Sigma Martingales: Foundations, Properties, and a New Proof of the Ansel-Stricker Lemma
Moritz Sohns · Preprints.org · 2025
Sigma martingales generalize local martingales through localizing sequences of predictable sets, which are essential in stochastic analysis and financial mathematics, particularly for arbitrage-free markets and portfolio theory. In this work, we present a new approach to sigma martingales that avoids using semimartingale characteristics. We develop all fundamental properties, provide illustrative examples, and establish the core structure of sigma martingales in a new, straightforward manner. This approach culminates in a new proof of the Ansel-Stricker lemma, which states that one-sided bounded sigma martingales are local martingales. This result, referenced in nearly every publication on mathematical finance, traditionally relies on the original French-language proof.