Stopping time

Summary

In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time) is a specific type of “random time”: a random variable whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a stopping rule, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some finite time. Stopping times occur in decision theory, and the optional stopping theorem is an important result in this context. Stopping times are also frequently applied in mathematical proofs to “tame the continuum of time”, as Chung put it in his book (1982). Definition Discrete time Let \tau be a random variable, which is defined on the filtered probability space (\Omega, \mat

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An optimal prediction problem in financial modelling

Violetta Bernyk

The subject of the present thesis is an optimal prediction problem concerning the ultimate maximum of a stable Lévy process over a finite interval of time. Such "optimal prediction" problems are of both theoretical and practical interest, in particular they have applications in finance. For instance, suppose that an investor has a long position in one financial asset, whose price is modelled by some stochastic process. The investor's objective is to determine a "best moment" at which to close out the position and to sell the asset at the highest possible price. This optimal decision must be based on continuous observations of the asset price performance and only on the information accumulated to date. Hence, the investor should use a prediction (forecasting) of the future evolution of the price of the financial security. We examine this problem in the case where the asset price is modelled by a Lévy process. Indeed, during the last several years, the application of Lévy processes in the modelling financial asset returns has become one of the active research directions in quantitative finance. Thus, this thesis contains suitable new results concerning Lévy processes. We derive the law of the supremum process associated with a strictly stable Lévy process with no negative jumps which is not a subordinator. We note that the latter problem dates back to 1973. In particular, we show that the probability density function of the supremum process can be expressed using an explicit power series representation or via an integral representation. We also derive the infinitesimal generator of the reflected process associated with a general strictly stable Lévy process. Throughout this thesis, we apply the theory of optimal stopping, the methods of fractional differential calculus, and some results from fluctuation theory. Implementing these theories in the context of Lévy processes requires the development of specific analytical results. In the case where the asset price is modelled by a spectrally positive stable Lévy process, we describe the optimal strategy under certain conditions on the model parameters. The optimal strategy is of the following form: the investor must stop the observation of the price process and sell the asset as soon as the associated reflected process crosses for the first time a particular stopping boundary. We also provide numerical estimates and simulation examples of the results obtained by using this strategy.

EPFL2007

Brain-Machine interfaces aim to create a direct neural link between user's brain and machines. This goal has pushed scientists to investigate a large spectrum of applications in the realm of assistive and rehabilitation technologies. However, despite great progress, the possibility of being in full control of a device with our brain is still far to be reached. One of the primary limitations of BMIs is the notion of self-paced control, which states that a user should be able to activate or deactivate such interface by mere modulation of brain activity. Many studies have investigated this question, focusing notably on the use of endogenous signals (such as Event-Related Desynchronization/Synchronization, ERD/S) to control the activation of these interfaces in natural interaction scenarios. This question is, however, largely ignored when looking at their interruption. Hence, in this first part of the thesis, my work was devoted to investigate how a BMI user would be able to control the interruption of non-invasive BMIs based on Motor Imagery (MI), a paradigm promoting natural and endogenous signals that can be used for BMI control. I investigated the decoding of motor termination as well as its correlates characterized by the post-movement ERS phenomenon. Specifically, this part of the thesis aimed to study (i) the feasibility to decode motor termination from a specific neural correlate as well as the adaptation from the user in closed-loop scenarios, (ii) the benefits of using motor termination correlates to detect the stopping process, and (iii) the effect of BMI effectors such as an exoskeleton on the detection of motor termination correlates. The obtained results provide new insights into closed-loop decoding of motor termination. In particular, I show that BMI users exhibit an adaptation of their EEG correlates enabling them to have a reliable control when switching off a BMI in closed-loop scenarios. Second, I show that the decoding of motor termination is a particular process different from a resting state and, hence, should be decoded independently so as to achieve a faster and more robust detection. Finally, I investigate the use of BMI effectors with respect to the decoding of motor termination showing an effect of the effectors on the correlates of motor termination. However, due to the nature of these correlates, motor termination can still be reliably decoded with a similar latency. In the second part of my thesis, I also investigate how the interoceptive system affects BMI and particularly how breathing signals affect BMI based on MI. Breathing has been shown to have a key effect on numerous human functions, including, perceptual, cognitive and motor functions. However, currently not much is known about the role of respiratory signals in BMI and such signals have rather been considered as physiological noise. In my thesis, I evaluated how the breathing process affects the correlates of MI (ERD in mu and beta bands) as well as actual BMI performance. The results provide an extensive analysis regarding the effect of the breathing cycle specifically on mu-ERDs, showing stronger ERD during the late expiration phase. Moreover, I identified a link between breathing and BMI performance and propose that breathing signals are a valuable predictor for BMI performance highlighting the importance of monitoring such signals and, more generally, present the interoceptive system as a key component of motor preparation and motor imagery

EPFL2021

Time-independent feedbacks for the null controllability of homogeneous quasilinear hyperbolic systems in one dimensional space

Hoài-Minh Nguyên

We consider the null controllability of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks for the null-controllability for

C^1

-solutions at any time larger than the optimal time for the linear system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.

2020