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This book presents innovations in the mathematical foundations of financial analysis and numerical methods for finance and applications to the modeling of risk. The topics selected include measures of risk, credit contagion, insider trading, information in finance, stochastic control and its applications to portfolio choices and liquidation, models of liquidity, pricing, and hedging. The models presented are based on the use of Brownian motion, Lévy processes and jump diffusions. Moreover, fractional Brownian motion and ambit processes are also introduced at various levels. The chosen blend of topics gives an overview of the frontiers of mathematics for finance. New results, new methods and new models are all introduced in different forms according to the subject. Additionally, the existing literature on the topic is reviewed. The diversity of the topics makes the book suitable for graduate students, researchers and practitioners in the areas of financial modeling and quantitative finance. The chapters will also be of interest to experts in the financial market interested in new methods and products. This volume presents the results of the European ESF research networking program Advanced Mathematical Methods for Finance.

Advanced Mathematical Methods for Finance

Dynamic risk measures.- Ambit processes and stochastic partial differential equations.- Fractional processes as models in stochastic finance.- Credit contagion in a long range dependent macroeconomic factor model.- Modeling information flows in financial markets.- Comonotonicity applied in finance.- A general maximum principle for anticipative stochastic control and applications to insider trading.- Analyticity of the Wiener-Hopf factors and valuation of exotic options in Levy models.- Optimal liquidation of a pairs trade.- A PDE-based approach or pricing mortgage-backed securities.- Nonparametric methods for pricing mortgage-backed securities.- Fractional smoothness and applications in finance.- Liquidity models in continuous and discrete times.- Some new BSDE results for an infinite-horizon stochastic control problem.- Functionals associated with gradient stochastic flows and nonlinear SPDEs.- Fractional smoothness and applications in Finance modeled by F-doubly stochastic Markov chains.- Exotic derivatives in dense class of stochastic volatility models with jumps.- Asymptotics of HARA utility from terminal wealth under proportional transaction costs with decision lag or execution delay and obligatory diversification.