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IMA Journal of Management Mathematics Advance Access published online on November 28, 2005
IMA Journal of Management Mathematics, doi:10.1093/imaman/dpi041
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© The authors 2005. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Received October 22, 2003
Accepted October 17, 2005

Article

Saddle points for maximin investment problems with observable but non-predictable parameters: solution via heat equation

Nikolai Dokuchaev 1 *

1 Department of Mathematics and Statistics, University of Limerick, Ireland

* To whom correspondence should be addressed.
Nikolai Dokuchaev, E-mail: nikolai.dokuchaev{at}ul.ie


  Abstract

We study optimal investment problem for a market model where the evolution of risky assets prices is described by Itô's equations. The risk-free rate, the appreciation rates and the volatility of the stocks are all random; they depend on a random parameter that is not adapted to the driving Brownian motion. The distribution of this parameter is unknown. The optimal investment problem is stated in a ‘maximin’ setting to ensure that a strategy is found such that the minimum of expected utility over all possible distributions of parameters is maximal. We show that a saddle point exists and can be found via a solution of the standard 1D heat equation with a Cauchy condition defined via one dimensional minimization. This solution even covers models with unknown solution for a given distribution of the market parameters.

Keywords: continuous time market models; uncertainty; minimax problems; optimal portfolio; saddle point.
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