  This paper provides convergence analysis for the approximation of a class of
path-dependent functionals underlying a continuous stochastic process. In the
first part, given a sequence of weak convergent processes, we provide a
sufficient condition for the convergence of the path-dependent functional
underlying weak convergent processes to the functional of the original process.
In the second part, we study the weak convergence of Markov chain approximation
to the underlying process when it is given by a solution of stochastic
differential equation. Finally, we combine the results of the two parts to
provide approximation of option pricing for discretely monitoring barrier
option underlying stochastic volatility model. Different from the existing
literatures, the weak convergence analysis is obtained by means of metric
computations in the Skorohod topology together with the continuous mapping
theorem. The advantage of this approach is that the functional under study may
be a function of stopping times, projection of the underlying diffusion on a
sequence of random times, or maximum/minimum of the underlying diffusion.
