It is well-known that modern stochastic calculus has been exhaustively developed under usual conditions. Despite such a well-developed theory, there is evidence to suggest that these very convenient technical conditions cannot necessarily be fulfilled in real-world applications. Optional Processes: Theory and Applications seeks to delve into the existing theory, new developments and applications of optional processes on "unusual" probability spaces. The development of stochastic calculus of optional processes marks the beginning of a new and more general form of stochastic analysis. This book aims to provide an accessible, comprehensive and up-to-date exposition of optional processes and their numerous properties. Furthermore, the book presents not only current theory of optional processes, but it also contains a spectrum of applications to stochastic differential equations, filtering theory and mathematical finance. Features Suitable for graduate students and researchers in mathematical finance, actuarial science, applied mathematics and related areas Compiles almost all essential results on the calculus of optional processes in unusual probability spaces Contains many advanced analytical results for stochastic differential equations and statistics pertaining to the calculus of optional processes Develops new methods in finance based on optional processes such as a new portfolio theory, defaultable claim pricing mechanism, etc.

It is well-known that modern stochastic calculus has been exhaustively developed under usual conditions. Despite such a well-developed theory, there is evidence to suggest that these very convenient technical conditions cannot necessarily be fulfilled in real-world applications. Optional Processes: Theory and Applications seeks to delve into the existing theory, new developments and applications of optional processes on "unusual" probability spaces. The development of stochastic calculus of optional processes marks the beginning of a new and more general form of stochastic analysis. This book aims to provide an accessible, comprehensive and up-to-date exposition of optional processes and their numerous properties. Furthermore, the book presents not only current theory of optional processes, but it also contains a spectrum of applications to stochastic differential equations, filtering theory and mathematical finance. Features Suitable for graduate students and researchers in mathematical finance, actuarial science, applied mathematics and related areas Compiles almost all essential results on the calculus of optional processes in unusual probability spaces Contains many advanced analytical results for stochastic differential equations and statistics pertaining to the calculus of optional processes Develops new methods in finance based on optional processes such as a new portfolio theory, defaultable claim pricing mechanism, etc.

The Egyptian revolution was a spark that enlightened the social media in a comprehensive manner among the nation, therefore online platforms are considered to be a marvelous opportunity to speed up the role of marketing in any start-up pharmacy. The objective of this research was to present a guideline for the Egyptian retail pharmacies on how to implement social media to grow in an online interactive market. Manifest anxieties for instance, concerns of information, safety over the social platforms, slow adaptation of technology by drug stores, inappropriate announcements generated by the customers, and the lack of privacy and security concerning the patients' health related information throughout the pharmacists' interaction presented dilemmas that required further careful handling for the success of the business results. Social marketing methodologies necessitated that the social platforms have to be set based on certain requirements, monitored and measured by suitable tools to be able to control the strategy performance and also to improve the process of planning. Undoubtedly, social media marketing has proven to be a better approach compared to traditional marketing. The market research is covered by the marketing facts and target groups that have brought to light the high prospects to establish and expand a pharmaceutical social media strategy in Egypt.