Stochastic Differential Equations, Backward Sdes, Partial Differential Equations
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Numar articol:205562581
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Specificatii
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. Ito 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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