  An efficient discrete time and space Markov chain approximation employing a
Brownian bridge correction for computing curvilinear boundary crossing
probabilities for general diffusion processes was recently proposed in Liang
and Borovkov (2021). One of the advantages of that method over alternative
approaches is that it can be readily extended to computing expectations of
path-dependent functionals over the event of the process trajectory staying
between two curvilinear boundaries. In the present paper, we extend the scheme
to compute expectations of the Feynman--Kac type that frequently appear in
option pricing. To illustrate our approximation scheme, we apply it in three
special cases. For sufficiently smooth integrands, numerical experiments
suggest that the proposed approximation converges at the rate $O(n^{-2})$,
where $n$ is the number of steps on the uniform time grid used
