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.
