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Accuracy of Approximation Schemes

Sampling from ItoProcess and StratonovichProcess are generically using strong approximation schemes of different orders of convergence.

    
Consider a vector‐valued Ito process with components and and convert it to a standard Ito process:
Wolfram Language code: pr = ItoProcess[ItoProcess[{0, 1, {w[t], w[t] ^ 4 - 6 w[t] ^ 2 t + 3 t ^ 2}}, w, t]]
Sample from the standard Ito process using different approximation schemes, and measure the residuals with the exact function of the Wiener process:
Wolfram Language code: residuals[mthd_] := Block[{td}, td = RandomFunction[pr, {0., 1, .01}, 1250, Method -> mthd]; Histogram[Function[{x, y}, y - (x ^ 4 - 6 x ^ 2 + 3)]@@@Part[td["States"], All, -1], ImageSize -> 250, PlotRange -> {{-1, 1}, Automatic}, PlotRangeClipping -> True, PlotLabel -> mthd, Frame -> True, Axes -> False]]
Wolfram Language code: {{residuals["EulerMaruyama"], residuals[Automatic]}, {residuals["Milstein"], residuals["StochasticRungeKutta"]}, {residuals["KloedenPlatenSchurz"], residuals["StochasticRungeKuttaScalarNoise"]}}//Grid