Minimizing the expected market time to reach a certain wealth level

TL;DR

This paper investigates strategies to minimize the expected time for an investor to reach a target wealth level across different market models, introducing the concept of market time and demonstrating the optimality of the growth-optimal portfolio.

Contribution

It extends the analysis of wealth reaching time minimization to exponential Levy and general semimartingale markets, introducing the concept of market time for the first time.

Findings

Growth-optimal portfolio is asymptotically optimal in exponential Levy markets.

Optimal portfolio minimizes expected market time in Ito markets.

Generalized the concept of market time to broader semimartingale models.

Abstract

In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Levy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Ito market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investor's point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.