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研究生:許冰瑜
研究生(外文):Xu, Bing-Yu
論文名稱:使用Maple的隨機微分方程之弱格式
論文名稱(外文):Weak schemes for stochastic differential equations using Maple
指導教授:黃炎坤
指導教授(外文):Huang, Yan-Kun
學位類別:碩士
校院名稱:國立成功大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:1998
畢業學年度:86
語文別:英文
中文關鍵詞:隨機微分方程式弱格式隨機微分方程應用數學數學
外文關鍵詞:Stochastic Differential EquationsWeak SchemesMapleItoAPPLIED-MATHEMATICSMATHEMATICS
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In this paper we write a Maple package for explicit and
implicit weakschemes of d-dimensional Ito diffusion processes
generated by an 1-parameter d-dimensional Ito stochastic
differential equation ( SDE ) ofthe form dX(t)=
a(t,X(t))dt+b(t,X(t))dW(t) ( 1.1 ) The key
role played by truncated stochastic Taylor expansions in
thesystematic construction of higher order numerical schemes for
the 1-parameter Ito SDEs was demonatrated in Platen [10],
Milstein [6,7,8],Talay [12]. We shall use the convergence result
proved in Platen [10] inwhich he constructed numerical schemes
for 1-parameter SDEs. An 1-parameter Ito stochastic
differential equation ( SDE ) of the form dX(t)= a(t,X(
t))dt+sum(b(j,t,X(t))dW(j,t),j=1..m) ( 1.2 )is
interpreted as the 1-parameter stochastic integral equation X(
t)=X(0)+Integral(a(s,X(s))ds,s=t0..t)+sum(Integral(b(j,s,X(s))
dW(j,s), s=t0..t),j=1..m).
( 1.3 )Conditions on the coefficients a and b's that assure the
existence anduniqueness of a solution of (1.2) were studied by
Gikhman & Skorokhod [1],Ikeda & Watanabe [2] and Protter [11].
It is often necessary to use implicit schemes to simulate the
solutions ofstochastic differential equations. These schemes
usually have a wide range ofstep sizes suitable for the
approximation of dynamical behavior, in particularwith vastly
different time scales, without the excessive accumulation of
unavoidable initial and roundoff errors. In implementing an
implicit scheme weneed to solve an additional algebraic equation
at each time step, which canusually be done without too much
additional computational effort by usingMaple. Explicit and
implicit weak approximations were studied by Talay [12],Milstein
[7,8],Platen [10],Mikulevicius and Platen [5],Pardoux & Talay [9
],Klauder & Petersen [3],Liske and Platen [4]. Results and
notations on the stochastic Taylor expansions andapproximations
as developed in Platen [10] that are needed in the sequelare
summarized in chapter 2. The purpose of this paper is to write a
package for explicit and implicit weak schemes of d-dimensional
Itodiffusion processes to compute their orders by making use of
Maple VRelease 3 in the workstation HP-9000/735. The Maple
package is presentedin chapter 3.

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