Stochastic Calculus, a Practical Introduction, by Richard Durrett.
This is a clear discussion of one of the traditional rigorous
approaches to diffusions and the Ito calculus.
Stochastic Calculus and Financial Applications, by Michael Steele.
Clear and rigorous treatment of diffusions and the Ito calculus
with applications to derivative pricing.
The approach relies more on martingales, which are central to
much financial modeling.
Stochastic Differential Equations, by Bernt Oksendal.
An alternative to the book of Durrett that emphasizes the
relationship to partial differential equations.