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

Author

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  • Moritz Sohns

    (Faculty of Economic Studies, University of Finance and Administration, 10100 Prague, Czech Republic
    Mathematical Institute, University of Oxford, Oxford OX1 2JD, UK
    Department of Mathematical Sciences, University of South Africa, Florida 0003, South Africa)

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.

Suggested Citation

  • Moritz Sohns, 2025. "σ -Martingales: Foundations, Properties, and a New Proof of the Ansel–Stricker Lemma," Mathematics, MDPI, vol. 13(4), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:4:p:682-:d:1594970
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