Ito process as numerical solution of stochastic differential equation

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That makes sense.

Unfortunately, according to the tests I’ve done, neither centered nor non-centered gives satisfactory performance for nonlinear models. This is not surprising, and is consistent with @Charles_Driver 's comment. The scheme that I’m trying to implement, on the other hand, requires a two-step sampling process:
For an SDE with param \theta, trajectory data X^{\text{obs}}, we augment data X by add k missing data points between observation i and i+1, so that the added missing data(as param in Stan) \tilde{X} and X^{\text{obs}} line up as

X^{\text{obs}}_1, \tilde{X}_{1,1}, \tilde{X}_{1,2}, \dots, \tilde{X}_{1,k}, X^{\text{obs}}_2, \tilde{X}_{2,1}, \tilde{X}_{2,2}, \dots, \tilde{X}_{2,k}, X^{\text{obs}}_3, \dots

chronologically.

The sampling scheme mentioned earlier(Ito process as numerical solution of stochastic differential equation) samples \theta with two steps in iteration j of MCMC: first sample \tilde{X}^j from f(\tilde{X} | X^{\text{obs}}, \theta^{j-1}), then sample \theta^j from f(\theta | \tilde{X}^j, X^{\text{obs}}).

I’ve scratched my head and don’t see how this Gibbs-type two-steps sampling is doable in Stan. Is it?