  We consider the task of generating discrete-time realisations of a nonlinear
multivariate diffusion process satisfying an It\^o stochastic differential
equation conditional on an observation taken at a fixed future time-point. Such
realisations are typically termed diffusion bridges. Since, in general, no
closed form expression exists for the transition densities of the process of
interest, a widely adopted solution works with the Euler-Maruyama
approximation, by replacing the intractable transition densities with Gaussian
approximations. However, the density of the conditioned discrete-time process
remains intractable, necessitating the use of computationally intensive methods
such as Markov chain Monte Carlo. Designing an efficient proposal mechanism
which can be applied to a noisy and partially observed system that exhibits
nonlinear dynamics is a challenging problem, and is the focus of this paper. By
partitioning the process into two parts, one that accounts for nonlinear
dynamics in a deterministic way, and another as a residual stochastic process,
we develop a class of novel constructs that bridge the residual process via a
linear approximation. In addition, we adapt a recently proposed construct to a
partial and noisy observation regime. We compare the performance of each new
construct with a number of existing approaches, using three applications.
