I have an $\mathrm{ARIMA}(1,1,0)$ process $X_t$, for which I know the values $X_0=a$ and $X_T=b$. I want to sample paths $(X_t)_{t=1..(T-1)}$ consistent with the boundary conditions.
One way to do it is sampling all the intermediate points at the same time: the process $\Delta X_t$ is Gaussian, with known covariance, and a constraint on the sum, and can be obtained as a linear combination of independent Gaussians and $(b-a)$. This is the approach used e.g. in this answer to a similar problem.
However I would like instead a non anticipative sampling method, i.e. a way to sample $X_i$ knowing just the values of $\{X_j\}_{j=0..(i-1)}$ and $X_T$.
If $X_t$ is just an $\mathrm{ARIMA}(0,1,0)$ process with unit variance (i.e. a random walk), this is possible by using the characterization of a Brownian bridge, that is by sampling $X_1$ from a Gaussian with mean $a+(b-a)/T$ and variance $(T-1)/T$, and then repeat the procedure replacing $X_0$ with the just sampled $X_1$.
Is a similar method available for the $\mathrm{ARIMA}(1,1,0)$ case?
PS I found a paper that considers the problem of sampling from an Ornstein-Uhlenbeck bridge with a constraint on the area behind the path. I don't know however whether and how the results can be applied to my case.
