Monte Carlo convergence rates for kth moments in Banach spaces

We formulate standard and multilevel Monte Carlo methods for the kth moment M_ε^k[ξ] of a Banach space valued random variable ξ:Ω→E, interpreted as an element of the k-fold injective tensor product space ⊗_ε^kE. For the standard Monte Carlo estimator of M_ε^k[ξ], we prove the k...

Efficient Estimation of the Expected Shortfall: In a Nested Simulation Framework

We analyze three different methods that can approximate the expected shortfall of a financial portfolio in a nested simulation. In this simulation process, the outer simulation generates risk scenarios, and the inner simulation approximates the value of the financial portfolio under each risk scenario. The first method is the most standard one,...

Multilevel Monte Carlo with Surrogate Models for Resource Adequacy Assessment

Monte Carlo simulation is often used for the reliability assessment of power systems, but it converges slowly when the system is complex. Multilevel Monte Carlo (MLMC) can be applied to speed up computation without compromises on model complexity and accuracy that are limiting real-world effectiveness. In MLMC, models with different complexity...

Numerical Solutions for the Stochastic Local Volatility Model

This thesis is about pricing European options and forward start options under the Heston LSV model. The impact of conditionally calibrating the Heston parameters on the satisfaction of the Feller condition and thereafter correcting with a local volatility surface is investigated here. The results show that this approach is computationally time...

Solving Stochastic PDEs with Approximate Gaussian Markov Random Fields using Different Programming Environments

This thesis is a study on the implementation of the Gaussian Markov Random Field (GMRF) for random sample generation and also the Multilevel Monte Carlo (MLMC) method to reduce the computational costs involved with doing uncertainty quantification studies. The GMRF method is implemented in different programming environments in order to evaluate...