A continuous selection for optimal portfolios under convex risk measures does not always exist

Risk control is one of the crucial problems in finance. One of the most common ways to mitigate risk of an investor's financial position is to set up a portfolio of hedging securities whose aim is to absorb unexpected losses and thus provide the investor with an acceptable level of security. In this respect, it is clear that investors will try to reach acceptability at the lowest possible cost. Mathematically, this problem leads naturally to considering set-valued maps that associate to each financial position the corresponding set of optimal hedging portfolios, i.e., of portfolios that ensure acceptability at the cheapest cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note, we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.