This is an introduction to stochastic integration and stochastic
differential equations written in an understandable way for a wide
audience, from students of mathematics to practitioners in biology,
chemistry, physics, and finances. The presentation is based on the
naive stochastic integration, rather than on abstract theories of
measure and stochastic processes. The proofs are rather simple for
practitioners and, at the same time, rather rigorous for
mathematicians. Detailed application examples in natural sciences
and finance are presented. Much attention is paid to simulation
The topics covered include Brownian motion; motivation of
stochastic models with Brownian motion; Ito and Stratonovich
stochastic integrals, Ito's formula; stochastic differential
equations (SDEs); solutions of SDEs as Markov processes;
application examples in physical sciences and finance; simulation
of solutions of SDEs (strong and weak approximations). Exercises
with hints and/or solutions are also provided.
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