Plotting the solution of a vector stochastic differential equation

Question

I have a vector stochastic differential equation,

$$\mathrm dq = p\,\mathrm dt\qquad q(0)=0$$ $$\mathrm dp = (-q -p)\mathrm dt+\mathrm dW\qquad p(0)=10$$

This can be entered to give me the process describing either p or q, using

proc = ItoProcess[
    {{p[t], -p[t]-q[t]}, {{0}, {1}}, XXX[t]},
    {{q, p}, {0, 10}},
    {t,0}
]

where XXX is either q or p. The solution can be plotted using the usual method (here for p) of

ListLinePlot@RandomFunction[proc, {0, 10, .02}, 10]

However, the situation is more difficult if I want to extract both q and p, as for each simulation it will give me a list of the type (obtained using Normal, and by setting XXX[t] to {q[t], p[t]} in the ItoProcess)

{{{t0, {q[t0], p[t0]}}, {t1, {q[t1], p[t1]}}, ...}

i.e. the times aren't properly distributed over the p/q, and as a consequence I'm having a hard time finding a good way of getting this into a plottable form.

So the questions are:

1. Is there a nice way of getting all components out of a vector stochastic differential equation to plot them alongside each other?
2. If there's none, what's the right hacky approach? Fiddling with Transpose and Flatten?

Answer 1

You could use "PathComponents" property of TemporalData to split the vector-valued temporal data into the list of TemporalData objects and plot those:

proc = ItoProcess[{{p[t], -p[t] - q[t]}, {{0}, {1}}}, {{q, p}, {0, 
     10}}, {t, 0}];

td = RandomFunction[proc, {0., 10., 0.02}, 10];

td["PathComponents"]

Answer 2

Please confirm if this is what you were looking for

proc = ItoProcess[{{p[t], -p[t] - q[t]}, {{0}, {1}}, {q[t], 
    p[t]}}, {{q, p}, {0, 10}}, {t, 0}];

data = RandomFunction[proc, {0, 10, 0.02}, 10];

You could do

Plot[Through@data["PathFunction", All][t], {t, 0, 10}, Evaluated -> True]

Answer 3

Another way to visualize this is in the parametric phase space:

ListLinePlot[td[[2, 1]], Frame -> True, AspectRatio -> 1, PlotRange -> All]

---------- Comment reponse ----------

We can check the structure of underlying expression with InputForm and then it is straightforward to use Part to extract the sub-expressions: