This is a concise introduction to stochastic calculus with some of its applications in mathematical finance, engineering and the sciences. Applications in finance include pricing of financial derivatives, such as options on stocks, exotic options and interest rate options. The filtering problem and its solution is presented as an application in engineering. Population models and randomly perturbed equations of physics are given as examples of applications in biology and physics. Only a basic knowledge of calculus and probability is required for reading this book. The text gradually takes the reader from a fairly low technical level to a sophisticated one. Heuristic arguments are often given before precise results are stated, and many ideas are illustrated by worked-out examples. Exercises are provided at the end of chapters to help test the readers' understanding.
Preliminaries from calculus; concepts of probability theory; basic stochastic processes; Brownian motion calculus; stochastic differential equations; diffusion processes; martingales; calculus for semimartingales; pure jump processes; change of probability measure; applications in finance; applications in biology; applications in engineering and physics.