Utility maximisation and change of variable formulas for time-changed dynamics

TL;DR

This paper develops new change of variable formulas for stochastic integrals involving time-changed Brownian motions with jumps, and applies these to optimize expected utility of terminal wealth in a complex stochastic setting.

Contribution

It introduces novel change of variable formulas for time-changed stochastic integrals with jumps and applies them to solve utility maximization problems in semimartingale models.

Findings

Derived new change of variable formulas for jump time-changed Brownian motions.

Solved utility maximization problems using filtration enlargement and martingale methods.

Provided explicit optimal strategies for power and logarithmic utilities.

Abstract

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.