σ -Martingales: Foundations, Properties, and a New Proof of the Ansel–Stricker Lemma

Moritz Sohns ()
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Moritz Sohns: Faculty of Economic Studies, University of Finance and Administration, 10100 Prague, Czech Republic

Mathematics, 2025, vol. 13, issue 4, 1-19

Abstract: σ -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 σ -martingales that avoids using semimartingale characteristics. We develop all fundamental properties, provide illustrative examples, and establish the core structure of σ -martingales in a new, straightforward manner. This approach culminates in a new proof of the Ansel–Stricker lemma, which states that one-sided bounded σ -martingales are local martingales. This result, referenced in nearly every publication on mathematical finance, traditionally relies on the original French-language proof. We use this result to prove a generalization, which is essential for defining the general semimartingale model in mathematical finance.

Keywords: ?-martingales; stochastic processes; stochastic integration; localization; semimartingales; Ansel–Stricker lemma (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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