Hierarchical and adaptive methods for accurate and efficient risk estimation

Abstract

Practical systems that depend on unknown factors are frequently well-represented through a stochastic model. By estimating statistics of the underlying model, critical features of the system can be inferred. When such inferences assist decision-making, accurate uncertainty quantification is crucial, meaning that robust error estimates or confidence intervals accompany the estimated parameters. Sufficiently accurate estimates can require several samples from the underlying model. When exact samples of the model are computationally infeasible or unavailable, one must carefully balance statistical errors with approximation bias to retain accurate uncertainty quantification. The multilevel Monte Carlo (MLMC) approach provides an efficient framework for accurately approximating expectations of quantities of interest given a hierarchy of increasingly accurate model approximations. Motivated by problems arising in financial credit risk management and option pricing, this thesis considers the development and analysis of novel MLMC estimators within two frameworks: Firstly, we develop a hierarchy of nested MLMC estimators to estimate systems of repeatedly nested expectations given approximate samples of the model conditioned an underlying filtration at a discrete progression of time points. Secondly, we consider an adaptive MLMC scheme to approximate point evaluations of the distribution of underlying quantities of interest. Both methods are combined to compute the probability of significant financial losses arising from credit risk factors. The method attains a specified error tolerance ε with an asymptotic cost of order ε −2 |log ε| 2 , reduced from order ε −5 using standard Monte Carlo estimation