Utility maximization in incomplete markets for unbounded processes
When the price processes of the financial assets are described by possibly unbounded semimartingales, the classical concept of admissible trading strategies may lead to a trivial utility maximization problem because the set of stochastic integrals bounded from below may be reduced to the zero process. However, it could happen that the investor is willing to trade in such a risky market, where potential losses are unlimited, in order to increase his/her expected utility. We translate this attitude into mathematical terms by employing a class <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\mathcal{H}^{W}$</EquationSource> </InlineEquation> of W-admissible trading strategies which depend on a loss random variable W. These strategies enjoy good mathematical properties and the losses they could generate in trading are compatible with the preferences of the agent. We formulate and analyze by duality methods the utility maximization problem on the new domain <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\mathcal{H}^{W}$</EquationSource> </InlineEquation>. We show that, for all loss variables W contained in a properly identified set <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\mathcal{W}$</EquationSource> </InlineEquation>, the optimal value on the class <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\mathcal{H}^{W}$</EquationSource> </InlineEquation> is constant and coincides with the optimal value of the maximization problem over a larger domain <InlineEquation ID="Equ5"> <EquationSource Format="TEX">${K} _{\Phi}.$</EquationSource> </InlineEquation> The class <InlineEquation ID="Equ6"> <EquationSource Format="TEX">${K}_{\Phi}$</EquationSource> </InlineEquation> does not depend on a single <InlineEquation ID="Equ7"> <EquationSource Format="TEX">$W\in \mathcal{W},$</EquationSource> </InlineEquation> but it depends on the utility function u through its conjugate function <InlineEquation ID="Equ8"> <EquationSource Format="TEX">$\Phi $</EquationSource> </InlineEquation>. By duality methods we show that the solution exists in <InlineEquation ID="Equ9"> <EquationSource Format="TEX">${K}_{\Phi}$</EquationSource> </InlineEquation> and can be represented as a stochastic integral that is a uniformly integrable martingale under the minimax measure. We provide an economic interpretation of the larger class <InlineEquation ID="Equ10"> <EquationSource Format="TEX">${K}_{\Phi}$</EquationSource> </InlineEquation> and analyze some examples to show that this enlargement of the class of trading strategies is indeed necessary. Copyright Springer-Verlag Berlin/Heidelberg 2005
