An Introduction to Stochastic Differential Equations
by Lawrence C. Evans · American Mathematical Society
Short 151-page text grown from Berkeley undergraduate lectures. Covers a probability crash course, Brownian motion and white noise, Itô integrals and Itô's formula, existence and uniqueness for SDEs, and applications to PDEs, stopping problems and option pricing. The standard short primer before Øksendal for readers who want the mathematics rather than the finance.
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More resources on Stochastic Calculus
Almost Sure: Stochastic Calculus Notes
Free series of rigorous blog lectures building general stochastic integration from filtrations and martingales. Covers quadratic variation, Itô's lemma, SDEs, semimartingales, Girsanov transforms, local times and Lévy processes, with worked counterexamples. The best free, rigorous treatment of general stochastic integration online. Many readers use it as a companion to Le Gall or Protter. It assumes measure-theoretic probability.
Brownian Motion, Martingales, and Stochastic Calculus
Rigorous graduate text (Springer GTM 274) on continuous-time stochastic calculus. Covers Gaussian processes, Brownian motion, continuous semimartingales, stochastic integration, Girsanov, Markov processes, SDEs and their links to PDEs, and local times. Endpoint for readers who want the full measure-theoretic theory. It is shorter and more modern than Karatzas & Shreve and is the assigned text at MIT's graduate stochastic calculus course.
Stochastic Differential Equations: An Introduction with Applications (6th ed.)
Classic graduate introduction to Itô calculus and SDEs. Chapters cover the Itô integral, Itô formula, martingale representation, diffusions, Girsanov's theorem, filtering (Kalman-Bucy), optimal stopping, stochastic control and mathematical finance. The sixth edition adds solutions to many exercises. It covers the filtering application, which Shreve does not. It is the standard second book after Shreve or Evans.