Solving SDE: $\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t) $ in Mathematica

Question

I want to solve this differential equation $\frac{dy(t)}{dt}=(c+\sigma_w W(t))y(t)+\epsilon(t) $. For details see https://math.stackexchange.com/questions/1385633/solving-sde-fracdytdt-c-sigma-wwtyt-epsilont. My approach is using Mathematica. However, I have not been able to specify this SDE in Mathematica. As far as I understand, the ItoProcess function is appropriate. I have come this far:

proc = 
  ItoProcess[
     {\[DifferentialD]y[t] == c*y[t]\[DifferentialD]t + 
       e[t]*\[DifferentialD]t + σ*y[t]*w[t]}, 
     y[t], {y, y0}, t, Distributed[w, WienerProcess[]]] 

How do I now tell Mathematica that e[t] is a gaussian white noise process, c is a scalar and σ a non-negative scalar?

Answer 1

Revision to accommodate multiple random processes

One approach is to define e[t] in the question as e*\[DifferentialD]w[t], with e a scale factor.

Then, ItoProcess yields

proc = ItoProcess[{\[DifferentialD]y[t] == c*y[t]*\[DifferentialD]t + σ*y[t]
    *\[DifferentialD]w1[t] + e*\[DifferentialD]w2[t]}, y[t], {y, y0}, t, 
    {w1 \[Distributed] WienerProcess[], w2 \[Distributed] WienerProcess[]}]

or, equivalently,

proc = ItoProcess[{\[DifferentialD]y[t] == y[t]*\[DifferentialD]w1[t] + 
    \[DifferentialD]w2[t]}, y[t], {y, y0}, t, 
    {w1 \[Distributed] WienerProcess[c, σ], w2 \[Distributed] WienerProcess[0, e]}]

(* ItoProcess[{{c y[t]}, {{σ y[t], e}}, y[t]}, {{y}, {y0}}, {t, 0}] *)

A typical result can be obtained by specifying the four parameters.

SeedRandom[1236];
ans = RandomFunction[proc /. {σ -> 0.1, y0 -> 1, c -> -0.01, e -> 0.1}, {0, 100, .1}];
ListLinePlot[ans, Filling -> Axis, PlotRange -> All, AxesLabel -> {t, y}]

Note that SeedRandom is included only to obtain this particular curve and otherwise is unnecessary.