# Impact of time illiquidity in a mixed market without full observation

## Author Info

• Salvatore Federico
• Paul Gassiat
• Fausto Gozzi

## Abstract

We study a problem of optimal investment/consumption over an infinite horizon in a market consisting of two possibly correlated assets: one liquid and one illiquid. The liquid asset is observed and can be traded continuously, while the illiquid one can be traded only at discrete random times corresponding to the jumps of a Poisson process with intensity $\lambda$, is observed at the trading dates, and is partially observed between two different trading dates. The problem is a nonstandard mixed discrete/continuous optimal control problem which we face by the dynamic programming approach. When the utility has a general form we prove that the value function is the unique viscosity solution of the HJB equation and, assuming sufficient regularity of the value function, we give a verification theorem that describes the optimal investment strategies for the illiquid asset. In the case of power utility, we prove the regularity of the value function needed to apply the verification theorem, providing the complete theoretical solution of the problem. This allows us to perform numerical simulation, so to analyze the impact of time illiquidity in this mixed market and how this impact is affected by the degree of observation.

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File URL: http://arxiv.org/pdf/1211.1285

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1211.1285.

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Handle: RePEc:arx:papers:1211.1285

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## References

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1. Alessandra Cretarola & Fausto Gozzi & Huyên Pham & Peter Tankov, 2008. "Optimal consumption policies in illiquid markets," Working Papers hal-00292673, HAL.
2. Paul Gassiat & Fausto Gozzi & Huyen Pham, 2011. "Investment/consumption problem in illiquid markets with regimes switching," Working Papers hal-00610214, HAL.
3. Marina Di Giacinto & Salvatore Federico & Fausto Gozzi, 2011. "Pension funds with a minimum guarantee: a stochastic control approach," Finance and Stochastics, Springer, vol. 15(2), pages 297-342, June.
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## Citations

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Cited by:
1. Salvatore Federico & Paul Gassiat, 2012. "Viscosity characterization of the value function of an investment-consumption problem in presence of illiquid assets," Papers 1211.1286, arXiv.org.
2. Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2013. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Papers 1301.0280, arXiv.org.

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