Utility maximisation and time-change

TL;DR

This paper addresses the problem of maximizing expected utility from terminal wealth in a semimartingale framework involving time-changed Brownian motion, providing new change of variable formulas and optimal strategies.

Contribution

It introduces change of variable formulas for stochastic integrals w.r.t. time-changed Brownian motion, enabling the solution of utility maximization problems in complex semimartingale models.

Findings

Derived change of variable formulas for stochastic integrals.

Reduced the utility maximization problem to a simpler setting.

Provided explicit optimal strategies under certain conditions.

Abstract

We consider the problem of maximising expected utility from terminal wealth in a semimartingale setting, where the semimartingale is written as a sum of a time-changed Brownian motion and a finite variation process. To solve this problem, we consider an initial enlargement of filtration and we derive change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion. The change of variable formulas allow us 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 could be solved by using martingale methods. Then applying again the change of variable formula, we derive the optimal strategy for the original problem for a power utility under certain assumptions on the finite variation process of the semimartingale.