Before revision:  Furthermore, motivated by a basket credit default swap (CDS ) pricing problem with counterparty risk and contagion risk, we show weak convergence for some functions of correlated first-passage times of the approximating processes. We also discuss an open question on the probability of hitting a fixed point by correlated Brownian motions in a fixed time-interval .
After revision:  Furthermore, motivated by a basket credit default swap (CDS ) pricing problem with counterparty risk under a credit contagion model of multinames. This model enables us to capture the systematic volatility increases in the market triggered by a particular bankruptcy. The drawback of this problem is its analytical complication due to its path-dependent functional, which bears a potential failure in its convergence of numerical approximation under standing assumptions. In this paper we find sufficient conditions for the desired convergence by means of the weak convergence method to a class of path-dependent stochastic differential equations .