Finance and Stochastics

, Volume 22, Issue 2, pp 297–326 | Cite as

An expansion in the model space in the context of utility maximization

  • Kasper Larsen
  • Oleksii Mostovyi
  • Gordan Žitković


In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian), an explicit second-order expansion formula for the power investor’s value function—seen as a function of the underlying market price of risk process—is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.


Continuous semimartingales Second-order expansion Incomplete markets Power utility Convex duality Optimal investment 

Mathematics Subject Classification (2010)

91G10 91G80 60K35 

JEL Classification

C61 G11 



The authors would like to thank the anonymous referees, the anonymous Associate Editor, the Editor Martin Schweizer, Milica Čudina, Claus Munk, Mihai Sîrbu, and Kim Weston for useful comments.

During the preparation of this work, the first author has been supported by the National Science Foundation under Grant No. DMS-1411809 (2014–2017). The second author has been supported by the National Science Foundation under grant No. DMS-1600307 (2015–2018). The third author has been supported by the National Science Foundation under Grant No. DMS-1107465 (2012–2017) and Grant No. DMS-1516165 (2015–2018). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Kasper Larsen
    • 1
  • Oleksii Mostovyi
    • 2
  • Gordan Žitković
    • 3
  1. 1.Department of MathematicsRutgers UniversityPiscatawayUSA
  2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA

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