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What is high dimensional probability? Under this broad name we
collect topics with a common philosophy, where the idea of high
dimension plays a key role, either in the problem or in the methods
by which it is approached. Let us give a specific example that can
be immediately understood, that of Gaussian processes. Roughly
speaking, before 1970, the Gaussian processes that were studied
were indexed by a subset of Euclidean space, mostly with dimension
at most three. Assuming some regularity on the covariance, one
tried to take advantage of the structure of the index set. Around
1970 it was understood, in particular by Dudley, Feldman, Gross,
and Segal that a more abstract and intrinsic point of view was much
more fruitful. The index set was no longer considered as a subset
of Euclidean space, but simply as a metric space with the metric
canonically induced by the process. This shift in perspective
subsequently lead to a considerable clarification of many aspects
of Gaussian process theory, and also to its applications in other
settings.
What is high dimensional probability? Under this broad name we
collect topics with a common philosophy, where the idea of high
dimension plays a key role, either in the problem or in the methods
by which it is approached. Let us give a specific example that can
be immediately understood, that of Gaussian processes. Roughly
speaking, before 1970, the Gaussian processes that were studied
were indexed by a subset of Euclidean space, mostly with dimension
at most three. Assuming some regularity on the covariance, one
tried to take advantage of the structure of the index set. Around
1970 it was understood, in particular by Dudley, Feldman, Gross,
and Segal that a more abstract and intrinsic point of view was much
more fruitful. The index set was no longer considered as a subset
of Euclidean space, but simply as a metric space with the metric
canonically induced by the process. This shift in perspective
subsequently lead to a considerable clarification of many aspects
of Gaussian process theory, and also to its applications in other
settings.
Taking continuous-time stochastic processes allowing for jumps as
its starting and focal point, this book provides an accessible
introduction to the stochastic calculus and control of
semimartingales and explains the basic concepts of Mathematical
Finance such as arbitrage theory, hedging, valuation principles,
portfolio choice, and term structure modelling. It bridges thegap
between introductory texts and the advanced literature in the
field. Most textbooks on the subject are limited to diffusion-type
models which cannot easily account for sudden price movements. Such
abrupt changes, however, can often be observed in real markets. At
the same time, purely discontinuous processes lead to a much wider
variety of flexible and tractable models. This explains why
processes with jumps have become an established tool in the
statistics and mathematics of finance. Graduate students,
researchers as well as practitioners will benefit from this
monograph.
Taking continuous-time stochastic processes allowing for jumps as
its starting and focal point, this book provides an accessible
introduction to the stochastic calculus and control of
semimartingales and explains the basic concepts of Mathematical
Finance such as arbitrage theory, hedging, valuation principles,
portfolio choice, and term structure modelling. It bridges thegap
between introductory texts and the advanced literature in the
field. Most textbooks on the subject are limited to diffusion-type
models which cannot easily account for sudden price movements. Such
abrupt changes, however, can often be observed in real markets. At
the same time, purely discontinuous processes lead to a much wider
variety of flexible and tractable models. This explains why
processes with jumps have become an established tool in the
statistics and mathematics of finance. Graduate students,
researchers as well as practitioners will benefit from this
monograph.
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