j𝒫[α_,x0_]=ItoProcess[\[DifferentialD]x[t]==(α-x[t])\[DifferentialD]t+(Sqrt[x[t](1-x[t])])\[DifferentialD]w[t],x[t],{x,x0},t,w\[Distributed]WienerProcess[]];

moments=Table[Moment[j𝒫[a,x0][t],r],{r,1,4}];

Column[moments,Background->{{LightOrange,LightGreen}},Dividers->All,Spacings->{1,2}]

Expectation[moments,x0\[Distributed]BetaDistribution[2a,2(1-a)]]

Expectation[ϕ[(Subscript[t, 1])]ϕ[(Subscript[t, 2])]ϕ[(Subscript[t, 3])],ϕ\[Distributed]j𝒫[a,x0]]

j𝒫[α_,x0_]=ItoProcess[\[DifferentialD]x[t]==(α-x[t])\[DifferentialD]t+(Sqrt[x[t](1-x[t])])\[DifferentialD]w[t],x[t],{x,x0},t,w\[Distributed]WienerProcess[]];
Framed[Column[Table[Moment[j𝒫[a,x0][t],r],{r,1,4}],Background->{{LightOrange,LightGreen}},Dividers->All,Spacings->{1,2},FrameStyle->Directive[Thick,White]],FrameStyle->None,RoundingRadius->10,Background->Hue[.15,.15,.95]]