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WienerProcess[μ,σ]

表示 drift 为 μ、波动率为 σ 的 Wiener 过程.

WienerProcess[]

表示 drift 为 0、波动率为 1 的标准 Wiener 过程.

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范例  
基本范例  
范围  
基本用途  
过程切片属性  
应用  
属性和关系  
巧妙范例  
参见
相关指南
历史
按以下格式引用：

WienerProcess

WienerProcess[μ,σ]

表示 drift 为 μ、波动率为 σ 的 Wiener 过程.

WienerProcess[]

表示 drift 为 0、波动率为 1 的标准 Wiener 过程.

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范例

打开所有单元 关闭所有单元

基本范例  (3)

模拟 Wiener 过程：

Wolfram Language code: data = RandomFunction[WienerProcess[.3, .5], {0, 1, 0.01}]
Wolfram Language code: ListLinePlot[%, Filling -> Axis]

均值和方差函数：

Wolfram Language code: Mean[WienerProcess[μ, σ][t]]
Wolfram Language code: Variance[WienerProcess[μ, σ][t]]

协方差函数：

Wolfram Language code: CovarianceFunction[WienerProcess[μ, σ], s, t]
Wolfram Language code: CovarianceFunction[WienerProcess[], s, t]
Wolfram Language code: Plot3D[CovarianceFunction[WienerProcess[], s, t], {s, 0, 5}, {t, 0, 5}, ColorFunction -> "Rainbow"]

范围  (12)

基本用途  (7)

模拟路径集合：

Wolfram Language code: data = RandomFunction[WienerProcess[.3, .5], {0, 1, 0.01}, 4]
Wolfram Language code: ListLinePlot[data, Filling -> Axis]

使用任意经度模拟：

Wolfram Language code: RandomFunction[WienerProcess[1 / 10, 1 / 3], {0, 1, 1 / 4}, WorkingPrecision -> 20]["Path"]

比较不同偏移值的路径：

Wolfram Language code: sample[μ_] := (SeedRandom[14];RandomFunction[WienerProcess[μ, 1], {0, 1, .01}])
Wolfram Language code: ListStepPlot[sample[#], Filling -> Axis, PlotLabel -> StringJoin["μ = ", ToString[#]]]& /@ {-2, 0, 2}

比较不同波动值的路径：

Wolfram Language code: sample[σ_] := (SeedRandom[4];RandomFunction[WienerProcess[0, σ], {0, 1, .01}])
Wolfram Language code: ListStepPlot[sample[#], Filling -> Axis, PlotRange -> {-1, 3}, PlotLabel -> StringJoin["σ = ", ToString[#]]]& /@ {.5, 1, 1.4}

过程参数估计：

Wolfram Language code: data = RandomFunction[WienerProcess[.4, .7], {0, 100, 0.01}];
Wolfram Language code: EstimatedProcess[data, WienerProcess[μ, σ]]

相关性函数：

Wolfram Language code: CorrelationFunction[WienerProcess[μ, σ], s, t]

绝对相关函数：

Wolfram Language code: AbsoluteCorrelationFunction[WienerProcess[μ, σ], s, t]

过程切片属性  (5)

单变量 SliceDistribution:

Wolfram Language code: SliceDistribution[WienerProcess[μ, σ], t]

一阶概率密度函数：

Wolfram Language code: Plot[Evaluate@Table[PDF[WienerProcess[][t], x], {t, {1 / 2, 1, 2}}], {x, -4, 4}, Filling -> Axis, PlotLegends -> {"t = 1/2", "t = 1", "t = 2"}]
Wolfram Language code: PDF[WienerProcess[μ, σ][t], x]

与正态分布的密度函数比较：

Wolfram Language code: PDF[NormalDistribution[μ t, σ Sqrt[t]], x]
Wolfram Language code: Simplify[% - %%]

多变量切片分布：

Wolfram Language code: SliceDistribution[WienerProcess[μ, σ], {s, t}]
Wolfram Language code: WienerProcess[][{1, 2, 3}]//Mean

二阶 PDF：

Wolfram Language code: PDF[WienerProcess[μ, σ][{s, t}], {x, y}]

高阶 PDF：

Wolfram Language code: PDF[WienerProcess[μ, σ][{1, 3, 7}], {x, y, z}]

计算表达式的期望：

Wolfram Language code: Expectation[x[t] ^ 2, xWienerProcess[μ, σ]]

计算事件的概率：

Wolfram Language code: Probability[x[t] < 6, xWienerProcess[μ, σ]]

偏度和峰度是常量：

Wolfram Language code: Skewness[WienerProcess[μ, σ][t]]
Wolfram Language code: Kurtosis[WienerProcess[μ, σ][t]]

r 阶矩量：

Wolfram Language code: Moment[WienerProcess[μ, σ][t], r]

母函数：

Wolfram Language code: CharacteristicFunction[WienerProcess[μ, σ][t], w]
Wolfram Language code: MomentGeneratingFunction[WienerProcess[μ, σ][t], w]

CentralMoment 和它的母函数：

Wolfram Language code: CentralMoment[WienerProcess[μ, σ][t], r]
Wolfram Language code: CentralMomentGeneratingFunction[WienerProcess[μ, σ][t], w]

FactorialMoment 没有符号式阶数的解析式：

Wolfram Language code: FactorialMoment[WienerProcess[μ, σ][t], 3]
Wolfram Language code: FactorialMomentGeneratingFunction[WienerProcess[μ, σ][t], w]

Cumulant 和它的母函数：

Wolfram Language code: Cumulant[WienerProcess[μ, σ][t], r]
Wolfram Language code: CumulantGeneratingFunction[WienerProcess[μ, σ][t], w]

应用  (7)

定义二维 Bessel 过程：

Wolfram Language code: Bessel2 = TransformedProcess[Sqrt[x[t]^2 + y[t]^2], {xWienerProcess[], yWienerProcess[]}, t];
Wolfram Language code: data = RandomFunction[Bessel2, {0, 10, 0.01}, 3];
Wolfram Language code: ListLinePlot[data, PlotRange -> All]

均值和方差函数：

Wolfram Language code: {Mean[Bessel2[t]], Variance[Bessel2[t]]}

使用二次 WienerProcess 定义 martingale 过程：

Wolfram Language code: proc = TransformedProcess[w[t] ^ 2 - t, wWienerProcess[], t];
Wolfram Language code: Mean[proc[t]]
Wolfram Language code: CovarianceFunction[proc, s, t]
Wolfram Language code: ListLinePlot[RandomFunction[proc, {0, 2, .01}, 3]]

定义随机指数函数：

Wolfram Language code: 𝒫 = TransformedProcess[E^w[t] - (t/2), wWienerProcess[], t];
Wolfram Language code: SeedRandom[5]; data = RandomFunction[𝒫, {0, 10, 0.01}, 3];
Wolfram Language code: ListLinePlot[data, PlotRange -> All]

切片属性：

Wolfram Language code: {Mean[𝒫[t]], Variance[𝒫[t]]}

相应的微分方程是 u[t] u[t] w[t]:

Wolfram Language code: 𝒫2 = ItoProcess[ⅆu[t] == u[t]ⅆw[t], u[t], {u, 1}, t, wWienerProcess[]];
Wolfram Language code: SeedRandom[5]; data = RandomFunction[𝒫2, {0, 10, 0.01}, 3];
Wolfram Language code: ListLinePlot[data, PlotRange -> All]

切片属性：

Wolfram Language code: {Mean[𝒫2[t]], Variance[𝒫2[t]]}

直接使用 WienerProcess 来模拟 GeometricBrownianMotionProcess:

Wolfram Language code: μ = 1;σ = 1 / 3;x0 = 2;
Wolfram Language code: proc = WienerProcess[μ - σ^2 / 2, σ]; SeedRandom[3]; data = RandomFunction[proc, {0, 1, s = .01}];

对随机样本应用变换：

Wolfram Language code: sample = x0 Exp[data];

与相应的 GeometricBrownianMotionProcess 比较:

Wolfram Language code: SeedRandom[3]; gbm = RandomFunction[GeometricBrownianMotionProcess[μ, σ, x0], {s, 1 + s, s}];
Wolfram Language code: ListLinePlot[{sample, gbm}, PlotStyle -> {Thick, Dashed}]

直接使用 WienerProcess 来模拟 BrownianBridgeProcess:

Wolfram Language code: μ = 0;σ = 1 / 3;t0 = 0;t1 = 2;x0 = 0;x1 = 0;
Wolfram Language code: proc = WienerProcess[μ, σ]; SeedRandom[3]; data = RandomFunction[proc, {t0, t1, .01}];

对随机样本应用变换：

Wolfram Language code: sample = TimeSeriesMapThread[x0(x1 - x0)(#1 - t0) / (t1 - t0) + #2 - (#1 - t0) Last[data["Values"]] / (t1 - t0) - (t1 - #1)First[data["Values"]] / (t1 - t0)&, data];

与相应的 BrownianBridgeProcess 比较:

Wolfram Language code: SeedRandom[3]; data = RandomFunction[BrownianBridgeProcess[σ, {t0, x0}, {t1, x1}], {t0, t1, .01}];
Wolfram Language code: ListLinePlot[{sample, data}, PlotStyle -> {Thick, Dotted}]

使用 Wiener 过程来模拟随机微分方程 的解:

Wolfram Language code: P[t_, w_] := p E^σ w + (μ - (σ^2/2)) t

使用模拟来绘制解的图线：

Wolfram Language code: μ = 2;σ = .1;p = 1;
Wolfram Language code: sim = RandomFunction[WienerProcess[], {0, 3, 0.01}, 10 ^ 2]; sol = TimeSeriesMapThread[P[#1, #2]&, sim];
Wolfram Language code: paths = ListLinePlot[sol, PlotStyle -> Directive[Opacity[.1]]]

求模拟路径的均值函数：

Wolfram Language code: meanFunction = TimeSeriesThread[Mean, sol];
Wolfram Language code: Show[paths, ListLinePlot[meanFunction, PlotStyle -> Directive[Black, Thick]]]

与相应的平滑解比较：

Wolfram Language code: smoothsol = NDSolve[P'[t] == μ P[t] && P[0] == p, P, {t, 0, 3}]
Wolfram Language code: Plot[P[t] /. smoothsol, {t, 0, 3}]

求具有正偏移的 WienerProcess 浮动到 2 所花的时间分布：

Wolfram Language code: sample = Map[FirstCase[#, _ ? (Last[#] ≥ 2&)]&, RandomFunction[WienerProcess[1, 1], {0, 15, 0.01}, 10 ^ 4]["Paths"]];

删除空列表，并且提取时间：

Wolfram Language code: data0 = DeleteCases[sample, {}][[All, 1]];

选择正值：

Wolfram Language code: data = Cases[data0, _ ? Positive];

对数据拟合 InverseGaussianDistribution：

Wolfram Language code: edist = EstimatedDistribution[data, InverseGaussianDistribution[μ, λ]]

将直方图与概率密度函数比较：

Wolfram Language code: Show[ Histogram[data, 20, "PDF"], Plot[PDF[edist, x], {x, 0, 10}, PlotStyle -> Thick]]

属性和关系  (12)

Wiener 过程不是弱稳态：

Wolfram Language code: WeakStationarity[WienerProcess[μ, σ]]

Wiener 过程具有独立增量：

Wolfram Language code: proc = WienerProcess[μ, σ];
Wolfram Language code: Expectation[(x[t2] - x[t1])(x[t4] - x[t3]), xproc, Assumptions -> 0 < t1 < t2 < t3 < t4]//Simplify

与期望乘积比较:

Wolfram Language code: Expectation[(x[t2] - x[t1]), xproc, Assumptions -> 0 < t1 < t2] * Expectation[(x[t4] - x[t3]), xproc, Assumptions -> 0 < t3 < t4]//Simplify
Wolfram Language code: % === %%

条件累积分布函数：

Wolfram Language code: Table[Plot[Evaluate@Probability[(x[5] <= Subscript[x, 2])(x[2] == Subscript[x, 1]), xWienerProcess[]], {Subscript[x, 2], -4, 6}, Filling -> Axis, PlotLabel -> Row[{"SubscriptBox[x, 1] = ", Subscript[x, 1]}]], {Subscript[x, 1], {-1.5, -.4, 1.3, 3}}]
Wolfram Language code: Probability[(x[Subscript[t, 2]] ≤ Subscript[x, 2])(x[Subscript[t, 1]] == Subscript[x, 1]), xWienerProcess[], Assumptions -> 0 < Subscript[t, 1] < Subscript[t, 2] && 0 ≤ Subscript[x, 1] ≤ Subscript[t, 1] && 0 ≤ Subscript[x, 2] ≤ Subscript[t, 2]]//Simplify

Wiener 过程的相关函数与 RandomWalkProcess 的相同：

Wolfram Language code: CorrelationFunction[RandomWalkProcess[p], s, t]
Wolfram Language code: CorrelationFunction[WienerProcess[μ, σ], s, t]

Wiener 过程是一个特殊 ItoProcess:

Wolfram Language code: ItoProcess[WienerProcess[μ, σ]]

以及 StratonovichProcess:

Wolfram Language code: StratonovichProcess[WienerProcess[μ, σ]]

模拟标准 WienerProcess 的正侧所花的时间比率：

Wolfram Language code: data[n_] := RandomFunction[WienerProcess[], {0, 1, 1 / n}];
Wolfram Language code: ratio[n_] := Count[data[n]["Values"], _ ? Positive] / n

在极限下，比率服从 ArcSinDistribution:

Wolfram Language code: sample[n_, walks_] := Table[ratio[n], {i, walks}]
Wolfram Language code: Show[Histogram[sample[10 ^ 3, 10 ^ 3], 20, "PDF"], Plot[PDF[ArcSinDistribution[{0, 1}], x], {x, 0, 1}, PlotStyle -> Thick]]

求在时间0和1之间 WienerProcess 最后改变符号的分布：

Wolfram Language code: data = RandomFunction[WienerProcess[], {0, 1, .01}, 10 ^ 4];

计算符号差值，以求解符号变化：

Wolfram Language code: diff = Differences[Sign[data]];

提取路径并且求解每条路径最后的符号变化的次数：

Wolfram Language code: paths = diff["Paths"];
Wolfram Language code: sample = Map[First[FirstCase[Reverse[#], _ ? (Last[#] =!= 0&)]]&, paths];

在极限下，次数服从 ArcSinDistribution:

Wolfram Language code: Show[Histogram[sample, 20, "PDF"], Plot[PDF[ArcSinDistribution[{0, 1}], x], {x, 0, 1}, PlotStyle -> Thick]]

求对应于 WienerProcess 最大值的时间分布，直至时间 1:

Wolfram Language code: data = RandomFunction[WienerProcess[], {0, 1, .01}, n = 10 ^ 4]["Paths"];

从每条路径，提取对应于路径最大值的次数：

Wolfram Language code: sample = {}; Do[ path = data[[i]]; m = Max[path[[All, 2]]];AppendTo[sample, First@FirstCase[path, _ ? (Last[#] == m&)]], {i, n}]

在极限下，次数服从 ArcSinDistribution:

Wolfram Language code: Show[Histogram[sample, 20, "PDF"], Plot[PDF[ArcSinDistribution [{0, 1}], x], {x, 0, 1}, PlotStyle -> Thick]]

Wiener 过程在缩放下不变：

Wolfram Language code: proc = TransformedProcess[Sqrt[c]x[t / c], xWienerProcess[], t];
Wolfram Language code: Mean[proc[t]]

协方差函数：

Wolfram Language code: CovarianceFunction[proc, s, t]

与 WienerProcess 比较：

Wolfram Language code: Mean[WienerProcess[][t]]
Wolfram Language code: CovarianceFunction[WienerProcess[], s, t]

Wiener 过程在差值变换下不变：

Wolfram Language code: proc = TransformedProcess[x[t + u] - x[u], xWienerProcess[], t];
Wolfram Language code: Mean[proc[t]]

协方差函数：

Wolfram Language code: CovarianceFunction[proc, s, t]

与 WienerProcess 比较：

Wolfram Language code: Mean[WienerProcess[][t]]
Wolfram Language code: CovarianceFunction[WienerProcess[], s, t]

GeometricBrownianMotionProcess 是 WienerProcess 的变换：

Wolfram Language code: proc[μ_, σ_, a_] := TransformedProcess[a Exp[(μ - σ^2 / 2)t + σ w[t]], wWienerProcess[], t]
Wolfram Language code: ListLinePlot[RandomFunction[proc[0, 1, 3], {0, 1, .001}]]

切片分布的概率密度函数：

Wolfram Language code: PDF[proc[0, 1, 3][t], x]

与相应 GeometricBrownianMotionProcess 的切片分布比较：

Wolfram Language code: PDF[GeometricBrownianMotionProcess[0, 1, 3][t], x]
Wolfram Language code: % - %%//FullSimplify

Wiener 过程是 BrownianBridgeProcess 的变换：

Wolfram Language code: 𝒫 = TransformedProcess[(t + 1)x[t / (t + 1)], xBrownianBridgeProcess[], t];
Wolfram Language code: Mean[𝒫[t]]

协方差函数：

Wolfram Language code: CovarianceFunction[𝒫, s, t]
Wolfram Language code: sample = RandomFunction[𝒫, {0, 1, .01}];
Wolfram Language code: ListLinePlot[sample, Filling -> Axis]

巧妙范例  (3)

模拟二维 Wiener 过程：

Wolfram Language code: SeedRandom[103];sample = RandomFunction[WienerProcess[], {0, 1, .001}, 2]["ValueList"];
Wolfram Language code: ListLinePlot[Transpose@sample, ColorFunction -> "FallColors"]

模拟三维 Wiener 过程：

Wolfram Language code: proc = WienerProcess[]; SeedRandom[123]; sample = Table[RandomFunction[proc, {0, 1, 0.01}, 3]["ValueList"], {6}];
Wolfram Language code: Graphics3D@Table[{ColorData["SolarColors"][RandomReal[]], Tube@Line@sample[[i]]}, {i, 6}]

模拟来自 Wiener 过程的500条路径：

Wolfram Language code: data = RandomFunction[WienerProcess[], {0, 1, .01}, 500];

获取1处的切片并且将分布可视化：

Wolfram Language code: sd = data["SliceData", 1];
Wolfram Language code: cf = ColorData["Rainbow"]; sliced = BarChart[Last[#], Axes -> False, BarOrigin -> Left, AspectRatio -> 4, ChartStyle -> (cf /@ Rescale[MovingAverage[First[#], 2], {Min[sd], Max[sd]}, {0, 1}]), ImageSize -> 70]&[HistogramList[sd, {Range[Min[sd], Max[sd], (Max[sd] - Min[sd]) / 20]}]];

绘制1处的切片分布的路径和直方图分布：

Wolfram Language code: ListLinePlot[data, ImageSize -> 400, PlotRange -> All, AspectRatio -> 3 / 4, Epilog -> Inset[sliced, {1.01, 0}, {0, 10}], PlotStyle -> (cf /@ Rescale[sd]), BaseStyle -> Directive[Thin, Opacity[0.5]], PlotRangePadding -> {{0, .25}, {.5, .5}}]
Wolfram Research (2012)，WienerProcess，Wolfram 语言函数，https://reference.wolfram.com/language/ref/WienerProcess.html.

文本

Wolfram Research (2012)，WienerProcess，Wolfram 语言函数，https://reference.wolfram.com/language/ref/WienerProcess.html.

CMS

Wolfram 语言. 2012. "WienerProcess." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/WienerProcess.html.

APA

Wolfram 语言. (2012). WienerProcess. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/WienerProcess.html 年

BibTeX

@misc{reference.wolfram_2026_wienerprocess, author="Wolfram Research", title="{WienerProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/WienerProcess.html}", note=[Accessed: 02-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_wienerprocess, organization={Wolfram Research}, title={WienerProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/WienerProcess.html}, note=[Accessed: 02-August-2026]}