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Is it possible to make a parametric ItoProcess ?

I'd like to write that equation :

V0 = 1
a = 0.1
i0[Vt0_] = 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 + 
     0.1*V0^(1/3) V[t]^(2/3)/3 \[DifferentialD]w[t])], V[t], {V, Vt0}, t, w \[Distributed] WienerProcess[]];

ListLinePlot[RandomFunction[i0[1], {0., 200, 1}]]

But it doesn't work, and ParametricItoProcess doesn't exist, so I was wondering how to deal with that question ?

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3
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That's no problem with SetDelayed (:=):

a = 0.1
V0 = 1.0;
ClearAll[i0];
i0[V0t_] := 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, V0t},
   t,
   w \[Distributed] WienerProcess[]
   ];

ListLinePlot[
 Table[
  RandomFunction[i0[V0t], {0., 5., 0.01}],
  {V0t, 1., 2., 0.25}
  ]
 ]

enter image description here

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