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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