Convert dX = a(X) dt + b(X) dW from Ito form to Stratonovich form. For b(x) = 0.3 x, compute the size of the drift adjustment at x = 50.

Convert dX = a(X) dt + b(X) dW from Ito form to Stratonovich form. For b(x) = 0.3 x, compute the size of the drift adjustment at x = 50.

Approach: Compare the left endpoint evaluation of the Ito integral with the midpoint evaluation, and express the difference as half the cross variation of the integrand with the driving process.

2.25. The Stratonovich integral uses midpoint evaluation of the integrand, so it exceeds the Ito integral by an Ito correction equal to half the cross variation of the integrand with the driver: ∫ b(X) o dW = ∫ b(X) dW + (1/2)[b(X), W]. Since d[b(X), W] = b'(X) b(X) dt, the Ito equation dX = a dt + b dW is the same process as dX = (a - (1/2) b b') dt + b o dW, so the drift adjustment is (1/2) b(x) b'(x). With b(x) = 0.3x, b'(x) = 0.3 and the adjustment is (1/2)(0.3)(50)(0.3) = 2.25. The Stratonovich form obeys the ordinary chain rule, which is why it is preferred in physical modelling, and the Ito form keeps the martingale property, which is why it is used in pricing. Neither form changes the process, only the bookkeeping of where the integrand is sampled.

Follow-up: Written in Stratonovich form, does the ordinary chain rule applied to ln S of a geometric Brownian motion reproduce the correct solution?

Key concepts: Stratonovich integral, Ito correction, chain rule, midpoint evaluation.