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Stochastic Differential Equations
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| Description |
| This lecture is designed for the persons who want to develop p
rofessional skill in stochastic cal-
culus and its application to problems in finance but don’t inv
olve in too much mathematics. As
well-known, the more experience we have with mathematical s
kills (especially real analysis and
probability), the more piece we can get. Nevertheless, the c
ourse will move quickly and students
can expect to acquire tools that are deep enough and rich enou
gh to be relied upon throughout
their professional careers.The course begins with basic concepts and theory of probabil
ity space,
needed in this lecture. This material is used to motivate the
theory of continuous time
stochastic processes, especially Brownian motion. The con
struction of Brownian motion is given
in detail, and enough material on the subtle properties of Br
ownian paths is developed so that
the student should evolve a good sense of when intuition can b
e trusted and when it cannot.
The course then takes up the Itô integral and aims to provide a
development that is honest and
complete without being pedantic. With the Itô integral in ha
nd, the course focuses more on
models. Stochastic processes of importance in Finance and E
conomics are developed in concert
with the tool. The financial notion of replication is develop
ed, and the Black-Scholes PDE will
be introduced with its importance. At the last part of this co
urse, we introduce enough of the
theory of the diffusion equation to be able to solve the Black-
Scholes PDE and prove the
uniqueness of the solution. |
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