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Simulation of Processes Driven by Vector Noise Process

    
Define a scalar process driven by two independent Wiener noise processes:
Wolfram Language code: pr = ItoProcess[ⅆx[t] == -(x[t]/Sqrt[1 + x[t] ^ 2])ⅆt + Sin[x[t]]ⅆw1[t] + Cos[x[t]]ⅆw2[t], x[t], {x, 0}, t, {w1WienerProcess[], w2WienerProcess[]}]
Simulate the process using a Milstein scheme:
Wolfram Language code: paths = RandomFunction[pr, {0, 4Pi, 0.05}, 250, Method -> "Milstein"]
Visualize five sample trajectories:
Wolfram Language code: ListLinePlot[paths["Part", 1 ;; 5]]
Visualize a slice distribution on the simulated data and overlay it over process trajectories:
Wolfram Language code: plot1 = ListLinePlot[paths["Part", 1 ;; 12]]; plot2 = DistributionChart[Transpose@(paths["SliceData", Range[0, 4Pi, 1.1]]), ChartStyle -> Opacity[0.7]]; Show[plot1, plot2, PlotRange -> {{0, 11}, Automatic}]
Wolfram Language code: Show[plot1, plot2, PlotRange -> {{0, 11}, Automatic}, ImageSize -> 400, PlotLabel -> Pane[Row[{HoldForm[ⅆx[t]], "==", HoldForm[-(x[t]/Sqrt[1 + x[t] ^ 2])ⅆt + Sin[x[t]]ⅆSubscript[w, 1][t] + Cos[x[t]]ⅆSubscript[w, 2][t]], ", ", HoldForm[x[0]], "==", 0}], 400, Alignment -> Center]]