  In this paper we derive novel change of variable formulas for stochastic
integrals w.r.t. a time-changed Brownian motion where we assume that the
time-change is a general increasing stochastic process with finitely many jumps
in a bounded set of the positive half-line and is independent of the Brownian
motion. As an application we consider the problem of maximising the expected
utility of the terminal wealth in a semimartingale setting, where the
semimartingale is written in terms of a time-changed Brownian motion and a
finite variation process. To solve this problem, we use an initial enlargement
of filtration and our change of variable formulas to shift the problem to a
maximisation problem under the enlarged filtration for models driven by a
Brownian motion and a finite variation process. The latter problem can be
solved by using martingale properties. Then applying again a change of variable
formula, we derive the optimal strategy for the original problem for a power
utility and for a logarithmic utility.
