Stochastic Differential Equations, Backward SDEs, Partial Differential Equations

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Springer, Jun 24, 2014 - Mathematics - 667 pages

This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter.

Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has become an important subject of Mathematics and Applied Mathematics, because of its mathematical richness and its importance for applications in many areas of Physics, Biology, Economics and Finance, where random processes play an increasingly important role. One important aspect is the connection between diffusion processes and linear partial differential equations of second order, which is in particular the basis for Monte Carlo numerical methods for linear PDEs. Since the pioneering work of Peng and Pardoux in the early 1990s, a new type of SDEs called backward stochastic differential equations (BSDEs) has emerged. The two main reasons why this new class of equations is important are the connection between BSDEs and semilinear PDEs, and the fact that BSDEs constitute a natural generalization of the famous Black and Scholes model from Mathematical Finance, and thus offer a natural mathematical framework for the formulation of many new models in Finance.

 

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Contents

Background of Stochastic Analysis
1
Itôs Stochastic Calculus
72
Stochastic Differential Equations
135
SDEs with Multivalued Drift
228
Backward Stochastic Differential Equations
353
Annexes
516
References
659
Index
663
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About the author (2014)

Etienne Pardoux: Born in 1947, graduated from Ecole Polytechnique (1970), Thesis Univ. Paris-Sud 1975, CNRS Research Assoc. 1970-79, Maître de Conférences, then Professor Univ. d’Aix-Marseille since 1979. Member of the Institute for Advanced Study, Princeton NJ, 1986-1987. Member of the Institut Universitaire de France, 1992-2002. Pardoux has published more than 140 papers on nonlinear filtering, stochastic partial differential equations, anticipating stochastic calculus, backward stochastic differential equations, homogenization and probabilistic models in evolutionary biology, and three books.

Aurel Rascanu: Born 1950, Graduated from “Alexandru Ioan Cuza” University of Iasi (UAIC-Iasi), Romania (1974), Thesis UAIC-Iasi 1983, Assistant (1978-1985), Lecturer (1985-1990), Doctor (1990-1997), Professor (since 1997) at UAIC-Iasi, Dean of Faculty of Mathematics (1990-1992) and Head of the Department of Applied Mathematics (2000-2004) at UAIC Iasi. He has written 35 scientific papers. His scientific research is in stochastic differential equations, stochastic variational inequalities, approximation and numerical simulation, stochastic optimal control, viability and invariance, and probabilistic methods in the study of partial differential equations.

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