Approximate Wiener–Hopf factorization and the Monte Carlo methods for Lévy processes

In the present paper, we justify the convergence formulas for approximate Wiener–Hopf factorization to exact formulas for factors from a broad class of Lévy processes. Another result obtained here is the analysis of the convergence of Monte Carlo methods that are based on time randomization and explicit Wiener–Hopf factorization formulas. The paper puts forward two generalized approaches to the construction of a Monte Carlo method in the case of Lévy models that do not admit explicit Wiener–Hopf factorization. Both methods depend on approximate formulas that do for Wiener–Hopf factors. In the first approach, the simulation of the supremum and infimum processes at exponentially distributed time moments is effected by inverting their approximate cumulative distribution functions. The second approach, which does not require a partition of the path, involves direct simulation of terminal values of the infimum (supremum) process, and can be used for the simulation of the joint distribution of a Lévy process and the corresponding extrema of the process.