Problems in Mathematical Finance Related to Transaction Costs and Model Uncertainty.

This thesis is devoted to the study of three problems in mathematical finance which involve either transaction costs or model uncertainty or both. In Chapter II, we investigate the Fundamental Theorem of Asset Pricing (FTAP) under both transaction costs and model uncertainty, where model uncertainty is described by a family of probability measures, possibly non-dominated. We first show that the recent results on the FTAP and the super-hedging theorem in the context of model uncertainty can be extended to the case where only options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. Next, we look at the more difficult case where the market consists of a money market and a dynamically traded stock with bid-ask spread. Under a continuity assumption, we prove using a backward-forward scheme that no-arbitrage is equivalent to the existence of a suitable family of consistent price systems. In Chapter III, we study the problem where an individual targets at a given consumption rate, invests in a risky financial market, and seeks to minimize the probability of lifetime ruin under drift uncertainty. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation. In Chapter IV, we adapt stochastic Perron's method to the lifetime ruin problem under proportional transaction costs which can be formulated as a singular stochastic control problem. Without relying on the Dynamic Programming Principle, we characterize the value function as the unique viscosity solution of an associated variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111560/1/yuchong_1.pd

A note on the Fundamental Theorem of Asset Pricing under model uncertainty

We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of Asset Pricing and the super-hedging theorem can be extended to the case in which the options available for static hedging (\emph{hedging options}) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of \emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are \emph{non-redundant}. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.Comment: Final version. To appear in Risk

From Industry to Practice: Can Users Tackle Domain Tasks with Augmented Reality?

Augmented Reality (AR) is a cutting-edge interactive technology. While Virtual Reality (VR) is based on completely virtual and immersive environments, AR superimposes virtual objects onto the real world. The value of AR has been demonstrated and applied within numerous industrial application areas due to its capability of providing interactive interfaces of visualized digital content. AR can provide functional tools that support users in undertaking domain-related tasks, especially facilitating them in data visualization and interaction by jointly augmenting physical space and user perception. Making effective use of the advantages of AR, especially the ability which augment human vision to help users perform different domain-related tasks is the central part of my PhD research.Industrial process tomography (IPT), as a non-intrusive and commonly-used imaging technique, has been effectively harnessed in many manufacturing components for inspections, monitoring, product quality control, and safety issues. IPT underpins and facilitates the extraction of qualitative and quantitative data regarding the related industrial processes, which is usually visualized in various ways for users to understand its nature, measure the critical process characteristics, and implement process control in a complete feedback network. The adoption of AR in benefiting IPT and its related fields is currently still scarce, resulting in a gap between AR technique and industrial applications. This thesis establishes a bridge between AR practitioners and IPT users by accomplishing four stages. First of these is a need-finding study of how IPT users can harness AR technique was developed. Second, a conceptualized AR framework, together with the implemented mobile AR application developed in an optical see-through (OST) head-mounted display (HMD) was proposed. Third, the complete approach for IPT users interacting with tomographic visualizations as well as the user study was investigated.Based on the shared technologies from industry, we propose and examine an AR approach for visual search tasks providing visual hints, audio hints, and gaze-assisted instant post-task feedback as the fourth stage. The target case was a book-searching task, in which we aimed to explore the effect of the hints and the feedback with two hypotheses: that both visual and audio hints can positively affect AR search tasks whilst the combination outperforms the individuals; that instant post-task feedback can positively affect AR search tasks. The proof-of-concept was demonstrated by an AR app in an HMD with a two-stage user evaluation. The first one was a pilot study (n=8) where the impact of the visual hint in benefiting search task performance was identified. The second was a comprehensive user study (n=96) consisting of two sub-studies, Study I (n=48) and Study II (n=48). Following quantitative and qualitative analysis, our results partially verified the first hypothesis and completely verified the second, enabling us to conclude that the synthesis of visual and audio hints conditionally improves AR search task efficiency when coupled with task feedback

The Magic of Vision: Understanding What Happens in the Process

How important is the human vision? Simply speaking, it is central for domain\ua0related users to understand a design, a framework, a process, or an application\ua0in terms of human-centered cognition. This thesis focuses on facilitating visual\ua0comprehension for users working with specific industrial processes characterized\ua0by tomography. The thesis illustrates work that was done during the past two\ua0years within three application areas: real-time condition monitoring, tomographic\ua0image segmentation, and affective colormap design, featuring four research papers\ua0of which three published and one under review.The first paper provides effective deep learning algorithms accompanied by\ua0comparative studies to support real-time condition monitoring for a specialized\ua0microwave drying process for porous foams being taken place in a confined chamber.\ua0The tools provided give its users a capability to gain visually-based insights\ua0and understanding for specific processes. We verify that our state-of-the-art\ua0deep learning techniques based on infrared (IR) images significantly benefit condition\ua0monitoring, providing an increase in fault finding accuracy over conventional\ua0methods. Nevertheless, we note that transfer learning and deep residual network\ua0techniques do not yield increased performance over normal convolutional neural\ua0networks in our case.After a drying process, there will be some outputted images which are reconstructed\ua0by sensor data, such as microwave tomography (MWT) sensor. Hence,\ua0how to make users visually judge the success of the process by referring to the\ua0outputted MWT images becomes the core task. The second paper proposes an\ua0automatic segmentation algorithm named MWTS-KM to visualize the desired low\ua0moisture areas of the foam used in the whole process on the MWT images, effectively\ua0enhance users\u27understanding of tomographic image data. We also prove its\ua0performance is superior to two other preeminent methods through a comparative\ua0study.To better boost human comprehension among the reconstructed MWT image,\ua0a colormap deisgn research based on the same segmentation task as in the second\ua0paper is fully elaborated in the third and the fourth papers. A quantitative\ua0evaluation implemented in the third paper shows that different colormaps can\ua0influence the task accuracy in MWT related analytics, and that schemes autumn,\ua0virids, and parula can provide the best performance. As the full extension of\ua0the third paper, the fourth paper introduces a systematic crowdsourced study,\ua0verifying our prior hypothesis that the colormaps triggering affect in the positiveexciting\ua0quadrant in the valence-arousal model are able to facilitate more precise\ua0visual comprehension in the context of MWT than the other three quadrants.\ua0Interestingly, we also discover the counter-finding that colormaps resulting in\ua0affect in the negative-calm quadrant are undesirable. A synthetic colormap design\ua0guideline is brought up to benefit domain related users.In the end, we re-emphasize the importance of making humans beneficial in every\ua0context. Also, we start walking down the future path of focusing on humancentered\ua0machine learning(HCML), which is an emerging subfield of computer\ua0science which combines theexpertise of data-driven ML with the domain knowledge\ua0of HCI. This novel interdisciplinary research field is being explored to support\ua0developing the real-time industrial decision-support system

A Mean Field Game of Sequential Testing

We introduce a mean field game for a family of filtering problems related to the classic sequential testing of the drift of a Brownian motion. To the best of our knowledge this work presents the first treatment of mean field filtering games with stopping and an unobserved common noise in the literature. We show that the game is well-posed, characterize the solution, and establish the existence of an equilibrium under certain assumptions. We also perform numerical studies for several examples of interest.Comment: 51 pages, 3 figure