Giulia Di Nunno - Böcker

The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory.In recent years we have witnessed a remarkable convergence between individual mathematical disciplines that approach deterministic and stochastic dynamical systems from mathematical analysis, computational mathematics and control theoretical perspectives. Breakthrough developments in these fields now provide a common mathematical framework for attacking many different problems related to differential geometry, analysis and algorithms for stochastic and deterministic dynamics. In the Abel Symposium 2016, which took place from August 16-19 in Rosendal near Bergen, leading researchers in the fields of deterministic and stochastic differential equations, control theory, numerical analysis, algebra and random processes presented and discussed the current state of the art in these diverse fields.The current Abel Symposia volume may serve as a point of departure for exploring these related but diverse fields of research, as well as an indicator of important current and future developments in modern mathematics.

Financial markets have extremely complex behavior that cannot be fully modeled using classical approaches. In particular, numerous empirical studies show that market volatility exhibits some form of long-range dependence and has time-varying Hölder regularity with prominent periods of “roughness” (i.e., of Hölder order ≈ 0.1). These two properties are far beyond the capabilities of classical Brownian diffusions and it is challenging to reproduce them simultaneously in one model. In the existing literature, the phenomenons of long-range dependence and roughness mentioned above are often addressed by using fractional Brownian motion. However, in this case, these two features turn out to be mutually exclusive and cannot be grasped simultaneously. Furthermore, existing stochastic models based on fractional Brownian motion pose additional challenges of the technical kind: they tend to produce prices with moment explosions (and hence are not applicable to pricing some widespread derivatives); they may have volatilities that hit zero (or even become negative) which results in problems with transitioning between physical and pricing measures; they often lack efficient numerical algorithms for derivative pricing, hedging, etc. In this book, we introduce a novel class of stochastic processes driven by general Hölder noises that allows for a very broad flexibility in the noises (to account for both roughness and long-range dependence simultaneously) and grasps the unconventional behavior of market volatility. We also present a variety of associated numerical methods and propose practically feasible algorithms for various applications, such as pricing of derivatives (including options with discontinuous payoffs) and quadratic hedging.

These Proceedings offer a selection of peer-reviewed research and survey papers by some of the foremost international researchers in the fields of finance, energy, stochastics and risk, who present their latest findings on topical problems.

These Proceedings offer a selection of peer-reviewed research and survey papers by some of the foremost international researchers in the fields of finance, energy, stochastics and risk, who present their latest findings on topical problems.

Kiyosi Ito, the founder of stochastic calculus, is one of the few central figures of the twentieth century mathematics who reshaped the mathematical world. Today stochastic calculus is a central research field with applications in several other mathematical disciplines, for example physics, engineering, biology, economics and finance.The Abel Symposium 2005 was organized as a tribute to the work of Kiyosi Ito on the occasion of his 90th birthday. Distinguished researchers from all over the world were invited to present the newest developments within the exciting and fast growing field of stochastic analysis. The present volume combines both papers from the invited speakers and contributions by the presenting lecturers. A special feature is the Memoirs that Kiyoshi Ito wrote for this occasion. These are valuable pages for both young and established researchers in the field.

There are already several excellent books on Malliavin calculus. However, most of them deal only with the theory of Malliavin calculus for Brownian motion, with [35] as an honorable exception. Moreover, most of them discuss only the applicationto regularityresults for solutions ofSDEs, as this wasthe original motivation when Paul Malliavin introduced the in?nite-dimensional calculus in 1978 in [158]. In the recent years, Malliavin calculus has found many applications in stochastic control and within ?nance. At the same time, L' evy processes have become important in ?nancial modeling. In view of this, we have seen the need for a book that deals with Malliavin calculus for L' evy processesin general,not just Brownianmotion, and that presentssome of the most important and recent applications to ?nance. It is the purpose of this book to try to ?ll this need. In this monograph we present a general Malliavin calculus for L' evy processes, covering both the Brownianmotioncaseand the purejump martingalecasevia Poissonrandom measures,and also some combination of the two.