FractionalBrownianMotionProcess
FractionalBrownianMotionProcess[μ,σ,h]
represents fractional Brownian motion process with drift μ, volatility σ, and Hurst index h.
FractionalBrownianMotionProcess[h]
represents fractional Brownian motion process with drift 0, volatility 1, and Hurst index h.
Details
- FractionalBrownianMotionProcess is also known as fractal Brownian motion or fractional Wiener process.
- FractionalBrownianMotionProcess is a continuous-time and continuous-state random process.
- FractionalBrownianMotionProcess is a Gaussian process with mean function
and covariance function ![sigma^2 (s^(2 h)+t^(2 h)-TemplateBox[{{t, -, s}}, Abs]^(2 h))/2](Files/FractionalBrownianMotionProcess.en/2.png "sigma^2 (s^(2 h)+t^(2 h)-TemplateBox[{{t, -, s}}, Abs]^(2 h))/2")
. It reduces to a WienerProcess for 
. - FractionalBrownianMotionProcess allows μ to be any real number, σ to be any positive real number, and h to be a real number between 0 and 1.
- FractionalBrownianMotionProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examples
open all close allBasic Examples (3)
Simulate a fractional Brownian motion process:
data = RandomFunction[FractionalBrownianMotionProcess[.3], {0, 1, 0.01}]ListLinePlot[data, Filling -> Axis]Mean[FractionalBrownianMotionProcess[μ, σ, h][t]]Variance[FractionalBrownianMotionProcess[μ, σ, h][t]]CovarianceFunction[FractionalBrownianMotionProcess[μ, σ, h], s, t]Plot3D[CovarianceFunction[FractionalBrownianMotionProcess[1 / 3], s, t], {s, 0, 5}, {t, 0, 5}, ColorFunction -> "Rainbow"]Scope (11)
Basic Uses (6)
Simulate an ensemble of paths:
data = RandomFunction[FractionalBrownianMotionProcess[.3], {0, 10, .1}, 4]ListLinePlot[data, Filling -> Axis]Simulate with arbitrary precision:
RandomFunction[FractionalBrownianMotionProcess[1 / 3], {0, 1, 1 / 4}, WorkingPrecision -> 20]["Path"]Compare paths for different Hurst indices:
sample[h_] := (SeedRandom[3];RandomFunction[FractionalBrownianMotionProcess[h], {0, 1, 0.01}])ListLinePlot[#, Filling -> Axis]& /@ {sample[.1], sample[.5], sample[.9]}SeedRandom[15];data = RandomFunction[FractionalBrownianMotionProcess[0.1], {0, 1, 0.1}];eproc = EstimatedProcess[data, FractionalBrownianMotionProcess[h]]CorrelationFunction[FractionalBrownianMotionProcess[μ, σ, h], s, t]Absolute correlation function:
AbsoluteCorrelationFunction[FractionalBrownianMotionProcess[μ, σ, h], s, t]Process Slice Properties (5)
Univariate SliceDistribution:
SliceDistribution[FractionalBrownianMotionProcess[μ, σ, h], t]First-order probability density function for the slice distribution:
Plot[Evaluate@Table[PDF[FractionalBrownianMotionProcess[1 / 3][t], x], {t, {1 / 2, 1, 2}}], {x, -4, 4}, Filling -> Axis, PlotLegends -> {"t = 1/2", "t = 1", "t = 2"}]PDF[FractionalBrownianMotionProcess[μ, σ, h][t], x]Multivariate slice distributions:
SliceDistribution[FractionalBrownianMotionProcess[1 / 4], {s, t}]dist = SliceDistribution[FractionalBrownianMotionProcess[.3], {1, 2, 3}]Mean[dist]PDF[FractionalBrownianMotionProcess[h][{s, t}], {x, y}]Compute the expectation of an expression:
Expectation[x[t] ^ 2 + 5x[t], xFractionalBrownianMotionProcess[μ, σ, h]]Calculate the probability of an event:
Probability[x[t] < 6, xFractionalBrownianMotionProcess[μ, σ, h]]Skewness and kurtosis are constant:
Skewness[FractionalBrownianMotionProcess[μ, σ, h][t]]Kurtosis[FractionalBrownianMotionProcess[μ, σ, h][t]]Moment[FractionalBrownianMotionProcess[μ, σ, h][t], r]CharacteristicFunction[FractionalBrownianMotionProcess[μ, σ, h][t], w]MomentGeneratingFunction[FractionalBrownianMotionProcess[μ, σ, h][t], w]CentralMoment and its generating function:
CentralMoment[FractionalBrownianMotionProcess[μ, σ, h][t], r]CentralMomentGeneratingFunction[FractionalBrownianMotionProcess[μ, σ, h][t], w]FactorialMoment[FractionalBrownianMotionProcess[μ, σ, h][t], 3]FactorialMomentGeneratingFunction[FractionalBrownianMotionProcess[μ, σ, h][t], w]Cumulant and its generating function:
Cumulant[FractionalBrownianMotionProcess[μ, σ, h][t], r]CumulantGeneratingFunction[FractionalBrownianMotionProcess[μ, σ, h][t], w]Generalizations & Extensions (1)
Properties & Relations (4)
FractionalBrownianMotionProcess is not weakly stationary:
WeakStationarity[FractionalBrownianMotionProcess[μ, σ, h]]Fractional Brownian motion does not have independent increments for 
:
proc = FractionalBrownianMotionProcess[μ, σ, h];Expectation[(x[t2] - x[t1])(x[t4] - x[t3]), xproc, Assumptions -> 0 < t1 < t2 < t3 < t4]//SimplifyCompare to the product of expectations:
Expectation[(x[t2] - x[t1]), xproc, Assumptions -> 0 < t1 < t2] * Expectation[(x[t4] - x[t3]), xproc, Assumptions -> 0 < t3 < t4]% === %%Conditional cumulative probability distribution:
Table[Plot[Evaluate@Probability[(x[3] <= Subscript[x, 2])(x[1] == Subscript[x, 1]), xFractionalBrownianMotionProcess[.4]], {Subscript[x, 2], -4, 6}, Filling -> Axis, PlotRange -> {0, 1}, PlotLabel -> StringJoin["SubscriptBox[x, 1] = ", ToString[Subscript[x, 1]]]], {Subscript[x, 1], {-1.5, -.3, 1.4, 3}}]Probability[(x[Subscript[t, 2]] ≤ Subscript[x, 2])(x[Subscript[t, 1]] == Subscript[x, 1]), xFractionalBrownianMotionProcess[h], Assumptions -> 0 < Subscript[t, 1] < Subscript[t, 2] && 0 ≤ Subscript[x, 1] ≤ Subscript[t, 1] && 0 ≤ Subscript[x, 2] ≤ Subscript[t, 2]]WienerProcess is a special case of fractional Brownian motion:
proc1 = FractionalBrownianMotionProcess[μ, σ, 1 / 2];
proc2 = WienerProcess[μ, σ];Mean[#[t]]& /@ {proc1, proc2}CovarianceFunction[proc1, s, t]//Simplify[#, s ≤ t || s ≥ t]&CovarianceFunction[proc2, s, t]//PiecewiseExpand//Simplify% - %%//FullSimplifyCompare univariate slice distributions:
PDF[proc1[t], x] === PDF[proc2[t], x]Neat Examples (3)
Simulate a fractional Brownian motion process in two dimensions:
SeedRandom[103];sample = RandomFunction[FractionalBrownianMotionProcess[.7], {0, 1, .001}, 2]["ValueList"];ListLinePlot[Transpose@sample, ColorFunction -> "FallColors"]Compare 3D behavior of fractional Brownian motion depending on the Hurst parameter:
plot[h_] := Graphics3D@Table[{ColorData["SolarColors"][RandomReal[]], Tube@BSplineCurve[Transpose[RandomFunction[FractionalBrownianMotionProcess[h], {0, 1, 0.01}, 3]["ValueList"]]]}, {6}]{plot[.2], plot[.8]}Simulate 500 paths from a fractional Brownian motion process:
SeedRandom[20];data = RandomFunction[FractionalBrownianMotionProcess[1 / 4], {0, 1, .01}, 500];Take a slice at 1 and visualize its distribution:
sd = data["SliceData", 1];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 -> 63]&[HistogramList[sd, {Range[Min[sd], Max[sd], (Max[sd] - Min[sd]) / 20]}]];Plot paths and histogram distribution of the slice distribution at 1:
ListLinePlot[data, ImageSize -> 400, PlotRange -> All,
AspectRatio -> 3 / 4, Epilog -> Inset[sliced, {1.01, 0}, {0, 10.3}], PlotStyle -> (cf /@ Rescale[sd]), BaseStyle -> Directive[Thin, Opacity[0.5]], PlotRangePadding -> {{0, .25}, {.5, .5}}]Text
Wolfram Research (2012), FractionalBrownianMotionProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.
CMS
Wolfram Language. 2012. "FractionalBrownianMotionProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html.
APA
Wolfram Language. (2012). FractionalBrownianMotionProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FractionalBrownianMotionProcess.html