On the structure of increasing profits in a 1D general diffusion market with interest rates

Abstract

In this paper, we investigate a financial market model consisting of a risky asset, modeled as a general diffusion parameterized by a scale function and a speed measure, and a bank account process with a constant interest rate. This flexible class of financial market models allows for features such as reflecting boundaries, skewness effects, sticky points, and slowdowns on fractal sets. For this market model, we study the structure of a strong form of arbitrage opportunity called increasing profits. Our main contributions are threefold. First, we characterize the existence of increasing profits in terms of an auxiliary deterministic signed measure $ν$ and a canonical trading strategy $θ$, both of which depend only on the deterministic parametric characteristics of our model, namely the scale function, the speed measure, and the interest rate. More precisely, we show that an increasing profit exists if and only if $ν$ is nontrivial, and that this is equivalent to $θ$ itself generating an increasing profit. Second, we provide a precise characterization of the entire set of increasing profits in terms of $ν$ and $θ$, and moreover characterize the value processes associated with increasing profits. Finally, we establish novel connections between no-arbitrage theory and the general theory of stochastic processes. Specifically, we relate the failure of the representation property for general diffusions to the existence of certain types of increasing profits whose value processes are dominated by the quadratic variation measure of a space-transformed version of the asset price process.