Optimal consumption and investment problem with random horizon in a BMAP model. (English) Zbl 1314.91192

Summary: In this paper, we consider the consumption and investment problem with random horizon in a Batch Markov Arrival Process (BMAP) model. The investor invests her wealth in a financial market consisting of a risk-free asset and a risky asset. The price processes of the riskless asset and the risky asset are modulated by a continuous-time Markov chain, which is the phase process of a BMAP. The possible consumption or investment are restricted to a sequence of random discrete time points which are determined by the same BMAP. The investor has only consumption opportunities at some of these random time points, has both consumption and investment opportunities at some other random time points, and can do nothing at the remaining random time points. The object of the investor is to select the consumption-investment strategy that maximizes the expected total discounted utility. The purpose of this paper is to analyze the impact of the consumption-investment opportunity and the economic state on the value functions and consumption-investment strategies. The general solution and the exact solution under the assumption that the consumption and the terminal wealth are evaluated by the power utility are obtained. Finally, a numerical example is presented.


91G10 Portfolio theory
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
90C40 Markov and semi-Markov decision processes
Full Text: DOI


[1] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scand. Actuar. J., 1-29 (2013) · Zbl 1401.91089
[2] Bäuerler, N.; Rieder, U., Markov Dicision Processes with Applications to Finance (2011), Springer
[3] Berdjane, B.; Pergamenshchikov, S., Optimal consumption and investment for markets with random coefficients, Finance Stoch., 17, 419-446 (2013) · Zbl 1278.91127
[4] Boyle, P. P.; Lin, X., Optimal portfolio selection with transaction costs, N. Am. Actuar. J., 1, 27-39 (1997) · Zbl 1080.91516
[5] Castaneda-Leyva, N.; Hernández-Hernández, D., Optimal consumption investment problems in incomplete markets with stochastic coefficients, SIAM J. Control Optim., 44, 1322-1344 (2005) · Zbl 1140.91381
[6] Davis, M. H.A.; Norman, A. R., Portfolio selection with transaction consts, Math. Oper. Res., 15, 4, 676-713 (1990) · Zbl 0717.90007
[7] Delong, L.; Klüppelberg, C., Optimal investment and consumption in a Black-Scholes market with Lévy-driven stochastic coefficients, Ann. Appl. Probab., 18, 879-908 (2008) · Zbl 1140.93048
[8] Engler, T.; Balmann, A., On investment consumption modeling with jump process extensions for productive sectors, J. Optim. Theory Appl. (2013), Publishen online: 05 September, http://dx.doi.org/10.10071s10957-013-0406-5 · Zbl 1336.91064
[9] Fleming, W. H.; Hernández-Hernández, An optimal consumption model with stochastic volatility, Finance Stoch., 7, 245-262 (2003) · Zbl 1035.60028
[11] Gennotte, G.; Jung, A., Investment strategies under transaction consts: the finite horizon case, Manage. Sci., 3, 385-404 (1994) · Zbl 0800.90065
[12] Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3, 4, 373-413 (1971) · Zbl 1011.91502
[13] Pham, H., Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints, Appl. Math. Optim., 46, 1, 55-78 (2002) · Zbl 1014.91038
[14] Rieder, U.; Wopperer, Ch., Robust consumption-investment problems with random market coefficients, Math. Financ. Econ., 6, 295-311 (2012) · Zbl 1279.91149
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.