DFG Project Hu1889/6-1 and Hu1889/6-2: On numerical approximations of high-dimensional nonlinear parabolic partial differential equations and of backward stochastic differential equations - Martin Hutzenthaler (2017-2023)

Parabolic partial differential equations (PDEs) are a fundamental tool in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high- dimensional and nonlinear. Since explicit solutions of such PDEs are typically not available, it is a very active topic of research to solve such PDEs approximately. However existing approximation methods are either computationally expensive or are only applicable to a (in our view) small class of PDEs. The goal of this project is to establish a class of generally applicable numerical approximations for semilinear parabolic partial differential equations and, in particular, to prove that the computational effort grows at most linearly in the dimension and cubic in the reciprocal of the prescribed accuracy. In particular, in the notation of information-based complexity, our goal is to show that the numerical problem of approximating the solution of a semilinear parabolic PDE at single space-time points is polynomially tractable.

DFG Project Hu1889/2-1: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients - Martin Hutzenthaler (2012-2014)

Stochastic differential equations (SDEs) are used in all areas for modeling dynamics with stochastic noise. As applied SDEs typically admit no explicit solution, it is crucial to solve SDEs numerically. The majority of applied SDEs have superlinearly growing coefficients and, therefore, do not satisfy the assumptions of the bulk of the literature. We have recently shown that algorithms developed for the case of global Lipschitz coefficients do in general not transfer to the non-global Lipschitz case without modifications. For this reason, we investigate the convergence behavior of suitably modified explicit Euler methods. More precisely we develop a thorough theory of numerical methods which are recursively defined as a general function of the previous state, of the time increment and of the increment of the Brownian motion. The convergence theory will apply to most of the stochastic ordinary differential equations with locally Lipschitz continuous coefficients having finite moments. Establishing the order of convergence will require additional assumptions such as local smoothness. Our main approach is to bring forward the successful Lyapunov technique to the theory of numerical approximations. Moreover, we extend our finite-dimensional results to stochastic partial differential equations. In particular we study a modified version of the exponential Euler method which hasrecently been proposed for the case of additive noise and which has a rather good order of convergence.

Project Hu1889/1-1: Competing selective sweeps - Martin Hutzenthaler and Peter Pfaffelhuber (Freiburg) (2011-2015)

The fixation of a positively selected allele reduces linked neutral sequence diversity, an effect known as a “selective sweep”. The literature on recurrent selective sweeps has so far mainly focused on the non-overlapping case where at most a single beneficial allele sweeps to fixation at any time. In this project, we study competing selective sweeps where further beneficial mutations arise at different recombining loci during the time-course of the first selective sweep. For example, such a scenario is realistic for a structured population with low migration rates or with previously isolated demes (subpopulations). Recombination events can then bring beneficial alleles to the same genetic background. If such recombination events happen more than once, we expect that competing selective sweeps leave the following distinct genetic footprint: Between selected loci, there is a strong haplotype structure; outside the selected loci, there is a severe reduction of genetic diversity, similar to the case of a single selective sweep. We analyze and apply this model of competing selective sweeps in three steps: (1) The genealogy at a neutral locus, linked to the beneficial alleles, can be studied using the ancestral selection graph. Using this genealogical picture we will quantify the genetic footprint. (2) Using simulation techniques based on the recently developed software MSMS and its proposed extension, we establish and analyze the distinct genetic footprint of competing sweeps in panmictic and structured populations. In particular, we explore possible sources of a strong haplotype structure. (3) In collaboration with empirical groups of the research unit, we scan for the signature of competing sweeps in SNP data of Drosophila melanogaster and in SNP data of Solanum chilense and Solanum peruvianum.