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Time Series and Stochastic Differential Equations
Accuracy of Approximation Schemes
Sampling from
ItoProcess
and
StratonovichProcess
generically uses strong approximation schemes of different orders of convergence.
Consider a vectorvalued Ito process with components
and
and convert it to a standard Ito process.
In[1]:=
X
pr = ItoProcess[ ItoProcess[{0, 1, {w[t], w[t]^4  6 w[t]^2 t + 3 t^2}}, w, t]]
Out[1]=
Sample from the standard Ito process using different approximation schemes, and measure the residuals with the exact function of the Wiener process.
In[2]:=
X
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]]
In[3]:=
X
{{residuals["EulerMaruyama"], residuals[Automatic]}, {residuals["Milstein"], residuals["StochasticRungeKutta"]}, {residuals["KloedenPlatenSchurz"], residuals["StochasticRungeKuttaScalarNoise"]}} // Grid
Out[3]=