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

In this paper, we investigate a financial market model with a risky asset represented by a general diffusion and a bank account process with a constant interest rate. This flexible class of 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 {\em increasing profits}. Our main contributions are threefold. First, we characterize the existence of increasing profits in terms of an auxiliary signed deterministic measure \(\nu\) and a canonical trading strategy \(\theta\). More precisely, we show that an increasing profit exists if and only if \(\nu\) is non-trivial, and that this is equivalent to the property that \(\theta\) itself generates an increasing profit. Second, we provide a precise characterization of the entire set of increasing profits in terms of \(\nu\) and \(\theta\), and, moreover, we 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.