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Time Series and Stochastic Differential Equations
Simulation of Processes Driven by a Vector Noise Process
Define a scalar process driven by two independent Wiener noise processes.
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pr = ItoProcess[\[DifferentialD]x[ t] == x[t]/(\[Sqrt](1 + x[t]^2)) \[DifferentialD]t + Sin[x[t]] \[DifferentialD]w1[t] + Cos[x[t]] \[DifferentialD]w2[t], x[t], {x, 0}, t, {w1 \[Distributed] WienerProcess[], w2 \[Distributed] WienerProcess[]}]
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Simulate the process using the Milstein scheme.
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paths = RandomFunction[pr, {0, 4 Pi, 0.01}, 250, Method > "Milstein"]
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Visualize five sample trajectories.
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ListLinePlot[paths["Part", 1 ;; 5], ImageSize > 300]
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Show the sample trajectories with slice distributions at various times.
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plot1 = ListLinePlot[paths["Part", 1 ;; 12]]; plot2 = DistributionChart[ Transpose@(paths["SliceData", Range[0, 4 Pi, 1.1]]), ChartStyle > Opacity[0.7]]; Show[plot1, plot2, PlotRange > {{0, 11}, Automatic}, ImageSize > 300]
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