Probabilistic Aspects of Arbitrage

Consider the logarithm log (1/U(T,z)) of the highest return on investment that can be achieved relative to a market with Markovian weights, over a given time-horizon [0,T] and with given initial market weight configuration (0)=z. We characterize this quantity (i) as the smallest amount of relative entropy with respect to the Föllmer exit measure, under which the market weight process (·) is a diffusion with values in the unit simplex Δ and the same covariance structure but zero drift; and (ii) as the smallest “total energy” expended during [0,T] by the respective drift, over a class of probability measures which are absolutely continuous with respect to the exit measure and under which (·) stays in the interior Δ o of the unit simplex at all times, almost surely. The smallest relative entropy, or total energy, corresponds to the conditioning of the exit measure on the event { (t)∈Δ°, ∀0≤t≤T; whereas, under this “minimal energy” measure, the portfolio $\widehat{\pi}(\cdot)$ generated by the function U(⋅ ,⋅) has the numéraire and relative log-optimality properties. This same portfolio $\widehat{\pi}(\cdot)$ also attains the highest possible relative return on investment with respect to the market.