This text introduces at a moderate speed and in a thorough way the
basic concepts of the theory of stochastic integrals and Ito
calculus for sem i martingales. There are many reasons to study
this subject. We are fascinated by the contrast between general
measure theoretic arguments and concrete probabilistic problems,
and by the own flavour of a new differential calculus. For the
beginner, a lot of work is necessary to go through this text in
detail. As areward it should enable her or hirn to study more
advanced literature and to become at ease with a couple of
seemingly frightening concepts. Already in this introduction, many
enjoyable and useful facets of stochastic analysis show up. We
start out having a glance at several elementary predecessors of the
stochastic integral and sketching some ideas behind the abstract
theory of semimartingale integration. Having introduced martingales
and local martingales in chapters 2 - 4, the stochastic integral is
defined for locally uniform limits of elementary processes in
chapter S. This corresponds to the Riemann integral in
one-dimensional analysis and it suffices for the study of Brownian
motion and diffusion processes in the later chapters 9 and 12."
General
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