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Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset

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Abstract

We study a problem of optimal investment/consumption over an infinite horizon in a market consisting of a liquid and an illiquid asset. The liquid asset is observed and can be traded continuously, while the illiquid one can only be traded and observed at discrete random times corresponding to the jumps of a Poisson process. The problem is a nonstandard mixed discrete/continuous optimal control problem, which we face by the dynamic programming approach. The main goal of the paper is the characterization of the value function as unique viscosity solution of an associated Hamilton–Jacobi–Bellman equation. We then use such a result to build a numerical algorithm, allowing one to approximate the value function and so to measure the cost of illiquidity.

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Notes

  1. Also in [13] the market is composed by a liquid and an illiquid asset. However, there the problem is over a finite horizon and the illiquid asset cannot be traded: the wealth held in the illiquid asset enters just in the optimization functional at the terminal date.

  2. Note that even though the HJB equation is nonlocal, the fact that \(\widehat{V}\) is a viscosity solution is standard. This is because the nonlocal term \(G[\mathcal{H}\widehat{V}]\) only enters as a running cost in the control problem (there are no jumps involved).

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Acknowledgements

The authors would like to thank two anonymous referees, who with their valuable comments have led to a significant improvement of the paper. Also they are grateful to Huyên Pham to have suggested to them the topic and to Fausto Gozzi for useful suggestions.

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Authors and Affiliations

  1. Dipartimento di Economia, Management e Metodi Quantitativi, Università di Milano, Via Conservatorio 7, 56122, Milano, Italy

    Salvatore Federico

  2. Institut für Mathematik, TU Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Paul Gassiat

Authors
  1. Salvatore Federico
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  2. Paul Gassiat
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Correspondence to Salvatore Federico.

Additional information

Communicated by Lars Grüne.

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Federico, S., Gassiat, P. Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset. J Optim Theory Appl 160, 966–991 (2014). https://doi.org/10.1007/s10957-013-0372-y

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