For a jump process Xt given by:

where Jt denotes the cumulative effect of jumps over the interval [0, t], and Xc/t the diffusive part of the process, the It¯o integral for the function f(Xt) is given by [Privault (2009)]:

where s− denotes the time just prior to s. Use this result to show that, for the process given in question 2, the process for log(St) is given by:

Jt X = X + Je = X + " modW, + [f" pads + J 0