Stratonovich

Stratonovich refers to a specific interpretation of stochastic calculus, particularly the Itō calculus. This interpretation, named after Ruslan Stratonovich, is an alternative to the more commonly used Itō calculus. The key difference lies in how the stochastic integral is defined. In the Stratonovich interpretation, the integrand is evaluated at the midpoint of the time interval, whereas in the Itō interpretation, it's evaluated at the beginning of the interval. This seemingly small difference leads to different rules for stochastic differentiation and integration.

A significant advantage of the Stratonovich calculus is that it preserves the ordinary rules of calculus, such as the chain rule. This makes it easier to work with when transforming stochastic differential equations. For example, if one has a function of a stochastic process defined by a Stratonovich equation, applying the chain rule directly yields the correct result. In contrast, the Itō calculus requires an additional "Itō correction term" when applying the chain rule, which can be a source of error for those accustomed to deterministic calculus.

Despite its theoretical advantages, the Itō calculus is often preferred in fields like finance because it results in martingales, a property that is crucial for certain pricing and hedging strategies. The Stratonovich interpretation is more commonly found in physics and engineering, where it can arise naturally from physical models involving noise that is not perfectly white. Ultimately, the choice between Itō and Stratonovich depends on the specific problem and the desired mathematical properties. One can often convert between the two interpretations by adding a correction term.