In biochemically reactive systems with small copy numbers of one or more reactant molecules, stochastic effects dominate the dynamics. In the first part of this thesis, we design novel efficient simulation techniques for a reliable and robust estimation of various statistical quantities for stochastic biological and chemical systems under the framework of Stochastic Reaction Networks (SRNs). In systems characterized by having simultaneously fast and slow timescales, existing discrete state-space stochastic path simulation methods can be very slow. In the first work in this part, we propose a novel hybrid multilevel Monte Carlo (MLMC), which uses a novel split-step implicit tau-leap scheme at the coarse levels, where the explicit scheme is not applicable due to numerical instability issues. Our analysis illustrates the advantage of our proposed method over MLMC combined with the explicit scheme.
In a second work related to the first part, we solve another challenge present in this context called the high kurtosis phenomenon. We address cases where the high kurtosis, observed for the MLMC estimator, is due to catastrophic coupling. We propose a novel method that provides a more reliable and robust multilevel estimator. Our approach combines the MLMC method with a pathwise-dependent importance sampling technique for simulating the coupled paths. Through our theoretical estimates and numerical analysis, we show that our approach not only improves the robustness of the multilevel estimator by reducing the kurtosis significantly but also improves the strong convergence rate, and consequently, the complexity rate of the MLMC method. We achieve all these improvements with a negligible additional cost.
In the second part of this thesis, we design novel numerical methods for pricing financial derivatives. Option pricing can be formulated as an integration problem, which is usually challenging due to a combination of two complications: 1) The high dimensionality of the input space, and 2) The low regularity of the integrand on the input parameters. We address these challenges by using different techniques for smoothing the integrand to uncover the available regularity and improve quadrature methods' convergence behavior. We develop different ways of smoothing that depend on the characteristics of the problem at hand. Then, we approximate the resulting integrals using hierarchical quadrature methods.
In the first work in this part, we design a fast method for pricing European options under the rough Bergomi model. This model exhibits several numerical and theoretical challenges. As a consequence, classical numerical methods for pricing become either inapplicable or computationally expensive. In our approach, we first smoothen the integrand analytically and then use quadrature methods. These quadrature methods are coupled with Brownian bridge construction and Richardson extrapolation, to approximate the resulting integral. Numerical examples with different parameter constellations exhibit the performance of our novel methodology. Indeed, our hierarchical methods demonstrate substantial computational gains compared to the MC method, which is the prevalent method in this context.
In the second work in this part, we consider cases where we cannot perform an analytic smoothing. Consequently, we perform a numerical smoothing based on root-finding techniques, with a particular focus on cases where discretization of the asset price dynamics is needed. We illustrate the advantage of our approach, which combines numerical smoothing with adaptive sparse grids' quadrature, over the MC approach. Furthermore, we demonstrate that our numerical smoothing procedure improves the robustness and the complexity rate of the MLMC estimator. Finally, our numerical smoothing, coupled with MLMC, enables us also to estimate density functions efficiently.
Chiheb Ben Hammouda is a Ph.D. candidate supervised by Prof. Raul Tempone in the Applied Mathematics program at King Abdullah University of Science and Technology. He received his Master of Applied Mathematics from King Abdullah University of Science and Technology in 2015 and his Bachelor in Multidisciplinary Engineering from Ecole Polytechnique de Tunisie in 2013. His research interests include stochastic modeling and simulation, Numerical analysis, quantitative finance in particular option pricing, stochastic networks theory, and computational biology.