Strong Convergence of the Split-Step θ

We develop a new split-step θ (SSθ) method for stochastic age-dependent capital system with random jump magnitudes. The main aim of this paper is to investigate the convergence of the SSθ method for a class of stochastic age-dependent capital system with random jump magnitudes. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory

Properties and advances of probabilistic and statistical algorithms with applications in finance

This thesis is concerned with the construction and enhancement of algorithms involving probability and statistics. The main motivation for these are problems that appear in finance and more generally in applied science. We consider three distinct areas, namely, credit risk modelling, numerics for McKean Vlasov stochastic differential equations and stochastic representations of Partial Differential Equations (PDEs), therefore the thesis is split into three parts. Firstly, we consider the problem of estimating a continuous time Markov chain (CTMC) generator from discrete time observations, which is essentially a missing data problem in statistics. These generators give rise to transition probabilities (in particular probabilities of default) over any time horizon, hence the estimation of such generators is a key problem in the world of banking, where the regulator requires banks to calculate risk over different time horizons. For this particular problem several algorithms have been proposed, however, through a combination of theoretical and numerical results we show the Expectation Maximisation (EM) algorithm to be the superior choice. Furthermore we derive closed form expressions for the associated Wald confidence intervals (error) estimated by the EM algorithm. Previous attempts to calculate such intervals relied on numerical schemes which were slower and less stable. We further provide a closed form expression (via the Delta method) to transfer these errors to the level of the transition probabilities, which are more intuitive. Although one can establish more precise mathematical results with the Markov assumption, there is empirical evidence suggesting this assumption is not valid. We finish this part by carrying out empirical research on non-Markov phenomena and propose a model to capture the so-called rating momentum. This model has many appealing features and is a natural extension to the Markov set up. The second part is based on McKean Vlasov Stochastic Differential Equations (MV-SDEs), these Stochastic Differential Equations (SDEs) arise from looking at the limit, as the number of weakly interacting particles (e.g. gas particles) tends to infinity. The resulting SDE has coefficients which can depend on its own law, making them theoretically more involved. Although MV-SDEs arise from statistical physics, there has been an explosion in interest recently to use MV-SDEs in models for economics. We firstly derive an explicit approximation scheme for MV-SDEs with one-sided Lipschitz growth in the drift. Such a condition was observed to be an issue for standard SDEs and required more sophisticated schemes. There are implicit and explicit schemes one can use and we develop both types in the setting of MV-SDEs. Another main issue for MVSDEs is, due to the dependency on their own law they are extremely expensive to simulate compared to standard SDEs, hence techniques to improve computational cost are in demand. The final result in this part is to develop an importance sampling algorithm for MV-SDEs, where our measure change is obtained through the theory of large deviation principles. Although importance sampling results for standard SDEs are reasonably well understood, there are several difficulties one must overcome to apply a good importance sampling change of measure in this setting. The importance sampling is used here as a variance reduction technique although our results hint that one may be able to use it to reduce propagation of chaos error as well. Finally we consider stochastic algorithms to solve PDEs. It is known one can achieve numerical advantages by using probabilistic methods to solve PDEs, through the so-called probabilistic domain decomposition method. The main result of this part is to present an unbiased stochastic representation for a first order PDE, based on the theory of branching diffusions and regime switching. This is a very interesting result since previously (Itô based) stochastic representations only applied to second order PDEs. There are multiple issues one must overcome in order to obtain an algorithm that is numerically stable and solves such a PDE. We conclude by showing the algorithm’s potential on a more general first order PDE

Spatial and stochastic epidemics : theory, simulation and control

It is now widely acknowledged that spatial structure and hence the spatial position of host populations plays a vital role in the spread of infection. In this work I investigate an ensemble of techniques for understanding the stochastic dynamics of spatial and discrete epidemic processes, with especial consideration given to SIR disease dynamics for the Levins-type metapopulation. I present a toolbox of techniques for the modeller of spatial epidemics. The highlight results are a novel form of moment closure derived directly from a stochastic differential representation of the epidemic, a stochastic simulation algorithm that asymptotically in system size greatly out-performs existing simulation methods for the spatial epidemic and finally a method for tackling optimal vaccination scheduling problems for controlling the spread of an invasive pathogen

Markov field models of molecular kinetics

Computer simulations such as molecular dynamics (MD) provide a possible means to understand protein dynamics and mechanisms on an atomistic scale. The resulting simulation data can be analyzed with Markov state models (MSMs), yielding a quantitative kinetic model that, e.g., encodes state populations and transition rates. However, the larger an investigated system, the more data is required to estimate a valid kinetic model. In this work, we show that this scaling problem can be escaped when decomposing a system into smaller ones, leveraging weak couplings between local domains. Our approach, termed independent Markov decomposition (IMD), is a first-order approximation neglecting couplings, i.e., it represents a decomposition of the underlying global dynamics into a set of independent local ones. We demonstrate that for truly independent systems, IMD can reduce the sampling by three orders of magnitude. IMD is applied to two biomolecular systems. First, synaptotagmin-1 is analyzed, a rapid calcium switch from the neurotransmitter release machinery. Within its C2A domain, local conformational switches are identified and modeled with independent MSMs, shedding light on the mechanism of its calcium-mediated activation. Second, the catalytic site of the serine protease TMPRSS2 is analyzed with a local drug-binding model. Equilibrium populations of different drug-binding modes are derived for three inhibitors, mirroring experimentally determined drug efficiencies. IMD is subsequently extended to an end-to-end deep learning framework called iVAMPnets, which learns a domain decomposition from simulation data and simultaneously models the kinetics in the local domains. We finally classify IMD and iVAMPnets as Markov field models (MFM), which we define as a class of models that describe dynamics by decomposing systems into local domains. Overall, this thesis introduces a local approach to Markov modeling that enables to quantitatively assess the kinetics of large macromolecular complexes, opening up possibilities to tackle current and future computational molecular biology questions

Numerical Methods for Random Parameter Optimal Control and the Optimal Control of Stochastic Differential Equations

This thesis considers the investigation and development of numerical methods for optimal control problems that are influenced by stochastic phenomena of various type. The first part treats tasks characterized by random parameters, while in the subsequent second part time-dependent stochastic processes are the basis of the dynamics describing the analyzed systems. In each case the investigations aim to transform the original problem into one that can be tackled by existing (direct) methods of deterministic optimal control - here we prefer Bock's direct multiple shooting approach. In the context of this transformation, in the first part approaches from stochastic programming as well as robust and probabilistic optimization are used. Regarding a specific application from mathematical economics, which considers pricing conspicuous consumption products in periods of recession, new numerical procedures are developed and analyzed with due regard to those techniques - in particular, a scenario tree approach, approximations of robust worst-case settings, and financial tools as the Value at Risk and Conditional Value at Risk. Furthermore, necessary reformulations of the resulting optimal control problems, in particular for Value at Risk and Conditional Value at Risk, as well as the discussion and interpretation of results determined depending on an uncertain recession duration, an uncertain recession strength, and control delays are in focus. The gained economic insight can be seen as an important step in the direction of a better understanding of real-world pricing strategies. In the second part of the thesis, based on the Wiener chaos expansion of a stochastic process and on Malliavin calculus, a system of coupled ordinary differential equations is developed that completely characterizes the stochastic differential equation describing the dynamics of the process. As in general this system includes infinitely many equations, a rigorous error estimation depending on the order of the chaos decomposition is proven in order to guarantee the numerical applicability. To transfer the generic procedure of the chaos expansion to stochastic optimal control problems, a method to preserve the feedback character of the occurring control process is shown. This allows the derivation of a novel direct method to solve finite-horizon stochastic optimal control problems. The appropriability and accuracy of this methodology are demonstrated by treating several problem instances numerically. Finally, the economic application of the first part is revisited under the viewpoint of dealing with a time-dependent recession strength, i.e., a stochastic process. In particular, those applications illustrate that the existing methods of deterministic optimal control can be extended to problems including stochastic differential equations

Essays on Energy Portfolio Management

Diese englischsprachige Dissertation behandelt ausgewählte Fragen zum Thema Portfoliomanagement in Energiemärkten. Im Kontext der modernen Portfoliotheorie werden theoretische Verteilungsannahmen untersucht, die einen optimalen Mittelwert-Varianz-Ansatz implizieren. Der Bereich zu Energiemärkten befasst sich einerseits mit Kurzfristprognosen von Day-Ahead-Preisen auf dem Strommarkt. Andererseits werden auf dem Erdgasmarkt die von komplexen Energiederivaten impliziten Volatilitäten analysiert. Einige interessante Beiträge, die diese Dissertation liefert, sind beispielsweise (i) die Erkenntnis, dass sich der Mittelwert-Varianz-Ansatz zur Bestimmung eines optimalen Portfolios von Vermögensgegenständen auch im Falle einer schiefen Renditeverteilung theoretisch rechtfertigen lässt, (ii) eine umfangreiche Vergleichsstudie mit verschiedenen Ansätzen zur Reduktion der Komplexität von multivariaten Strompreisprognosen und (iii) die Entwicklung eines theoretischen Rahmens und effizienten Algorithmus zur Übersetzung von Preisen für Swing-Optionen in implizite Volatilitäten