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

## Author Info

Listed author(s):
• Salvatore Federico
• Paul Gassiat
• Fausto Gozzi
Registered author(s):

## 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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 Length: Date of creation: Nov 2012 Date of revision: Mar 2015 Handle: RePEc:arx:papers:1211.1285 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Alessandra Cretarola & Fausto Gozzi & Huyên Pham & Peter Tankov, 2011. "Optimal consumption policies in illiquid markets," Finance and Stochastics, Springer, vol. 15(1), pages 85-115, January.
2. Paul Gassiat & Fausto Gozzi & Huyen Pham, 2011. "Investment/consumption problem in illiquid markets with regimes switching," Working Papers hal-00610214, HAL.
3. Schwartz, Eduardo S & Tebaldi, Claudio, 2004. "Illiquid Assets and Optimal Portfolio Choice," University of California at Los Angeles, Anderson Graduate School of Management qt7q65t12x, Anderson Graduate School of Management, UCLA.
4. 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.
5. Huyên Pham & Peter Tankov, 2008. "A Model Of Optimal Consumption Under Liquidity Risk With Random Trading Times," Mathematical Finance, Wiley Blackwell, vol. 18(4), pages 613-627.
6. Erhan Bayraktar & Mike Ludkovski, 2009. "Optimal Trade Execution in Illiquid Markets," Papers 0902.2516, arXiv.org.
7. Andrew Ang & Dimitris Papanikolaou & Mark M. Westerfield, 2014. "Portfolio Choice with Illiquid Assets," Management Science, INFORMS, vol. 60(11), pages 2737-2761, November.
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