Randomized Quasi-Monte Carlo with Importance Sampling for Functions under Generalized Growth Conditions and Its Applications in Finance

Abstract: Many problems can be formulated as high-dimensional integrals of discontinuous functions that often exhibit significant growth, challenging the error analysis of randomized quasi-Monte Carlo (RQMC) methods. This paper studies RQMC methods for functions with generalized exponential growth conditions, with a special focus on financial derivative pricing. The main contribution of this work is threefold. First, by combining RQMC and importance sampling (IS) techniques, we derive a new error bound for a class of integrands with the critical growth condition $e^{A|\boldsymbol{x}|^2}$ where $A = 1/2$. This theory extends existing results in the literature, which are limited to the case $A < 1/2$, and we demonstrate that by imposing a light-tail condition on the proposal distribution in the IS, the RQMC method can maintain its high-efficiency convergence rate even in this critical growth scenario. Second, we verify that the Gaussian proposals used in Optimal Drift Importance Sampling (ODIS) satisfy the required light-tail condition, providing rigorous theoretical guarantees for RQMC-ODIS in critical growth scenarios. Third, for discontinuous integrands from finance, we combine the preintegration technique with RQMC-IS. We prove that this integrand after preintegration preserves the exponential growth condition. This ensures that the preintegrated discontinuous functions can be seamlessly incorporated into our RQMC-IS convergence framework. Finally, numerical results validate our theory, showing that the proposed method is effective in handling these problems with discontinuous payoffs, successfully achieving the expected convergence rates.