For scalar SDEs with a one-sided reflection we study pathwise approximation, globally on a compact time interval or at a single time point. We consider algorithms based on sequential evaluations of the driving Brownian motion and establish upper and lower bounds for the minimal errors. Exploiting the relation to a reflected Ornstein-Uhlenbeck process, we also provide a new upper bound for a Cox-Ingersoll-Ross process.

Nowadays, high-speed computations are mandatory for financial and insurance institutes to survive in competition and to fulfill the regulatory reporting requirements that have just toughened over the last years. A majority of these computations are carried out on huge computing clusters, which are an ever increasing cost burden for the financial industry. There, state-of-the-art CPU and GPU architectures execute arithmetic operations with pre-defined precisions only, that may not meet the actual requirements for a specific application. Reconfigurable architectures like field programmable gate arrays (FPGAs) have a huge potential to accelerate financial simulations while consuming only very low energy by exploiting dedicated precisions in optimal ways. In this work we present a novel methodology to speed up multilevel Monte Carlo (MLMC) simulations on reconfigurable architectures. The idea is to aggressively lower the precisions for different parts of the algorithm without loosing any accuracy at the end. For this, we have developed a novel heuristic for selecting an appropriate precision at each stage of the simulation that can be executed with low costs at runtime. Further, we introduce a cost model for reconfigurable architectures and minimize the cost of our algorithm without changing the overall error. We consider the showcase of pricing Asian options in the Heston model. For this setup we improve one of the most advanced simulation methods by a factor of 3-9x on the same platform.

We wish to solve an ill-posed problem whose solution operator S is a measurable unbounded linear transformation of a Banach space into a Hilbert space. Let ε > 0. It is known that the e-complexity of this problem is infinite in the worst case setting. Suppose we turn to an average case or probabilistic setting, the domain of S being equipped with a zero-mean Gaussian measure μ. It is known that the problem has finite ε-complexity iff S ϵ L2(μ), and optimal information and algorithms are essentially the same as if S were bounded. We show that any such unbounded operator S belongs to L2(μ). Hence, the ε-complexity of any such illposed problem is finite in the average case and probabilistic settings.