Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I

The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in \cite{ETZ2}. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equation.Comment: 42 page

Optimal Stopping under Nonlinear Expectation

Let $X$ be a bounded c\`adl\`ag process with positive jumps defined on the canonical space of continuous paths. We consider the problem of optimal stopping the process $X$ under a nonlinear expectation operator \cE defined as the supremum of expectations over a weakly compact family of nondominated measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an \cE-supermartingale, and an \cE-martingale up to its first hitting time of the obstacle $X$. This result is obtained under an additional uniform continuity property of $X$. We also extend the result in the context of a random horizon optimal stopping problem. This result is crucial for the newly developed theory of viscosity solutions of path-dependent PDEs as introduced in Ekren et al., in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren, Touzi, and Zhang, parts I and II).Comment: 36 page

Critical Literacy: Conflicts, Challenges, Adaptations and Transformation?

This case study explored how the involvement of two ESL instructors in critical literacy research, including master’s thesis, made them experience different challenges, ideological conflicts and successes. One of them was teaching ESL in secondary classes when she carried out her thesis about critical literacy. She also cooperated with the researcher in a critical literacy study after she had finished data collection for her thesis. The other participant taught ESL in elementary classes and is currently teaching at universities. The study, which took place in Lebanon, revealed how the varied ideological positions, views and contexts of the two instructors made them go through different transformations. The data suggests that involving teachers and masters’ students in critical literacy research constitutes and important platform to train them in the various complex dimensions of critical literacy, particularly in settings where this approach faces significant resistance

Can Cognitive Strategies be Incorporated in a Critical Reading Course?

This study examined the effect of training in cognitive reading strategies in the context of critical literacy. Thirty students enrolled in a critical analysis course at a Lebanese university participated in the study. For three months, the participants received instruction based on a critical literacy model during which they were trained in summarizing, paraphrasing, using key words and discourse markers to find details and guessing meaning from context so that they comprehend the texts and read them critically. The Mixed Research Method was used in the study. In the quantitative part, the One-group, Posttest Pretest design was employed. For qualitative data, all the participants filled out a questionnaire. Eleven participants were interviewed, before which they filled out a survey. Also some class interaction was documented. The study also examined how students engaged texts critically, but this data was not presented in this report, which only discusses the results related to the targeted reading strategies. Statistical analysis showed a significant improvement in paraphrasing and summarizing but not in answering detail questions and guessing meaning from context. The qualitative data revealed interesting insights into why students did well in the first two strategies and badly in the last two ones

Politics and Identity in an EFL Class: Did You Say a Suicide Bomb Attack?

This article explores the possibility of critical analysis of terrorist suicide bomb attacks in an EFL critical literacy course. It reports on a part of a broader study that explored how 24 EFL high school Lebanese students were engaged in critical literacy activities during the whole 2013-2014 academic year. Most participants belonged to Lebanese communities that support Hezbollah, a Lebanese political party that plays an important role in the regional politics of the Middle East. The school was also located in an area dwelled mainly by supporters of this party. The participants wrote critical analysis essays about a double suicide bomb attack that took place close to their school, were involved in a focus group and were interviewed, all of which constituted the data. The study revealed that critical literacy events around hot political topics of significance to the students stimulate the students’ desires to learn the target language and give rise to complex discourses that reflect their multiple identities. In addition, critical analysis involved students in reconsidering their ideas and their ways of reacting to critical militarized political events

Critical Literacy: Performance and Reactions

This study examined how the socio-political and socio-cultural backgrounds of Lebanese university students affected their critical engagement with texts and how these students reacted to critical reading. It also investigated the effect of critical reading instruction on the participants’ performance in critical text analysis. It was a part of a broader research that employed the mixed research method. Twenty one participants majoring in Teaching English to Elementary Students at a university in Lebanon received instruction based on a critical reading model for three months. All the participants filled out a questionnaire. Eleven participants were interviewed, before which they filled out a survey. Also some class interaction was documented. Moreover, the One-group, Posttest Pretest design was employed. The data showed how the participants’ socio-political and religious backgrounds affected their critical engagement with texts. Some participants were less critical or more critical depending on their ideological positions towards the topics of the texts. Interestingly, most participants said that they enjoyed questioning the texts’ assumptions although it annoyed them that this questioning made them contemplate their long-held beliefs.  In addition, T-test and Mann-whitney test showed a significant improvement in critical text analysis at the end of the course. However, a few participants resisted critical reading

On viscosity solutions of path dependent PDEs

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman-Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional It\^{o} calculus recently introduced by Dupire [Functional It\^{o} calculus (2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/12-AOP788 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

ANALISIS KEMAMPUAN LITERASI MATEMATIS SISWA KELAS VIII MTs NEGERI KUPANG

This research aims to find out the mathematical literacy skills of grade VIII students of MTs Kupang State School Year 2019/2020. This type of research is descriptive research with a qualitative approach. The subjects of the study were 30 students grade  MTs Kupang State School Year 2019/2020. The object of this research is the mathematical literacy ability of students grade  MTs Kupang State School Year 2019/2020. Data collection techniques with tests using essay tests. Instruments used in research are essay type tests, interviews and documentation. The data analysis techniques used are three, namely, data reduction, data presentation, and drawing conclusions. Test the validity of the data in this study using triangulation techniques. From the results of the study shows that the results of the calculation of the percentage of students who are able to reach KKM as much as 40% while students who do not reach KKM as much as 60%, thus it is concluded that the mathematical literacy ability of grade  MTs Kupang state is still low because in this class there are still many students who are weak on formulating and planning strategies in solving problems