define one- or multi-dimensional Brownian motion process

はじめる前に
新機能一覧
Maple ワークシートの作成
Mapleワークシートを共有
Maple ウィンドウのカスタマイズ
Maple システムのカスタマイズ
コネクティビティ
Mathematics
数学
- テンソル解析
- パッケージ
- ファンクションアドバイザ
- ベキ級数
- ベクトル解析
- 不活性関数
- 代数
- 因数分解と方程式解法
- 基本数学
- 基本的な注意事項
- 変分法
- 変換
- 幾何
- 微分代数方程式
- 微分幾何学
- 微分方程式
- 微積分
- 数
- 数値計算
- 数学関数
- 数論
- 最適化
- 特殊関数
- 統計
- 線形代数
- 群論
- 評価
- 論理
- 過去のパッケージ
- 金融
  - はじめに
  - キャッシュフロー分析
  - パーソナルファイナンス
  - ラティス法
  - 日付計算
  - 短期料率モデル
  - 確率過程
  - 金利
  - 金融商品
  - 概要
- 離散数学
- piecewise examples
- グラフ電卓
- 区分関数
物理
プログラミング
グラフィックス
Student パッケージ
Science and Engineering
科学・エンジニアリング
アプリケーションと例題
リファレンス
システム
MapleSim
MapleSim ツールボックス
MapleSim ヘルプ
Tasks
Toolboxes
エラーメッセージガイド
マニュアル
数学アプリ

Finance[BrownianMotion] - define one- or multi-dimensional Brownian motion process

	Calling Sequence
	BrownianMotion(, mu, sigma, opts) BrownianMotion(, mu, sigma, t, opts) BrownianMotion(, Mu, Sigma)

Parameters

	-	real constant; initial value
mu	-	algebraic expression, operator or procedure; drift parameter
sigma	-	algebraic expression, operator, procedure or a one-dimensional stochastic process; volatility
t	-	time parameter
	-	Vector; initial value
Mu	-	Vector; drift parameter
Sigma	-	Matrix; covariance matrix
opts	-	(optional) equation(s) of the form option = value where option is scheme; specify options for the BrownianMotion command

Description

•	The BrownianMotion(, mu, sigma) and BrownianMotion(, mu, sigma, t) commands create a new one-dimensional Brownian motion process. This is a stochastic process , which is governed by the stochastic differential equation (SDE)

where

–	is the drift,

–	is the volatility,

and

–	is the standard Wiener process.

•	The parameter defines the initial value of the underlying stochastic process. It must be a real constant.

•

In the simplest case of a constant drift, mu is a real number (i.e. any expression of type ). Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. If mu is given as an algebraic expression, then the parameter must be passed to specify which variable in mu should be used as a time variable. Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

•	Similar to the drift, the volatility parameter can be constant or time dependent. In addition to this, the volatility can involve other (one-dimensional) stochastic variables. Note that stochastic drift is not supported.

•	The BrownianMotion(, Mu, Sigma) defines an -dimensional Brownian motion with drift Mu and covariance Sigma. This process is defined by the SDE

where

–	is a vector of size ,

–	is a -matrix ,

and

–	is the standard -dimensional Wiener process.

•	In this case the drift vector and the covariance matrix are time-independent. The drift parameter Mu must be given as a Vector and the covariance matrix Sigma must be a given as a symmetric matrix (see Matrix).

•

The scheme option specifies the discretization scheme used for simulation of this process. By default the standard Euler scheme is used. When scheme is set to unbiased the transition density will be used to simulate a value given . This scheme is appropriate in the case of a time-dependent drift and/or volatility.

Options

•	scheme = unbiased or Euler -- This option specifies which discretization scheme should be used for simulating this process.

Compatibility

•	The Finance[BrownianMotion] command was introduced in Maple 15.

•	For more information on Maple 15 changes, see Updates in Maple 15.

Examples

(1)

First consider the case of a one-dimensional Brownian motion with constant drift and volatility.

A := RTABLE(18446790986282351662, float[8], Array, rectangular, C_order, [], 2, 1 .. 10, 1 .. 101)

(2)

Compute the drift and diffusion for functions of .

(3)

(4)

(5)

(6)

Here is an example of a one-dimensional Brownian motion with time-dependent parameters given in algebraic form.

mu := PIECEWISE([0.2e-1, t < 1], [0.2e-1/t, otherwise])

(7)

sigma := PIECEWISE([.500-1.40*t+1.20*t^3, t < 1.], [-1.90+2.20*t+3.60*(-1.+t)^2-6.40*(-1.+t)^3, t < 1.5], [1.*t-6.*(t-1.5)^2+4.*(t-1.5)^3, otherwise])

(8)

Here is the same example but with drift and volatility given in the form of Maple procedures.

(9)

sigma := proc (t) options operator, arrow; piecewise(t < 1., .500-1.40*t+1.20*t^3, t < 1.5, -1.90+2.20*t+3.60*(-1.+t)^2-6.40*(-1.+t)^3, 1.*t-6.*(t-1.5)^2+4.*(t-1.5)^3) end proc

(10)

Here is an example of a two-dimensional Brownian motion.

X0 := Vector[column]([[0.], [0.]])

(11)

Mu := Vector[column]([[.2], [.3]])

(12)

Sigma := Matrix([[1.0, 0.5e-1], [.5, 1.0]])

(13)

(14)

RTABLE(18446790986282339494, float[8], Array, rectangular, C_order, [], 3, 1 .. 100000, 1 .. 2, 1 .. 2)

(15)

You can simulate values for any path function given as a Maple procedure.

[mean = .988749213777756, standarddeviation = 2.04881151111321, skewness = 0.386575235144395e-2, kurtosis = 2.99903802870409, minimum = -7.10227512307098, maximum = 9.60382569520030, cumulativeweight = 100000.]

(16)

Here are examples involving stochastic volatility.

(17)

(18)

Here is the same using different discretization schemes. For presentation purposes let us consider a Brownian motion with very low volatility and time-dependent drift. Compare the simulated results with the corresponding solution of an ordinary (non-stochastic) differential equation.

(19)

(20)

(21)

(22)

	See Also
	Finance[BlackScholesProcess], Finance[CEVProcess], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[GeometricBrownianMotion], Finance[ItoProcess], Finance[PathPlot], Finance[SamplePath], Finance[SampleValues], Finance[StochasticProcesses], Finance[WienerProcess]