A Quasilinear Parabolic Equation with Quadratic Growth of the Gradient Modeling Incomplete Financial Markets

University of Konstanz Discussion Paper No. 04/01

27 Pages Posted: 22 Mar 2004

See all articles by Bertram Düring

Bertram Düring

University of Warwick - Mathematics Institute

Ansgar Jüngel

Fachbereich Mathematik und Informatik, University of Mainz

Date Written: February 4, 2004

Abstract

We consider a quasilinear parabolic equation with quadratic gradient terms. It arises in the modeling of an optimal portfolio which maximizes the expected utility from terminal wealth in incomplete markets consisting of risky assets and non-tradable state variables. The existence of solutions is shown by extending the monotonicity method of Frehse. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution. The in influence of the non-tradable state variables on the optimal value function is illustrated by a numerical example.

Keywords: Quasilinear PDE, quadratic gradient, existence and uniqueness of solutions, optimal portfolio, incomplete market

JEL Classification: G11

Suggested Citation

Düring, Bertram and Jüngel, Ansgar, A Quasilinear Parabolic Equation with Quadratic Growth of the Gradient Modeling Incomplete Financial Markets (February 4, 2004). University of Konstanz Discussion Paper No. 04/01, Available at SSRN: https://ssrn.com/abstract=520462 or http://dx.doi.org/10.2139/ssrn.520462

Bertram Düring (Contact Author)

University of Warwick - Mathematics Institute ( email )

Zeeman Building
Coventry, CV4 7AL
United Kingdom

Ansgar Jüngel

Fachbereich Mathematik und Informatik, University of Mainz ( email )

D-55099 Mainz
Germany

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