Mario Hefter

Mixed precision multilevel Monte Carlo on hybrid computing systems

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.

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L 1 or L

We consider weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in L 1 or L ∞ norms. We provide conditions on the weights under which the corresponding spaces have equivalent norms with constants independent of, or only polynomially dependent on the number of variables.

On embeddings of weighted tensor product Hilbert spaces

We study embeddings between tensor products of weighted reproducing kernel Hilbert spaces. The setting is based on a sequence of weights γ j 0 and sequences 1 + γ j k and 1 + l γ j of reproducing kernels k such that H ( 1 + γ j k ) = H ( 1 + l γ j ) , in particular. We derive necessary and sufficient conditions for the norms on ? j = 1 s H ( 1 + γ j k ) and ? j = 1 s H ( 1 + l γ j ) to be equivalent uniformly in s . Furthermore, we study relaxed versions of uniform equivalence by modifying the sequence of weights, e.g., by constant factors, and by analyzing embeddings of the respective spaces. Likewise, we analyze the limiting case s = ∞ .

On arbitrarily slow convergence rates for strong numerical approximations of Cox–Ingersoll–Ross processes and squared Bessel processes

Cox–Ingersoll–Ross (CIR) processes are extensively used in state-of-the-art models for the pricing of financial derivatives. The prices of financial derivatives are very often approximately computed by means of explicit or implicit Euler- or Milstein-type discretization methods based on equidistant evaluations of the driving noise processes. In this article, we study the strong convergence speeds of all such discretization methods. More specifically, the main result of this article reveals that each such discretization method achieves at most a strong convergence order of δ/2

Adaptive approximation of the minimum of Brownian motion

\delta /2

Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

, where 0<δ<2

Exploiting Mixed-Precision Arithmetics in a Multilevel Monte Carlo Approach on FPGAs

0<\delta <2

Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

is the dimension of the squared Bessel process associated to the considered CIR process.

Strong convergence rates for Cox–Ingersoll–Ross processes — Full parameter range

We study the error in approximating the minimum of a Brownian motion on the unit interval based on finitely many point evaluations. We construct an algorithm that adaptively chooses the points at which to evaluate the Brownian path. In contrast to the

Optimal Strong Approximation of the One-dimensional Squared {B}essel Process

1/2