Stochastic Integration

Abstract

This chapter is at the core of the present book. We start by defining the stochastic integral with respect to a square-integrable continuous martingale, considering first the integral of elementary processes (which play a role analogous to step functions in the theory of the Riemann integral) and then using an isometry between Hilbert spaces to deal with the general case. It is easy to extend the definition of stochastic integrals to continuous local martingales and semimartingales. We then derive the celebrated Itô’s formula, which shows that the image of one or several continuous semimartingales under a smooth function is still a continuous semimartingale, whose canonical decomposition is given in terms of stochastic integrals. Itô’s formula is the main technical tool of stochastic calculus, and we discuss several important applications of this formula, including Lévy’s theorem characterizing Brownian motion as a continuous local martingale with quadratic variation process equal to t, the Burkholder–Davis–Gundy inequalities and the representation of martingales as stochastic integrals in a Brownian filtration. The end of the chapter is devoted to Girsanov’s theorem, which deals with the stability of the notions of a martingale and a semimartingale under an absolutely continuous change of probability measure. As an application of Girsanov’s theorem, we establish the famous Cameron–Martin formula giving the image of the Wiener measure under a translation by a deterministic function.