the Stratonovich integral
/ strah-tuh-NOH-vich /
The Stratonovich integral is the alternative stochastic integral that obeys the ordinary chain rule of calculus, with no second-order correction term. It exists because, for an integrator of unbounded variation, the value of the integral genuinely depends on where in each subinterval you evaluate the integrand: the Ito integral uses the left endpoint, and the Stratonovich integral uses the midpoint (or the average of the endpoints). This single change of convention buys you classical calculus at the price of the martingale property.
The Stratonovich integral of H against a continuous semimartingale X, written integral_0^t H_s o dX_s (the circle distinguishes it from the Ito d), is defined as the limit of Riemann sums sum (H_{t_k} + H_{t_{k+1}})/2 * (X_{t_{k+1}} - X_{t_k}) using the symmetric (midpoint-average) evaluation. Equivalently, it is the Ito integral plus half the quadratic covariation of the integrand process and the integrator: integral_0^t H o dX = integral_0^t H dX + (1/2) [H, X]_t. The decisive consequence is the ordinary chain rule: for f in C^3 (or C^2 with mild conditions), df(X) = f'(X) o dX with NO (1/2) f'' term — the symmetric convention has absorbed the Ito correction into the definition of the integral itself. So Stratonovich calculus looks exactly like ordinary calculus, which is why it is the natural choice in physics and geometry, where coordinate-change invariance (the chain rule) is paramount.
Where it is used and what it costs. Stratonovich is preferred in three settings: modelling physical noise as a limit of smooth (correlated) noise, because the Wong-Zakai theorem says SDEs driven by smooth approximations of white noise converge to the STRATONOVICH SDE, not the Ito one; in stochastic differential geometry, where the chain rule must respect coordinate changes on a manifold; and wherever one wants intuition from ordinary calculus to transfer directly. The honest caveats: (1) The Stratonovich integral is NOT in general a martingale, even against Brownian motion — that is the price of the chain rule, and you lose the optional-stopping and isometry tools. (2) It requires the integrand H to itself be a semimartingale (so that [H, X] makes sense), a stronger requirement than the merely predictable, possibly very rough, integrands the Ito integral allows. (3) For modelling you must choose deliberately: Ito (left endpoint, non-anticipating, martingale, the right convention when increments represent genuine fresh randomness, as in finance) versus Stratonovich (midpoint, chain rule, the right convention as an idealization of smooth physical noise). They describe different models, related by the explicit conversion formula.
Compute integral_0^t B o dB. By the chain rule it should be B_t^2/2, just as for an ordinary integral of x dx. Check via the conversion: integral B o dB = integral B dB + (1/2)[B, B]_t = (B_t^2 - t)/2 + (1/2) t = B_t^2/2. The Ito version had the -t/2 correction; the Stratonovich version restores the naive answer — the (1/2)[B,B] = t/2 term exactly cancels the Ito correction.
Stratonovich gives the ordinary-calculus answer B^2/2; the midpoint convention absorbs the Ito correction.
The Stratonovich integral obeys the chain rule but is generally NOT a martingale, and it needs the integrand to be a semimartingale; choose Ito when you want the martingale/no-arbitrage structure and fresh-randomness modelling, Stratonovich when you want coordinate-invariant calculus or a smooth-noise limit.