Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint *

Abstract : This paper provides an extension of the notion of consistent progressive utilities U to consistent progressive utilities of investment and consumption (U, V). It discusses the notion of market consistency in this forward framework, compared to the classic backward setting with a given terminal utility, and whose value function is an example of such consistent forward utility. To ensure the consistency with the market model or a given set of test processes, we establish a stochastic partial differential equation (SPDE) of Hamilton-Jacobi-Bellman (HJB)-type that U has to satisfy. This SPDE highlights the link between the utility of wealth U and the utility of consumption V, and between the drift and the volatility characteristics of the utility U. By associating with the HJB-SPDE two SDEs, we discuss the existence and the uniqueness of a concave solution. Finally, we provide explicit regularity conditions and characterize the consistent pairs of consistent utilities of investment and consumption. Some examples, such as power utilities, illustrate the theory.
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Soumis le : lundi 6 février 2017 - 19:50:57
Dernière modification le : vendredi 28 avril 2017 - 01:07:57
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  • HAL Id : hal-01458419, version 1

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Nicole El Karoui, Caroline Hillairet, Mohamed Mrad. Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint *. 2017. <hal-01458419>

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