Stochastic continuous-time cash flows: A coupled linear-quadratic model

<p>The focal point of this dissertation is stochastic continuous-time cash flow models. These models, as underpinned by the results of this study, prove to be useful to describe the rich and diverse nature of trends and fluctuations in cash flow randomness. Firstly, this study considers an important preliminary question: can cash flows be fully described in continuous time? Theoretical and empirical evidence (e.g. testing for jumps) show that under some not too stringent regularities, operating cash flow processes can be well approximated by a diffusion equation, whilst investing processes -preferably- will first need to be rescaled by a system-size variable. Validated by this finding and supported by a multitude of theoretical considerations and statistical tests, the main conclusion of this dissertation is that an equation consisting of a linear drift function and a complete quadratic diffusion function (hereafter: “the linear-quadratic model”) is a specification preferred to other specifications frequently found in the literature. These so-called benchmark processes are: the geometric and arithmetic Brownian motions, the mean-reverting Vasicek and Cox, Ingersoll and Ross processes, and the modified Square Root process. Those specifications can all be considered particular cases of the generic linear-quadratic model. The linear-quadratic model is classified as a hybrid model since it is shown to be constructed from the combination of geometric and arithmetic Brownian motions. The linear-quadratic specification is described by a fundamental model, rooted in well-studied and generally accepted business and financial assumptions, consisting of two coupled, recursive relationships between operating and investing cash flows. The fundamental model explains the positive feedback mechanism assumed to exist between the two types of cash flows. In a stochastic environment, it is demonstrated that the linear-quadratic model can be derived from the principles of the fundamental model. There is no (known) general closed-form solution to the hybrid linear-quadratic cash flow specification. Nevertheless, three particular and three approximated exact solutions are derived under not too stringent parameter restrictions and cash flow domain limitations. Weak solutions are described by (forward or backward) Fokker-Planck- Kolmogorov equations. This study shows that since the process is converging in time (that is, approximating a stable probability distribution), (uncoupled) investing cash flows can be described by a Pearson diffusion process approaching a stationary Person-IV probability density function, more appropriately a Student diffusion process. In contrast, (uncoupled) operating cash flow processes are diverging in time, that is exploding with no stable probability density function, a dynamic analysis in a bounded cash flow domain is required. A suggested solution method normalises a general hypergeometric differential equation, after separation of variables, which is then transformed into a Sturm-Liouville specification, followed by a choice of three separate second transformations. These second transformations are the Jacobi, the Hermitian and the Schrödinger, each yielding a homonymous equation. Only the Jacobi transformation provides an exact solution, the other two transformations lead to approximated closed-form general solutions. It turns out that a space-time density function of operating cash flow processes can be construed as the multiplication of two (independent) time-variant probability distributions: a stationary family of distributions akin to Pearson’s case 2, and the evolution of a standard normal distribution. The fundamental model and the linear-quadratic specification are empirically validated by three different statistical tests. The first test provides evidence that the fundamental model is statistically significant. Parameter values support the conclusion that operating and investing processes are converging to overall long-term stable values, albeit with significant stochastic variation of individual firms around averages. The second test pertains to direct estimation from approximated SDE solutions. Parameter values found, are not only plausible but agree with theoretical considerations and empirical observations elaborated in this study. The third test relates to an approximated density function and its associated approximated maximum likelihood estimator. The Ait-Sahalia- method, in this study adapted to derive the Fourier coefficients (of the Hermite expansion) from a (closed) system of moment ODEs, is considered a superior technique to derive an approximated density function associated with the linear-quadratic model. The maximum likelihood technique employed, proper for high-parametrised estimations, includes re-parametrisation (based on the extended invariance principle) and stepwise maximisation. Reported estimation results support the hypothesised superiority of the linear-quadratic cash flow model, either in complete (five-parameter form) or in a reduced-parameter form, in comparison to the examined five benchmark processes.</p>

Stochastic continuous-time cash flows: A coupled linear-quadratic model

<p>The focal point of this dissertation is stochastic continuous-time cash flow models. These models, as underpinned by the results of this study, prove to be useful to describe the rich and diverse nature of trends and fluctuations in cash flow randomness. Firstly, this study considers an important preliminary question: can cash flows be fully described in continuous time? Theoretical and empirical evidence (e.g. testing for jumps) show that under some not too stringent regularities, operating cash flow processes can be well approximated by a diffusion equation, whilst investing processes -preferably- will first need to be rescaled by a system-size variable. Validated by this finding and supported by a multitude of theoretical considerations and statistical tests, the main conclusion of this dissertation is that an equation consisting of a linear drift function and a complete quadratic diffusion function (hereafter: “the linear-quadratic model”) is a specification preferred to other specifications frequently found in the literature. These so-called benchmark processes are: the geometric and arithmetic Brownian motions, the mean-reverting Vasicek and Cox, Ingersoll and Ross processes, and the modified Square Root process. Those specifications can all be considered particular cases of the generic linear-quadratic model. The linear-quadratic model is classified as a hybrid model since it is shown to be constructed from the combination of geometric and arithmetic Brownian motions. The linear-quadratic specification is described by a fundamental model, rooted in well-studied and generally accepted business and financial assumptions, consisting of two coupled, recursive relationships between operating and investing cash flows. The fundamental model explains the positive feedback mechanism assumed to exist between the two types of cash flows. In a stochastic environment, it is demonstrated that the linear-quadratic model can be derived from the principles of the fundamental model. There is no (known) general closed-form solution to the hybrid linear-quadratic cash flow specification. Nevertheless, three particular and three approximated exact solutions are derived under not too stringent parameter restrictions and cash flow domain limitations. Weak solutions are described by (forward or backward) Fokker-Planck- Kolmogorov equations. This study shows that since the process is converging in time (that is, approximating a stable probability distribution), (uncoupled) investing cash flows can be described by a Pearson diffusion process approaching a stationary Person-IV probability density function, more appropriately a Student diffusion process. In contrast, (uncoupled) operating cash flow processes are diverging in time, that is exploding with no stable probability density function, a dynamic analysis in a bounded cash flow domain is required. A suggested solution method normalises a general hypergeometric differential equation, after separation of variables, which is then transformed into a Sturm-Liouville specification, followed by a choice of three separate second transformations. These second transformations are the Jacobi, the Hermitian and the Schrödinger, each yielding a homonymous equation. Only the Jacobi transformation provides an exact solution, the other two transformations lead to approximated closed-form general solutions. It turns out that a space-time density function of operating cash flow processes can be construed as the multiplication of two (independent) time-variant probability distributions: a stationary family of distributions akin to Pearson’s case 2, and the evolution of a standard normal distribution. The fundamental model and the linear-quadratic specification are empirically validated by three different statistical tests. The first test provides evidence that the fundamental model is statistically significant. Parameter values support the conclusion that operating and investing processes are converging to overall long-term stable values, albeit with significant stochastic variation of individual firms around averages. The second test pertains to direct estimation from approximated SDE solutions. Parameter values found, are not only plausible but agree with theoretical considerations and empirical observations elaborated in this study. The third test relates to an approximated density function and its associated approximated maximum likelihood estimator. The Ait-Sahalia- method, in this study adapted to derive the Fourier coefficients (of the Hermite expansion) from a (closed) system of moment ODEs, is considered a superior technique to derive an approximated density function associated with the linear-quadratic model. The maximum likelihood technique employed, proper for high-parametrised estimations, includes re-parametrisation (based on the extended invariance principle) and stepwise maximisation. Reported estimation results support the hypothesised superiority of the linear-quadratic cash flow model, either in complete (five-parameter form) or in a reduced-parameter form, in comparison to the examined five benchmark processes.</p>

Free Educational Android Mobile Application for Radiobiology

The use of mobile devices and applications dedicated to different medical fields has improved the quality and facilitated medical care, especially in the last 10 years. The number of applications running on the software platforms of smart phones or other smart devices is constantly growing. Radiotherapy also benefits from applications (apps) for TNM staging of cancers, for target volume delineation and toxicity management but also from radiobiological apps for calculating equivalent dose schemes for different dose fractionation regimens. In the context of the increasingly frequent use of altered fractionation schemes, the use of radiobiological models and calculations based on the linear quadratic model (LQ) becomes a necessity. We aim to evaluate free radiobiology apps for the Android software platform. Given the global educational deficit, the lack of experts and the concordance between radiobiology education and the need to use basic clinical notions of modern radiotherapy, the existence of free apps for the Android platform running on older generation processors can transform even an old smart device in a powerful “radiobiology station.” Apps for radiobiology can help the radiation oncologist and medical physicist with responsibilities in radiotherapy treatment planning in the context of accelerated adoption of hypo-fractionation regimens and calculation of the effect of treatment gaps, a topic of interest in the COVID-19 pandemic context. Radiobiology apps can also partially fill the educational gap in radiobiology by arousing the interest of young radiation oncologists to deepen the growing universe of fundamental and clinical radiobiology.

Analytical solution to the radiotherapy fractionation problem including dose bound constraints

This paper deals with the classic radiotherapy dose fractionation problem for cancer tumors concerning the following goals: a) To maximize the effect of radiation on the tumor, restricting the effect produced to the organs at risk (healing approach). b) To minimize the effect of radiation on the organs at risk, while maintaining enough effect of radiation on the tumor (palliative approach). We will assume the linear-quadratic model to characterize the radiation effect and consider the stationary case (that is, without taking into account the timing of doses and the tumor growth between them). The main novelty with respect to previous works concerns the presence of minimum and maximum dose fractions, to achieve the minimum effect and to avoid undesirable side effects, respectively. We have characterized in which situations is more convenient the hypofractionated protocol (deliver few fractions with high dose per fraction) and in which ones the hyperfractionated regimen (deliver a large number of lower doses of radiation) is the optimal strategy. In all cases, analytical solutions to the problem are obtained in terms of the data. In addition, the calculations to implement these solutions are elementary and can be carried out using a pocket calculator.

High Precision Tracking of Roll-to-Roll Manufacturing Processes

Methods for controlling tension and velocity in Roll-to-Roll (R2R) processes have been widely studied in literature. Such methods are very effective under normal operating conditions, but the performance will degrade when control efforts saturate the physical capability of systems. This paper presents a constrained linear-quadratic model predictive control (LQ-MPC) scheme for the purpose of reference tracking. Firstly, a discrete-time linearized model for a R2R process is derived based on the governing equations from prior studies. The model is augmented by integrating an incremental behavior into it. Then a model predictive controller is specifically designed to reach zero-offset tracking of tension and velocity references, while the encountered process constraints are enforced at the same time. The constrained control problem is reduced into a quadratic programming problem and solved by using gradient projection method. It is proved that the proposed controller can guarantee local closed-loop stability where process constraints are inactive. Simulation of a five-roller R2R system is conducted to compare the performance of a proposed controller and a typical decentralized PI controller. Results show that the proposed controller achieves better performance in terms of quick response to changing of set points and capability of decoupling the subsystems. It also demonstrates good robustness when reasonable parametric uncertainties are introduced in the system.