Abstract:
We consider the following stochastic control problem, inspired by finance: an agent is trading one risky asset vs. one numéraire; and her goal is to make her portfolio grow with the fastest possible rate, i.e. to maximise the logarithm of her wealth on the long term. We want to know how the optimal trading strategy behaves. In absence of bid-ask spread (i.e., if one sells and buys at the same price), this is an easy problem. Yet things become more interesting when we introduce (proportional) transaction costs, which discourage us from trading back and forth too quickly… Gerhold, Muhle-Karbe & Schachermayer already investigated the case of the Back-Scholes model; while Czichowsky & al. studied the solution for an Ornstein-Uhlenbeck process.
Together with Ch. Czichowsky and W. Schachermayer, we are currently considering the case when the price process is a fractional Brownian motion (which does not contradict no-arbitrage, since there are transaction costs). Then we found that a surprising phenomenon occurs: unlike for the aforementioned semimartingale processes, where optimal selling or buying was to be done in a continuous way following some local time, here optimal trading happens to be discrete, with nonzero amounts being traded instantly! In this talk I will explain the reasons why this phenomenon occurs, by a mix of rigorous and heuristic arguments.