Stochastic Calculus Under Sublinear Expectation and Volatility Uncertainty

Christian Bannasch

• Publisher: GRIN Verlag
• Year: 2020
• eBook ISBN:
  9783346105240
• Edition: 1

Research Paper (postgraduate) from the year 2017 in the subject Mathematics - Stochastics, grade: 1,7, LMU Munich, language: English, abstract: Venture into the uncharted territories of probability with a groundbreaking exploration of stochastic calculus under model uncertainty, guided by the pioneering work of Shige Peng. This book unveils the profound implications of sublinear expectations, offering a revolutionary approach to financial modeling, risk management, and stochastic analysis where classical probabilistic assumptions falter. Discover how sublinear expectation spaces provide a robust framework for handling ambiguity and Knightian uncertainty, surpassing the limitations of traditional linear expectations. Delve into the heart of G-Brownian motion, a cornerstone of this theory, and witness its unique characteristics unfold, challenging conventional notions of Brownian motion. Unravel the intricacies of Itô's integral in the G-setting, extending its power to a broader class of stochastic processes and paving the way for innovative applications in areas such as derivative pricing and hedging under uncertainty. The book meticulously constructs the mathematical machinery required to navigate this new landscape, from the representation theorem for sublinear expectations to the completion of functional spaces, ensuring a rigorous and accessible treatment of the subject. Explore the concept of capacity related to G-Expectation, essential for understanding the behavior of G-Brownian motion. Witness the extension of Itô's integral to stopping times, further enhancing the applicability of the theory. Master the intricacies of Itô's formula in the context of G-Brownian motion, a vital tool for solving stochastic differential equations and analyzing complex systems. This book is not merely a theoretical treatise; it is a practical guide for researchers, practitioners, and graduate students seeking to push the boundaries of stochastic analysis and unlock the potential of sublinear expectations in a world of ever-increasing uncertainty. Keywords: sublinear expectation, G-Brownian motion, Itô's integral, G-normal distribution, stochastic calculus, probability model uncertainty, capacity, stopping times, Itô's formula. Prepare to challenge your assumptions and redefine your understanding of probability in the face of uncertainty. This exploration of stochastic calculus transcends traditional boundaries, providing a robust framework for navigating the complexities of modern finance and beyond. Unearth the potential of G-Brownian motion, delve into the intricacies of Itô's integral, and master the art of stochastic analysis in a world where uncertainty reigns supreme, bridging the gap between theoretical concepts and practical applications in finance and risk management, offering a powerful toolkit for tackling real-world problems where traditional probabilistic models fall short.