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IAI Actuarial Core Principles · Economic Modelling · Stochastic models for security prices

In the Ito multiplication rules used with Ito's lemma, which set of results is correct for small dt?

The standard rules are (dW)² = dt, dW·dt = 0 and (dt)² = 0. Because the squared Brownian increment is of order dt, it contributes a second-order term in Ito's lemma, while the other products are negligible.

  1. A(dW)² = dt, dW·dt = 0, (dt)² = 0Correct
  2. B(dW)² = 0, dW·dt = dt, (dt)² = 0
  3. C(dW)² = dt, dW·dt = dt, (dt)² = dt
  4. D(dW)² = dt², dW·dt = 0, (dt)² = 0
  5. (dW)² = dt, dW·dt = 0, (dt)² = dt

Explanation

Brownian increments have variance dt, so (dW)² behaves like dt. Cross term dW·dt and (dt)² are of higher order and vanish. This yields the extra second-order term in Ito's lemma.

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