Suppose that $\{Z_t : t \in [0,1]\}$ is a standard Brownian Motion process. It's well known that $X_t = Z_t - tZ_1$ is a Brownian Bridge, because it's a continuous Gaussian process, with mean function $E[Z_t] = 0$ and covariance function $\text{Cov}(Z_t, Z_s) = \min(s,t) - st.$

I am interested in letting the right endpoint of the Brownian bridge being randomly distributed. What is the name of the following process: $$ Y_t = X_t + t N, \hspace{10mm} N \sim \text{Normal}(\mu, \sigma^2)? $$

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    $\begingroup$ Obviously, it will no longer be a bridge process. In the case you consider, the end point is again Gaussian, but it is not clear whether the $\sigma^2$ is the same as those of the original Gaussian that formed the Brownian. If it is, then it seems the process reverts to this original Brownian (but nonbridge). If it isn't not, you arrive at a process with a distribution-jump at $t=1$. Not sure whether any name exists for that. But it's an interesting thought. $\endgroup$ – Lucozade Oct 13 '18 at 16:40

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