For continuous semimartingales, the Stratonovich integral is defined by , where the first term is an Itô integral and the correction is quadratic covariation. It obeys the ordinary chain rule and is the limit in probability of symmetric endpoint sums.
For a continuous semimartingale and a sufficiently smooth , the Stratonovich integral obeys . In general is a semimartingale, so the definition must allow this larger class of integrands.
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The Stratonovich integral is a type of stochastic integral used in the theory of stochastic calculus, particularly in the context of stochastic differential equations (SDEs). It is named after the Russian mathematician Rostislav Stratonovich. The Stratonovich integral is specifically designed to handle the integration of stochastic processes where the integrators are often modeled as continuous-time martingales or Wiener processes (Brownian motion).