(FtXt) =Ftf(t,Xt)dt.( )Proof .SupposeYt( ) =Ft( )Xt( ). ThenXt=F 1tYt,( )anddYt=d(FtXt) =Ftf(t,Xt)dt=Ftf(t,F 1tYt)dt.( )From (3. 13) we obtaindYtdt=Ftf(t,F 1tYt).( )Note that this is just a deterministic differential equation in the functiont Yt( ) for each . Example + XtdBt,X0=x >0,( )where is constant. Wheref(t,Xt) =1 Xtandg(t,Xt) = Xt, equation ( ) is a SDE to form( ). SoFt= exp( t0C(s)dWs+12 t0C2(s)ds) = exp( Bt+12 2t),( ) Stochastic Differential Equations .. 4 (2013) No. 2,62-6767andXt=F 1tYt= exp( Bt+12 2t)Yt,( )dYtdt=Ftf(t,F 1tYt) = exp( Bt+12 2t)1exp( Bt 12 2t)Yt,( )YtdYt= exp(2( Bt+12 2t))dt.( )Then,Xt= exp( Bt 12 2t)[x2+ 2 t0exp(2( Bt+12 2t))dt]12.( )4. ConclusionWe introduced first-order nonlinear SDEs and it s applications. Also, we defined an integratingfactor for the large class of special nonlinear SDEs.

10 By multiplying both sides of the integratingfactor, we obtain a deterministic differential equation. By an example, we show that the accuracy ofthis [1] B. Oksendal, Stochastic diferential Equations : An introduction with applications, Springer-Verlage, 2000.[2] P. E. Kloeden and E. Platen,Numerical solution of Stochastic differential Equations , Springer, Berlin, 1995.[3] R. E. Mortensen,Mathematical problems of modeling Stochastic nonlinear dynamic systems, Int. J. Stat. Physics,Vol68, No2 (1999) 1079-1095.[4] R. Rezaeyan and R. Farnoosh,On The support of an Ito diffusion process in the SDEs, The 23th InternationalConference of Janggion mathematical Society, February 8-10, (2010).[5] R. Rezaeyan, R. Farnoosh and E. Baloui Jamkhaneh,Application of the Kalman-Bucy filter in the StochasticDifferential Equation for the modeling of RL Circuit, Int.