From portfolio theory to optimal transport and Schrodinger bridge in-between

A large part of stochastic portfolio theory, as initiated by Robert Fernholz in the 1990s, is concerned with construction of practical equity portfolios that can beat the stock market index by active rule-based trading. The truly remarkable part of the theory is that it requires no probabilistic modeling on the future behavior of stock prices. There is a Monge-Kantorovich optimal transport problem that naturally arises in the construction of such portfolios. This transport problem is a multiplicative analog of the well-studied quadratic Kantorovich- Wasserstein transport with equally striking properties. We will see aspects of this transport problem from theoretical uses such as defining gradient flows in a non-metric setting to practical uses such as in determining the right frequency of trading. Interesting probability theory comes in as we consider entropic relaxation of this problem giving rise to multiplicative Schrodinger bridges.