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Deterministic Income with Deterministic and Stochastic Interest Rates

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  • Julia Eisenberg

Abstract

We consider an individual or household endowed with an initial capital and an income, modeled as a deterministic process with a continuous drift rate. At first, we model the discounting rate as the price of a zero-coupon bond at zero under the assumption of a short rate evolving as an Ornstein-Uhlenbeck process. Then, a geometric Brownian motion as the preference function and an Ornstein-Uhlenbeck process as the short rate are taken into consideration. It is assumed that the primal interest of the economic agent is to maximise the cumulated value of (expected) discounted consumption from a given time up to a finite deterministic time horizon $T\in\R_+$ or, in a stochastic setting, infinite time horizon. We find an explicit expression for the value function and for the optimal strategy in the first two cases. In the third case, we have to apply the viscosity ansatz.

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  • Julia Eisenberg, 2016. "Deterministic Income with Deterministic and Stochastic Interest Rates," Papers 1603.09519, arXiv.org.
    • Handle: RePEc:arx:papers:1603.09519
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    References listed on IDEAS

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    1. Julia Eisenberg & Peter Grandits & Stefan Thonhauser, 2014. "Optimal Consumption Under Deterministic Income," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 255-279, January.
    2. Pablo Azcue & Nora Muler, 2005. "Optimal Reinsurance And Dividend Distribution Policies In The Cramér‐Lundberg Model," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 261-308, April.
    3. Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October.
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