arXiv subjects

Ralf Wunderlich

Publications and source records attributed to Ralf Wunderlich.

At least 19 recordsLinked to original sources

arXiv2025

Convergence of the extended Kalman filter with small and state-dependent noise

Nonlinear filtering problems are encountered in many applications, and one solution approach is the extended Kalman filter, which is not always convergent. Therefore, it is crucial to identify conditions under which the extended Kalman filter provides accurate approximations. This paper generalizes two significant results of Picard (1991) on the efficiency of the continuous-time extended Kalman filter for a filtering system with small noise, to a more general setting where the observation noise may be state-dependent but does not allow signal reconstruction from the quadratic variation of the observation process as for example in epidemic models. First, we show that if the drift of the signal process and the observation process becomes nearly linear when the parameter $\epsilon$, which scales the diffusion coefficients, approaches zero, and the drift coefficient of the observation process is strongly injective, then the estimation error is of the order of $\sqrt{\epsilon}$. We then establish conditions under which the impact of the initial filtering error decays exponentially fast.

math.PR↗

arXiv2025

Stochastic Models and Estimation of Undetected Infections in the Transmission of Zika Virus

Zika fever, a mosquito-borne viral disease with potential severe neurological complications and birth defects, remains a significant public health concern. The epidemiological models often oversimplify the dynamics of Zika transmission by assuming immediate detection of all infected cases. This study provides an enhanced SEIR (Susceptible-Exposed-Infectious-Recovered) model to incorporate partial information by distinguishing between detected and undetected Zika infections (also known as "dark figures"). By distinguishing the compartments, the model captures the complexities of disease spread by accounting for uncertainties about transmission and the number of undetected infections. This model implements the Kalman filter technique to estimate the hidden states from the observed states. Numerical simulations were performed to understand the dynamics of Zika transmission and real-world data was utilized for parameterization and validation of the model. The study aims to provide information on the impact of undetected Zika infections on disease spread within the population, which will contribute to evidence-based decision making in public health policy and practice.

q-bio.PE↗

arXiv2025

Estimating Unobservable States in Stochastic Epidemic Models with Partial Information

This article investigates stochastic epidemic models with partial information and addresses the estimation of current values of not directly observable states. The latter is also called nowcasting and related to the so-called "dark figure" problem, which concerns, for example, the estimation of unknown numbers of asymptomatic and undetected infections. The study is based on Ouabo Kamkumo et al. (2025), which provides detailed information about stochastic multi-compartment epidemic models with partial information and various examples. Starting point is a description of the state dynamics by a system of nonlinear stochastic recursions resulting from a time-discretization of a diffusion approximation of the underlying counting processes. The state vector is decomposed into an observable and an unobservable component. The latter is estimated from the observations using the extended Kalman filter approach in order to take into account the nonlinearity of the state dynamics. Numerical simulations for a Covid-19 model with partial information are presented to verify the performance and accuracy of the estimation method.

q-bio.PE↗

arXiv2025

Stochastic Epidemic Models with Partial Information

Mathematical models of epidemics often use compartmental models dividing the population into several compartments. Based on a microscopic setting describing the temporal evolution of the subpopulation sizes in the compartments by stochastic counting processes one can derive macroscopic models for large populations describing the average behavior by associated ordinary differential equations such as the celebrated SIR model. Further, diffusion approximations allow to address fluctuations from the average and to describe the state dynamics also for smaller populations by stochastic differential equations. In general, not all state variables are directly observable, and we face the so-called "dark figure" problem, which concerns, for example, the unknown number of asymptomatic and undetected infections. The present study addresses this problem by developing stochastic epidemic models that incorporate partial information about the current state of the epidemic, also known as nowcast uncertainty. Examples include a simple extension of the SIR model, a model for a disease with lifelong immunity after infection or vaccination, and a Covid-19 model. For the latter, we propose a ``cascade state approach'' that allows to exploit the information contained in formally hidden compartments with observable inflow but unobservable outflow. Furthermore, parameter estimation and calibration are performed using ridge regression for the Covid-19 model. The results of the numerical simulations illustrate the theoretical findings.

q-bio.PE↗

arXiv2025

Stochastic Optimal Control of an Epidemic Under Partial Information

In this paper, we address a social planner's optimal control problem for a partially observable stochastic epidemic model. The control measures include social distancing, testing, and vaccination. Using a diffusion approximation for the state dynamics of the epidemic, we apply filtering arguments to transform the partially observable stochastic optimal control problem into an optimal control problem with complete information. This transformed problem is treated as a Markov decision process. The associated Bellman equation is solved numerically using optimal quantization methods for approximating the expectations involved to mitigate the curse of dimensionality. We implement two approaches, the first involves state discretization coupled with linear interpolation of the value function at non-grid points. The second utilizes a parametrization of the value function with educated ansatz functions. Extensive numerical experiments are presented to demonstrate the efficacy of both methods.

math.OC↗

arXiv2025

Cost-optimal Management of a Residential Heating System With a Geothermal Energy Storage Under Uncertainty

In this paper, we consider a residential heating system with renewable and non-renewable heat generation and different consumption units and investigate a stochastic optimal control problem for its cost-optimal management. As a special feature, the heating system is equipped with a geothermal storage that enables the intertemporal transfer of thermal energy by storing surplus heat for later use. In addition to the numerous technical challenges, economic issues such as cost-optimal control also play a central role in the design and operation of such systems. The latter leads to challenging mathematical optimization problems, as the response of the storage to charging and discharging decisions depends on the spatial temperature distribution in the storage. We take into account uncertainties regarding randomly fluctuating heat generation from renewable energies and the environmental conditions that determine heat demand and supply. The dynamics of the multidimensional controlled state processes is governed by a partial, a random ordinary and two stochastic differential equations. We first apply a spatial discretization to the partial differential equation and use model reduction techniques to reduce the dimension of the associated system of ordinary differential equations. Finally, a time-discretization leads to a Markov decision process for which we apply a state discretization to determine approximations of the cost-optimal control and the associated value function.

math.OC↗

arXiv2024

Reinforcement Learning Methods for the Stochastic Optimal Control of an Industrial Power-to-Heat System

The optimal control of sustainable energy supply systems, including renewable energies and energy storage, takes a central role in the decarbonization of industrial systems. However, the use of fluctuating renewable energies leads to fluctuations in energy generation and requires a suitable control strategy for the complex systems in order to ensure energy supply. In this paper, we consider an electrified power-to-heat system which is designed to supply heat in form of superheated steam for industrial processes. The system consists of a high-temperature heat pump for heat supply, a wind turbine for power generation, a sensible thermal energy storage for storing excess heat and a steam generator for providing steam. If the system's energy demand cannot be covered by electricity from the wind turbine, additional electricity must be purchased from the power grid. For this system, we investigate the cost-optimal operation aiming to minimize the electricity cost from the grid by a suitable system control depending on the available wind power and the amount of stored thermal energy. This is a decision making problem under uncertainties about the future prices for electricity from the grid and the future generation of wind power. The resulting stochastic optimal control problem is treated as finite-horizon Markov decision process for a multi-dimensional controlled state process. We first consider the classical backward recursion technique for solving the associated dynamic programming equation for the value function and compute the optimal decision rule. Since that approach suffers from the curse of dimensionality we also apply reinforcement learning techniques, namely Q-learning, that are able to provide a good approximate solution to the optimization problem within reasonable time.

math.OC↗

arXiv2024

Optical and spin properties of nitrogen vacancy centers formed along the tracks of high energy heavy ions

Exposure of matter to high energy, heavy ions induces defects along the trajectories of the ions through electronic and nuclear energy loss processes. Defects, including color centers, can recombine or form along latent damage tracks in many materials, such as insulators and semiconductors. Latent tracks in diamond were only recently observed. Here, we report on color center formation in diamond along the latent tracks of 1 GeV gold and uranium ions. Using depth-resolved photoluminescence, we observe direct formation of single vacancy related color centers (GR1-centers) along the ion tracks. Mobile vacancies can form NV-centers with native nitrogen atoms during thermal annealing. Molecular dynamics simulations show that isolated vacancies and vacancy clusters form through electronic stopping processes, leading to color center formation along ion trajectories from the sample surface to a depth of about 25 microns. We further report on the creation of individually isolated quasi-1D chains of NV-centers by using 1 GeV Au ions with a dilute fluence. The individual 1D NV-chains appear as isolated bright luminescence strings and present competitive electron spin properties compared to a background of NV-centers. Such spin textures can be explored as building blocks for applications in quantum information processing and quantum sensing.

quant-ph↗

arXiv2023

Discretization of continuous-time arbitrage strategies in financial markets with fractional Brownian motion

This study evaluates the practical usefulness of continuous-time arbitrage strategies designed to exploit serial correlation in fractional financial markets. Specifically, we revisit the strategies of Shiryaev (1998) and Salopek (1998) and transfer them to a real-world setting by distretizing their dynamics and introducing transaction costs. In Monte Carlo simulations with various market and trading parameter settings as well as a formal analysis of discretization error, we show that both are promising with respect to terminal portfolio values and loss probabilities. These features and complementary sparsity make them worth serious consideration in the toolkit of quantitative investors.

q-fin.PM↗

arXiv2023

Portfolio Optimization in a Market with Hidden Gaussian Drift and Randomly Arriving Expert Opinions: Modeling and Theoretical Results

This paper investigates the optimal selection of portfolios for power utility maximizing investors in a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving randomly over time. The arrival dates are modeled as the jump times of a homogeneous Poisson process. Applying Kalman filter techniques we derive estimates of the hidden drift which are described by the conditional mean and covariance of the drift given the observations. The utility maximization problem is solved with dynamic programming methods. We derive the associated dynamic programming equation and study regularization arguments for a rigorous mathematical justification.

q-fin.PM↗

arXiv2023

Power Utility Maximization with Expert Opinions at Fixed Arrival Times in a Market with Hidden Gaussian Drift

In this paper we study optimal trading strategies in a financial market in which stock returns depend on a hidden Gaussian mean reverting drift process. Investors obtain information on that drift by observing stock returns. Moreover, expert opinions in the form of signals about the current state of the drift arriving at fixed and known dates are included in the analysis. Drift estimates are based on Kalman filter techniques. They are used to transform a power utility maximization problem under partial information into an optimization problem under full information where the state variable is the filter of the drift. The dynamic programming equation for this problem is studied and closed-form solutions for the value function and the optimal trading strategy of an investor are derived. They allow to quantify the monetary value of information delivered by the expert opinions. We illustrate our theoretical findings by results of extensive numerical experiments.

q-fin.PM↗

arXiv2022

On the Input-Output Behavior of a Geothermal Energy Storage: Approximations by Model Order Reduction

In this paper we consider a geothermal energy storage in which the spatio-temporal temperature distribution is modeled by a heat equation with a convection term. Such storages often are embedded in residential heating systems and control and management require the knowledge of some aggregated characteristics of that temperature distribution in the storage. They describe the input-output behaviour of the storage and the associated energy flows and their response to charging and discharging processes. We aim to derive an efficient approximative description of these characteristics by a low-dimensional system of ODEs. This leads to a model order reduction problem for a large scale linear system of ODEs arising from the semi-discretization of the heat equation combined with a linear algebraic output equation. In a first step we approximate the non time-invariant system of ODEs by a linear time-invariant system. Then we apply Lyapunov balanced truncation model order reduction to approximate the output by a reduced-order system with only a few state equations but almost the same input-output behavior. The paper presents results of extensive numerical experiments showing the efficiency of the applied model order reduction methods. It turns out that only a few suitable chosen ODEs are sufficient to produce good approximations of the input-output behaviour of the storage.

math.NA↗

arXiv2022

Well Posedness of Utility Maximization Problems Under Partial Information in a Market with Gaussian Drift

This paper investigates well posedness of utility maximization problems for financial markets where stock returns depend on a hidden Gaussian mean-reverting drift process. Since that process is potentially unbounded, well posedness cannot be guaranteed for utility functions which are not bounded from above. For power utility with relative risk aversion smaller than that of log-utility this leads to restrictions on the choice of model parameters such as the investment horizon and parameters controlling the variance of the asset price and drift processes. We derive sufficient conditions to the model parameters leading to bounded maximum expected utility of terminal wealth for models with full and partial information.

q-fin.PM↗

arXiv2022

Defect engineering of silicon with ion pulses from laser acceleration

Defect engineering is foundational to classical electronic device development and for emerging quantum devices. Here, we report on defect engineering of silicon single crystals with ion pulses from a laser accelerator with ion flux levels up to 10^22 ions/cm^2/s. Low energy ions from plasma expansion of the laser-foil target are implanted near the surface and then diffuse into silicon samples that were locally pre-heated by high energy ions. We observe low energy ion fluences of ~10^16 cm^-2, about four orders of magnitude higher than the fluence of high energy (MeV) ions. In the areas of highest energy deposition, silicon crystals exfoliate from single ion pulses. Color centers, predominantly W and G-centers, form directly in response to ion pulses without a subsequent annealing step. We find that the linewidth of G-centers increase in areas with high ion flux much more than the linewidth of W-centers, consistent with density functional theory calculations of their electronic structure. Laser ion acceleration generates aligned pulses of high and low energy ions that expand the parameter range for defect engineering and doping of semiconductors with tunable balances of ion flux, damage rates and local heating.

cond-mat.mtrl-sci↗

arXiv2021

Short-Term Behavior of a Geothermal Energy Storage: Modeling and Theoretical Results

This paper investigates numerical methods for simulations of the short-term behavior of a geothermal energy storage. Such simulations are needed for the optimal control and management of residential heating systems equipped with an underground thermal storage. There a given volume under or aside of a building is filled with soil and insulated to the surrounding ground. The thermal energy is stored by raising the temperature of the soil inside the storage. It is charged and discharged via pipe heat exchangers filled with a moving fluid. Simulations of geothermal energy storages aim to determine how much energy can be stored in or taken from the storage within a given short period of time. The latter depends on the dynamics of the spatial temperature distribution in the storage which is governed by a linear heat equation with convection and appropriate boundary and interface conditions. We consider semi- and full discretization of that PDE using finite difference schemes and study associated stability problems. Numerical results based on the derived methods are presented in the companion paper [17].

math.NA↗

arXiv2021

Short-Term Behavior of a Geothermal Energy Storage: Numerical Applications

This paper is devoted to numerical simulations of the short-term behavior of the spatial temperature distribution in a geothermal energy storage. Such simulations are needed for the optimal control and management of residential heating systems equipped with an underground thermal storage. We apply numerical methods derived in our companion paper [15] in which we study the governing initial boundary value problem for a linear heat equation with convection. Further, we perform extensive numerical experiments in order to investigate properties of the spatio-temporal temperature distribution and of its aggregated characteristics.

math.NA↗

arXiv2020

Asymptotic Filter Behavior for High-Frequency Expert Opinions in a Market with Gaussian Drift

This paper investigates a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poisson process. Drift estimates are based on Kalman filter techniques and described by the conditional mean and covariance matrix of the drift given the observations. We study the filter asymptotics for increasing arrival intensity of expert opinions and prove that the conditional mean is a consistent drift estimator, it converges in the mean-square sense to the hidden drift. Thus, in the limit as the arrival intensity goes to infinity investors have full information about the drift.

q-fin.MF↗

arXiv2020

Diffusion Approximations for Expert Opinions in a Financial Market with Gaussian Drift

This paper investigates a financial market where returns depend on an unobservable Gaussian drift process. While the observation of returns yields information about the underlying drift, we also incorporate discrete-time expert opinions as an external source of information. For estimating the hidden drift it is crucial to consider the conditional distribution of the drift given the available observations, the so-called filter. For an investor observing both the return process and the discrete-time expert opinions, we investigate in detail the asymptotic behavior of the filter as the frequency of the arrival of expert opinions tends to infinity. In our setting, a higher frequency of expert opinions comes at the cost of accuracy, meaning that as the frequency of expert opinions increases, the variance of expert opinions becomes larger. We consider a model where information dates are deterministic and equidistant and another model where the information dates arrive randomly as the jump times of a Poisson process. In both cases we derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. We use our limit theorems to derive so-called diffusion approximations of the filter for high-frequency discrete-time expert opinions. These diffusion approximations are extremely helpful for deriving simplified approximate solutions of utility maximization problems.

q-fin.PM↗