# Puzzling behavior of ItoProcess with Abs

Hi I am simulating a Fisher-Wright diffusion, it works

\[Mu] = ( # (1 - #)) &; \[Sigma] = (# (1 - #)/4) &;

FW = ItoProcess[\[DifferentialD]x[t] == \[Mu][t] \[DifferentialD]t +
Sqrt[\[Sigma][t]] \[DifferentialD]w[t], x[t], {x, 1/10}, t,
w \[Distributed] WienerProcess[]];

path = RandomFunction[FW //. ca, {0., 10, 0.01 }, 6];

lp = ListLinePlot[%]


but the negative values look suspicious (this could be investigated by Feller's boundary classification, but first thought is to add Abs under the square root, and hell breaks loose)

FW = ItoProcess[\[DifferentialD]x[t] == \[Mu][t] \[DifferentialD]t +
Sqrt[Abs[\[Sigma][t]]] \[DifferentialD]w[t], x[t], {x, 1/10}, t,
w \[Distributed] WienerProcess[]];

path = RandomFunction[FW //. ca, {0., 10, 0.01 }, 6];

lp = ListLinePlot[%]

• Expression for ca is needed. – Edmund Nov 8 '18 at 22:31
• Do you want \[Mu][t] and \[Sigma][t], or \[Mu][x[t]] and \[Sigma][x[t]]? – Chris K Nov 9 '18 at 0:53
• Thanks, Chris K. Indeed, I was solving wrong equation. The results with and without abs are very plausible and quick now :) – florin Nov 15 '18 at 15:51