Ito Process with Piecewise

Question

I have the following ItoProcess :

V0=1.5
a = 0.1

i0 = ItoProcess[\[DifferentialD]V[t] == a*Piecewise[{{V[t] *\[DifferentialD]t, 
   V[t] < V0}, {a*V[t]^(2/3)*V0^(1/3)* \[DifferentialD]t + 
    V0^(1/3)*V[t]^(2/3)/3* \[DifferentialD]w[t], V[t] > V0}}], 
   V[t], {V, 1}, t, w \[Distributed] WienerProcess[]]

RandomFunction[i0, {0., 5., 0.01}]

But I'm getting the error message :

RandomFunction::unsproc: The specification ItoProcess[[DifferentialD]V[t]==0.1 ([Piecewise] [DifferentialD]t V[t] V[t]<1 0.1 [DifferentialD]<<1>> Power[<<2>>]+1/3 [DifferentialD]<<1>> Power[<<2>>] V[t]>=1 0 True ),V[t],{V,1},t,w[Distributed]WienerProcess[0,1]] is not a random process recognized by the system.

Is it because of the Piecewise function ? What can I do to solve that issue ?

Answer 1

Try this with UnitStep and a numeric value for V0:

a = 0.1
V0 = 1.0;
i0 = ItoProcess[
   \[DifferentialD]V[t] == a*Plus[
    UnitStep[V0 - V[t]] V[t]*\[DifferentialD]t,
    (1 - UnitStep[V0 - V[t]]) (a V[t]^(2/3) V0^(1/3) \[DifferentialD]t + V0^(1/3) V[t]^(2/3)/3 \[DifferentialD]w[t])
    ],
   V[t],
   {V, 1},
   t,
   w \[Distributed] WienerProcess[]
   ];

RandomFunction[i0, {0., 5., 0.01}]