  In this paper we derive stochastic representations for the finite dimensional
distributions of a multidimensional diffusion on a fixed time interval,
conditioned on the terminal state. The conditioning can be with respect to a
fixed point or more generally with respect to some subset. The representations
rely on a reverse process connected with the given (forward) diffusion as
introduced in Milstein, Schoenmakers and Spokoiny [Bernoulli 10 (2004) 281-312]
in the context of a forward-reverse transition density estimator. The
corresponding Monte Carlo estimators have essentially root-$N$ accuracy, and
hence they do not suffer from the curse of dimensionality. We provide a
detailed convergence analysis and give a numerical example involving the
realized variance in a stochastic volatility asset model conditioned on a fixed
terminal value of the asset.
