Refine
Document Type
- Doctoral thesis (4)
Has Fulltext
- yes (4)
Is part of the Bibliography
- no (4)
Language
- English (4) (remove)
Keywords
Institute
- FG Wirtschaftsmathematik (4) (remove)
Stochastic optimal control problems of residential heating systems with a geothermal energy storage
(2023)
In this thesis we consider a residential heating system equipped with several heat production and consumption units and investigate the stochastic optimal control problem for its cost-optimal management. As a special feature the manager has access to a geothermal storage (GS) which allows for inter-temporal transfer of heat energy by storing leftover solar thermal energy generated in summer for satisfying demand later. It is charged and discharged via heat exchanger pipes filled with a moving fluid. Further, the manager of that system faces uncertainties about the future fuel price and heat demand. The main goal is to minimize the expected aggregated cost for generating heat and running the system. This leads to a challenging mathematical optimization problem. The problem is formulated first as a non-standard continuous-time stochastic optimal control problem for a controlled state process whose dynamics is described by a system of ordinary differential equations (ODEs), stochastic differential equations and a partial differential equation (PDE). The PDE, which describes the temperature distribution in the GS, is first converted into a high-dimensional system of ODEs by semi-discretizing the space variables and its stability is investigated. This makes it possible to compute some aggregated characteristics which are useful for the operation of the GS embedded in the residential heating system. Second, the linear time-varying system of ODEs is approximated by a suitable linear time-invariant system. This allows the Lyapunov balanced truncation model order reduction method to be applied. Finally, we investigate the solution of the resulting standard optimal control problem for a controlled multi-dimensional diffusion process using dynamic programming methods and derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation. However, no analytical solution of the HJB equation can be expected for the control problem under investigation. Therefore, we transform the continuous-time optimal control problem into a discrete-time control problem for a controlled Markov chain with finitely many states by discretizing both the time and the states. After determining the transition probabilities, the problem is solved using methods from the theory of Markovian decision processes. The thesis presents results of extensive numerical experiments carried out with the developed methods which reveal typical properties of the value function and the optimal strategy of the optimization problem. We end this thesis by describing some alternative methods to overcome the curse of dimensionality.
Motivated by computing functionals of high-dimensional, potentially metastable diffusion processes, this thesis studies robustness issues appearing in the numerical approximation of expectation values and their gradients. A major challenge being high variances of corresponding estimators, we investigate importance sampling of stochastic processes for improving statistical properties and provide novel nonasymptotic bounds on the relative error of corresponding estimators depending on deviations from optimality. Numerical strategies that aim to come close to those optimal sampling strategies can be encompassed in the framework of path space measures, and minimizing suitable divergences between those measures suggests a variational formulation that can be addressed in the spirit of machine learning. A key observation is that while several natural choices of divergences have the same unique minimizer, their finite sample properties differ vastly. We provide the novel log-variance divergence, which turns out to have favorable robustness properties that we investigate theoretically and apply in the context of path space measures as well as in the context of densities, for instance offering promising applications in Bayesian variational inference.
Aiming for optimal importance sampling of diffusions is (more or less) equivalent to solving Hamilton-Jacobi- Bellman PDEs and it turns out that our numerical methods can be equally applied for the approximation of rather general high-dimensional semi-linear PDEs. Motivated by stochastic representations of elliptic and parabolic boundary value problems we refine variational methods based on backward SDEs and provide the novel diffusion loss, which can be related to other state-of-the-art attempts, while offering certain numerical advantages.
This thesis is concerned with stochastic models to manage financial risks. The first part deals with market risk and considers an investor facing a classical portfolio problem of optimal investment in log-Brownian stocks and a fixed-interest bond, but constrained to choose portfolio and consumption strategies which reduce the corresponding shortfall risk. Risk limits are formulated in terms of Value at Risk, Tail Conditional Expectation and Expected Loss and are dynamically imposed on the strategy as a risk constraint. The resulting stochastic optimal control problem is tackled using the dynamic programming approach. For both continuous-time and discrete-time financial markets the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint is investigated. The presented numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. Furthermore, the loss resulting from infrequent trading due to time discretization effects is typically bigger than the loss of portfolio performance resulting from imposing a risk constraint.
The second part deals with credit risk and sets up a first-passage model of corporate default risk. The default event is specified in terms of the evolution of the total value of the firm's asset and the default barrier. Short-term default risk is incorporated by modeling the default barrier at which the firm is liquidated as a random variable which is time-dependent and allowed to switch. This setup combines the two classical modeling approaches and enables to model changes in the economy or the appointment of a new firm management. Different information levels on the firm's assets are distinguished and explicit formulas for the conditional default probability given the accessible information are derived. The impact of asymmetric information on the default probability and credit yield spread is investigated. Numerical results are presented indicating that the information on the firm value has a considerable impact on the estimate of the conditional survival probability and the associated credit yield spread.
This work is devoted to the problem of liquidity that draws a lot of attention after the global financial crisis. We consider an optimization problem for a portfolio with an illiquid, a risky and a riskless liquid asset. We work in Merton’s optimal consumption framework with continuous time. The liquid part of the investment is described by a standard Black-Scholes market. The illiquid asset is sold at an exogenous random moment with prescribed distribution and generates additional liquid wealth dependent on its paper value. We show that one can consider a problem with infinite time horizon and special weight function that is characterized by the probability distribution of the liquidation time instead of a problem with an exogenous random liquidation time. Using the viscosity solution techniques, developed for the problem of optimization in presence of a random income, we prove the existence and uniqueness of the solution for the considered problem with logarithmic utility and modest restrictions on the liquidation time distribution. We find asymptotic bounds for the value function when liquidation time has exponential or Weibull distribution. We find optimal policies in a feedback form and illustrate how they differ from classical Merton’s policies. Through a Lie group analysis we find the admitted Lie algebra for a problem with general liquidation time distribution in cases of HARA and log utility functions and formulate corresponding theorems for all these cases. Using these Lie algebras we obtain reduced equations of the lower dimension for the studied three dimensional partial differential equations. Several of similar substitutions were used in other works before, whereas others are new to our knowledge. The applied method of Lie group analysis gives us the possibility to provide a complete set of non-equivalent substitutions and reduced equations that was not provided for the problem of such type so far. Further research of these equations with numerical and quantitative methods is expected to benefit from such analysis.