Compute Moments Symbolically 

Compute moments of the Jacobi diffusion process symbolically.

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X j\[ScriptCapitalP][\[Alpha]_, x0_] = ItoProcess[\[DifferentialD]x[ t] == (\[Alpha] - x[t]) \[DifferentialD]t + \[Sqrt](x[ t] (1 - x[t])) \[DifferentialD]w[t], x[t], {x, x0}, t, w \[Distributed] WienerProcess[]];

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X moments = Table[Moment[j\[ScriptCapitalP][a, x0][t], r], {r, 1, 4}];

Compute the Expectation of the above moments assuming the initial value x0 follows a BetaDistribution. The expected value is time independent, which suggests that the beta distribution is the stationary distribution of the Jacobi diffusion process.

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X moments = Table[Moment[j\[ScriptCapitalP][a, x0][t], r], {r, 1, 4}];; Expectation[%, x0 \[Distributed] BetaDistribution[2 a, 2 (1 - a)]]

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Compute the three-point correlation function.

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X Expectation[\[Phi][Subscript[t, 1]] \[Phi][Subscript[t, 2]] \[Phi][ Subscript[t, 3]], \[Phi] \[Distributed] j\[ScriptCapitalP][a, x0]]

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