Stratonovich integral

Remarks

Equivalently, it is the limit in probability of the sums $\sum_i \tfrac12 (Y_{t_i} + Y_{t_{i+1}})(X_{t_{i+1}} - X_{t_i})$ over partitions $0 = t_0 < \cdots < t_n = t$ whose mesh tends to $0$. Example: let $B$ be a Brownian motion with $B_0 = 0$. Then $\int_0^t B_s \, dB_s = \tfrac12 (B_t^2 - t)$ and $[B, B]_t = t$, so $\int_0^t B_s \circ dB_s = \tfrac12 B_t^2$. Non-example: the left-point sums $\sum_i B_{t_i} (B_{t_{i+1}} - B_{t_i})$ converge to $\tfrac12 (B_t^2 - t)$, not to the Stratonovich integral, because they omit the covariation term. For $f$ three times continuously differentiable, the chain rule takes the classical form $f(X_t) = f(X_0) + \int_0^t f'(X_s) \circ dX_s$. The notation $\circ \, dX$, with a small circle, marks the Stratonovich integral.