Finance and Stochastics

, Volume 22, Issue 4, pp 1007–1036 | Cite as

Weak time-derivatives and no-arbitrage pricing

  • Massimo Marinacci
  • Federico SeverinoEmail author


We prove a risk-neutral pricing formula for a large class of semimartingale processes through a novel notion of weak time-differentiability that permits to differentiate adapted processes. In particular, the weak time-derivative isolates drifts of semimartingales and is null for martingales. Weak time-differentiability enables us to characterize no-arbitrage prices as solutions of differential equations, where interest rates play a key role. Finally, we reformulate the eigenvalue problem of Hansen and Scheinkman (Econometrica 77:177–234, 2009) by employing weak time-derivatives.


No-arbitrage pricing Weak time-derivative Martingale component Special semimartingales Stochastic interest rates 

Mathematics Subject Classification (2010)

60G07 91G80 49J40 

JEL Classification




We thank Anna Battauz, Francesco Caravenna, Andrea Carioli, Simone Cerreia-Vioglio, Lars Peter Hansen, Ioannis Karatzas, Luigi Montrucchio, Fulvio Ortu, Emanuela Rosazza-Gianin, Martin Schweizer and two anonymous referees for useful comments. We also thank seminar participants at XL AMASES Annual Meeting in Catania (2016) and at XVIII Quantitative Finance Workshop at Università Bicocca, Milan (2017).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Decision Sciences & IGIERUniversità BocconiMilanItaly
  2. 2.Department of EconomicsUniversità della Svizzera Italiana (USI)LuganoSwitzerland
  3. 3.Department of FinanceUniversità BocconiMilanItaly

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