Control theory provides a large set of theoretical and computational tools
with applications in a wide range of fields, running from ”pure” branches of
mathematics, like geometry, to more applied areas where the objective is to
find solutions to ”real life” problems, as is the case in robotics, control of
industrial processes or finance.
The ”high tech” character of modern business has increased the need for
advanced methods. These rely heavily on mathematical techniques and seem
indispensable for competitiveness of modern enterprises. It became essential
for the financial analyst to possess a high level of mathematical skills. Conversely,
the complex challenges posed by the problems and models relevant to
finance have, for a long time, been an important source of new research topics
for mathematicians.
The use of techniques from stochastic optimal control constitutes a well
established and important branch of mathematical finance. Up to now, other
branches of control theory have found comparatively less application in financial
problems.
To some extent, deterministic and stochastic control theories developed as
different branches of mathematics. However, there are many points of contact
between them and in recent years the exchange of ideas between these fields
has intensified. Some concepts from stochastic calculus (e.g., rough paths)
have drawn the attention of the deterministic control theory community. Also,
some ideas and tools usual in deterministic control (e.g., geometric, algebraic
or functional-analytic methods) can be successfully applied to stochastic control.