Recall that Ito’s lemma expresses a twice differentiable function applied to a continuous semimartingale in terms of stochastic integrals, according to the following formula

(1) |

In this form, the result only applies to continuous processes but, as I will show in this post, it is possible to generalize to arbitrary noncontinuous semimartingales. The result is also referred to as Ito’s lemma or, to distinguish it from the special case for continuous processes, it is known as the *generalized Ito formula* or generalized Ito’s lemma.

If equation (1) is to be extended to noncontinuous processes then, there are two immediate points to be considered. The first is that if the process is not continuous then it need not be a predictable process, so need not be predictable either. So, the integrands in (1) will not be -integrable. To remedy this, we should instead use the left limits in the integrands, which is left-continuous and adapted and therefore is predictable. The second point is that the jumps of the left hand side of (1) are equal to and, on the right, they are . There is no reason that these should be equal, and (1) cannot possibly hold in general. To fix this, we can simply add on the correction to the jump terms on the right hand side,

(2) |