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Investment/consumption problem in illiquid markets with regime-switching

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  • Paul Gassiat
  • Fausto Gozzi
  • Huy^en Pham

Abstract

We consider an illiquid financial market with different regimes modeled by a continuous-time finite-state Markov chain. The investor can trade a stock only at the discrete arrival times of a Cox process with intensity depending on the market regime. Moreover, the risky asset price is subject to liquidity shocks, which change its rate of return and volatility, and induce jumps on its dynamics. In this setting, we study the problem of an economic agent optimizing her expected utility from consumption under a non-bankruptcy constraint. By using the dynamic programming method, we provide the characterization of the value function of this stochastic control problem in terms of the unique viscosity solution to a system of integro-partial differential equations. We next focus on the popular case of CRRA utility functions, for which we can prove smoothness $C^2$ results for the value function. As an important byproduct, this allows us to get the existence of optimal investment/consumption strategies characterized in feedback forms. We analyze a convergent numerical scheme for the resolution to our stochastic control problem, and we illustrate finally with some numerical experiments the effects of liquidity regimes in the investor's optimal decision.

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  • Paul Gassiat & Fausto Gozzi & Huy^en Pham, 2011. "Investment/consumption problem in illiquid markets with regime-switching," Papers 1107.4210, arXiv.org, revised Apr 2012.
    • Handle: RePEc:arx:papers:1107.4210
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    References listed on IDEAS

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    1. Koichi Matsumoto, 2006. "Optimal portfolio of low liquid assets with a log-utility function," Finance and Stochastics, Springer, vol. 10(1), pages 121-145, January.
    2. Traian A. Pirvu & Huayue Zhang, 2011. "On Investment-Consumption with Regime-Switching," Papers 1107.1895, arXiv.org.
    3. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    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. 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.
    6. Michael Ludkovski & Hyekyung Min, 2010. "Illiquidity Effects in Optimal Consumption-Investment Problems," Papers 1004.1489, arXiv.org, revised Sep 2010.
    7. 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, October.
    8. Luz Rocío Sotomayor & Abel Cadenillas, 2009. "Explicit Solutions Of Consumption‐Investment Problems In Financial Markets With Regime Switching," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 251-279, April.
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    Cited by:

    1. Michael Ludkovski & Qunying Shen, 2012. "European Option Pricing with Liquidity Shocks," Papers 1205.1007, arXiv.org.
    2. Yuki SHIGETA, 2022. "A Continuous-Time Utility Maximization Problem with Borrowing Constraints in Macroeconomic Heterogeneous Agent Models:A Case of Regular Controls under Markov Chain Uncertainty," Discussion papers e-22-009, Graduate School of Economics , Kyoto University.
    3. Baojun Bian & Nan Wu & Harry Zheng, 2012. "Optimal Liquidation in a Finite Time Regime Switching Model with Permanent and Temporary Pricing Impact," Papers 1212.3145, arXiv.org, revised Oct 2014.
    4. Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2017. "Impact Of Time Illiquidity In A Mixed Market Without Full Observation," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 401-437, April.

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