A nonlinear partial integro-differential equation from mathematical finance

Frederic Abergel; Remi Tachet

doi:10.3934/dcds.2010.27.907

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2010, Volume 27, Issue 3: 907-917. Doi: 10.3934/dcds.2010.27.907

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A nonlinear partial integro-differential equation from mathematical finance

Frederic Abergel^1, and
Remi Tachet^1,

1.
Grande Voie des Vignes, Chatenay-Malabry, 92295

Received: June 30, 2009

Revised: December 31, 2009

Published: August 2010

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Abstract

Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.

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Mathematics Subject Classification: Primary: 35A01, 35K55, 35R09, 47G20; Secondary: 60H15.

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