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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization
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by Yves Achdou, Bruno Franchi and Nicoletta Tchou PDF
Math. Comp. 74 (2005), 1291-1322 Request permission

Abstract:

This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein–Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.
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Additional Information
  • Yves Achdou
  • Affiliation: UFR Mathématiques, Université Paris 7, 2 place Jussieu, 75251 Paris cedex 05, France; and Laboratoire J.L. Lions, Université Paris 6, 4 place Jussieu, 75252 Paris cedex 05, France
  • Email: achdou@math.jussieu.fr
  • Bruno Franchi
  • Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy
  • Email: bfranchi@dm.unibo.it
  • Nicoletta Tchou
  • Affiliation: IRMAR, Université de Rennes 1, Rennes, France
  • Email: nicoletta.tchou@univ-rennes1.fr
  • Received by editor(s): April 16, 2003
  • Received by editor(s) in revised form: March 3, 2004
  • Published electronically: October 5, 2004
  • Additional Notes: The second author was partially supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy, project “Analysis in metric spaces”.
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1291-1322
  • MSC (2000): Primary 35K65, 65M15, 65M60, 65N30
  • DOI: https://doi.org/10.1090/S0025-5718-04-01714-4
  • MathSciNet review: 2137004