Stochastic calculus

The Itô integral is central to the study of stochastic calculus. The integral \int H\,dX is defined for a semimartingale X and locally bounded predictable process H. == Stratonovich integral ==

The Stratonovich integral or Fisk–Stratonovich integral of a semimartingale X against another semimartingale Y can be defined in terms of the Itô integral as :\int_0^t X_{s-} \circ d Y_s : = \int_0^t X_{s-} d Y_s + \frac{1}{2} \left [ X, Y\right]_t^c, where [X, Y]tc denotes the optional quadratic covariation of the continuous parts of X and Y, which is the optional quadratic covariation minus the jumps of the processes X and Y, i.e. :\left [ X, Y\right]_t^c:= [X,Y]_t - \sum\limits_{s\leq t}\Delta X_s\Delta Y_s. The alternative notation :\int_0^t X_s \, \partial Y_s is also used to denote the Stratonovich integral. == Applications ==

An important application of stochastic calculus is in mathematical finance, in which asset prices are often assumed to follow stochastic differential equations. For example, the Black–Scholes model prices options as if they follow a geometric Brownian motion, illustrating the opportunities and risks from applying stochastic calculus. == Stochastic integrals ==

Besides the classical Itô and Fisk–Stratonovich integrals, many other notions of stochastic integrals exist, such as the Hitsuda–Skorokhod integral, the Marcus integral, and the Ogawa integral. ==See also==