Saddle points for maximin investment problems with observable but non-predictable parameters: solution via heat equation.
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This is a pre-copy-editing, author produced PDF of an article accepted for publication in IMA Journal of Management Mathematics following peer review. The definitive publisher-authenticated version: Dokuchaev, Nikolai. 2006. Saddle points for maximin investmentproblems with observable but non-predictable parameters: solutionvia heat equation. IMA Journal of Management Mathematics. 17 (3): pp. 257-276. is available online at http://dx.doi.org/10.1093/imaman/dpi041
We study optimal investment problem for a market model where the evolution of risky assets prices is described by Itoˆs 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 mazimin 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 solution of the standard one-dimensional 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.
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