Complex systems in finance: Monte Carlo evaluation of first passage time density functions

Many examples of complex systems are provided by applications in finance and economics areas. Some of intrinsic features of such systems lie with the fact that their parts are interacting in a non-trivial dynamic manner and they can be subject to stochastic forces and jumps. The mathematical models for such systems are often based on stochastic differential equations and efficient computational tools are required to solve them. Here, on an example from the credit risk analysis of multiple correlated firms, we develop a fast Monte-Carlo type procedure for the analysis of complex systems such as those occurring in the financial market. Our procedure is developed by combining the fast Monte-Carlo method for one-dimensional jump-diffusion processes and the generation of correlated multidimensional variates. As we demonstrate on the evaluation of first passage time density functions in credit risk analysis, this allows us to analyze efficiently multivariate and correlated jump-diffusion processes.