Change of drift in one-dimensional diffusions

TL;DR

This paper characterizes when a measure change transforms one-dimensional diffusions into others with different drifts, and applies this to analyze arbitrage opportunities in a generalized Heston model, especially when the Feller condition fails.

Contribution

It provides a complete characterization of the conditions under which the measure change local martingale is a true martingale, enabling analysis of drift changes and arbitrage in complex models.

Findings

Complete characterization of when the measure change is a true martingale.

Application to absence of arbitrage in generalized Heston models.

Analysis includes cases where the Feller condition is violated.

Abstract

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this we have to know that the change-of-measure local martingale that we write down is a true martingale; we provide a complete characterization of when this happens. This is then used to discuss absence of arbitrage in a generalized Heston model including the case where the Feller condition for the volatility process is violated.