the Ito-Stratonovich conversion
/ EE-toh / strah-tuh-NOH-vich /
Since the Ito and Stratonovich integrals genuinely differ, you need a dictionary to translate between them — and you do, all the time, because models are written in one convention but analyzed (or simulated, or hedged) in the other. The Ito-Stratonovich conversion is that dictionary: it is an explicit formula relating the two integrals, and the difference is exactly one half of a quadratic covariation. Knowing it lets you take a physically-motivated Stratonovich SDE and rewrite it as an Ito SDE so you can use the martingale machinery (and vice versa).
At the integral level the conversion is integral_0^t H o dX = integral_0^t H dX + (1/2) [H, X]_t, where the circle is Stratonovich, the bare d is Ito, and [H, X] is the quadratic covariation of the integrand process H with the integrator X. The practically important form is at the SDE level. A Stratonovich SDE dX = b(X) dt + sigma(X) o dB is equivalent to the Ito SDE dX = (b(X) + (1/2) sigma(X) sigma'(X)) dt + sigma(X) dB; the extra drift term (1/2) sigma sigma' (the Ito-Stratonovich correction, also called the spurious or noise-induced drift) arises because, in the Stratonovich integral, the integrand sigma(X) is itself a function of the diffusing X, so [sigma(X), B] = sigma'(X) sigma(X) dt is nonzero. In several dimensions the correction becomes (1/2) sum_{j,k} sigma^{jk} partial_j sigma^{ik}, the same kind of trace-against-the-diffusion structure as in the multidimensional Ito formula. Going the other way (Ito to Stratonovich) you subtract the same term.
Why it matters in practice. The Wong-Zakai theorem says that an SDE driven by smooth approximations to white noise converges to the STRATONOVICH version, so physical and engineering models built from 'real', band-limited noise are naturally Stratonovich; but the martingale, no-arbitrage, and numerical (Euler-Maruyama) machinery is naturally Ito, so you convert. The two SDEs have the same solution paths only after adding the correction drift — they are NOT the same equation. The honest caveats: (1) The conversion requires sigma to be differentiable (the formula contains sigma'), whereas the Ito SDE itself can make sense for merely Lipschitz sigma — so a well-posed Ito SDE need not have a Stratonovich form at all. (2) The two conventions agree (zero correction) precisely when sigma is constant (additive noise), because then sigma' = 0; the discrepancy is a pure feature of multiplicative noise. (3) The correction is a drift, not a diffusion change: it shifts the mean behaviour, leaving the noise coefficient sigma the same.
Geometric Brownian motion is usually written in Ito form dS = mu S dt + sigma S dB. Its Stratonovich form is dS = (mu - sigma^2/2) S dt + sigma S o dB: the Stratonovich drift is smaller by (1/2) sigma sigma' = (1/2) sigma * sigma = sigma^2/2 (here the noise coefficient is sigma(S) = sigma S, so sigma'(S) = sigma). Equivalently, converting the Stratonovich equation back to Ito ADDS the sigma^2/2 term to the drift — the explicit dictionary in action.
Same diffusion coefficient, drift differs by (1/2) sigma sigma'; the conversion term is a pure multiplicative-noise effect, zero for additive noise.
The conversion needs sigma differentiable and only changes the DRIFT (by (1/2) sigma sigma'), never the diffusion; for additive (constant sigma) noise the two conventions coincide, so the Ito-vs-Stratonovich choice only matters with multiplicative noise.