  We propose a discrete time discrete space Markov chain approximation with a
Brownian bridge correction for computing curvilinear boundary crossing
probabilities of a general diffusion process on a finite time interval. For
broad classes of curvilinear boundaries and diffusion processes, we prove the
convergence of the constructed approximations in the form of products of the
respective substochastic matrices to the boundary crossing probabilities for
the process as the time grid used to construct the Markov chains is getting
finer. Numerical results indicate that the convergence rate for the proposed
approximation with the Brownian bridge correction is $O(n^{-2})$ in the case of
$C^2$-boundaries and a uniform time grid with $n$ steps.
