Chapter 19: Stochastic Differential Equations. In the chapter mind map this icon labels Itô vs Stratonovich Integrals. The discussion below is excerpted and lightly edited from § Itô vs Stratonovich Integrals in Mathematics for AI and Machine Learning.

A Stratonovich SDE uses the Stratonovich integral, where the integrand is evaluated at the midpoint:

Key property: Stratonovich integrals satisfy the ordinary chain rule, making them more intuitive for some applications. However, they are not adapted (they "look into the future"), which makes them less suitable for modeling.

the book figure illustrates the difference between Itô and Stratonovich integration for a single time step. The black curve shows a highly volatile stochastic path $X_t$ over the interval $[t, t+\Delta t]$. The blue rectangle represents the Itô integral, which evaluates the integrand at the left endpoint $X_t$ (shown as a blue dot). The red dashed rectangle represents the Stratonovich integral, which evaluates the integrand at the midpoint $\frac{X_t + X_{t+\Delta t}}{2}$ (shown as a red dot). Because the path is so volatile, the choice of evaluation point significantly changes the area under the curve. This difference is exactly the Itô correction term that appears when converting between Itô and Stratonovich SDEs. The visualization makes concrete why stochastic integrals require careful definition: unlike smooth functions where the choice of evaluation point doesn't matter, the infinite variation of Brownian motion makes this choice crucial.