JMF  Vol.5 No.2 , May 2015
Optimal Investment under Dual Risk Model and Markov Modulated Financial Market
ABSTRACT
In this paper, the optimal investment problem for an agent with dual risk model is studied. The financial market is assumed to be a diffusion process with the coefficients modulated by an external process, which is specified by the solution to a kind of stochastic differential equation. The object of the agent is to maximize the expected utility from terminal wealth. Together with the regularity property of the value function, by dynamic programming principle, the value function of our control problem is turned to be the unique solution to the associated Hamilton-Jacob-Bellman (HJB for short) equation. When the utility is an exponential function with constant risk aversion, close form expressions for value function and optimal investment policy are obtained.

Cite this paper
Xu, L. , Zhang, L. and Zhu, D. (2015) Optimal Investment under Dual Risk Model and Markov Modulated Financial Market. Journal of Mathematical Finance, 5, 157-171. doi: 10.4236/jmf.2015.52015.
References
[1]   Asmussen, S. and Albrecher, H. (2010) Ruin Probabilities. World Scientific, Singapore.
http://dx.doi.org/10.1142/9789814282536

[2]   Rolski, T., Schmidli, H., Schmidt V. and Teugels, J. (2009) Stochastic Processes for Insurance and Finance. John Wiley & Sons, Hoboken.

[3]   Albrecher, H., Badescu, A. and Landriault, D. (2008) On the Dual Risk Model with Tax Payments. Insurance: Mathematics and Economics, 42, 1086-1094.
http://dx.doi.org/10.1016/j.insmatheco.2008.02.001

[4]   Cheung, E.C.K. and Drekic, S. (2008) Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches. Astin Bulletin, 38, 399-422.
http://dx.doi.org/10.2143/AST.38.2.2033347

[5]   Yao, D.J., Yang, H.L. and Wang, R.M. (2010) Optimal Financing and Dividend Strategies in a Dual Model with Proportional costs. Journal of Industrial and Management Optimization, 6, 761-777.
http://dx.doi.org/10.3934/jimo.2010.6.761

[6]   Zhu, J.X. and Yang, H.L. (2008) Ruin Probabilities of a Dual Markov—Modulated Risk Model. Communications in Statistics—Theory and Methods, 37, 3298-3307.
http://dx.doi.org/10.1080/03610920802117080

[7]   Bai, L.H. and Guo, J.Y. (2008) Optimal Proportional Reinsurance and Investment with Multiple Risky Assets and No-Shorting Constraint. Insurance: Mathematics and Economics, 42, 968-975.
http://dx.doi.org/10.1016/j.insmatheco.2007.11.002

[8]   Browne, S. (1995) Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20, 937-958.
http://dx.doi.org/10.1287/moor.20.4.937

[9]   Fleming, W.H. and Hernández, D. (2005) The Tradeoff between Consumption and Investment in Incomplete Financial Markets. Applied Mathematics and Optimization, 52, 219-235.
http://dx.doi.org/10.1007/s00245-005-0826-1

[10]   Hipp, C. and Plum, M. (2000) Optimal Investment for Insurers. Insurance: Mathematics and Economics, 27, 215-228.
http://dx.doi.org/10.1016/S0167-6687(00)00049-4

[11]   Li, Z.F., Zeng, Y. and Lai, Y.Z. (2012) Optimal Time-Consistent Investment and Reinsurance Strategies for Insurers under Heston’s SV Model. Insurance: Mathematics and Economics, 51, 191-203.
http://dx.doi.org/10.1016/j.insmatheco.2011.09.002

[12]   Zhang, X. and Siu, T.K. (2009) Optimal Investment and Reinsurance of an Insurer with Model Uncertainty. Insurance: Mathematics and Economics, 45, 81-88.
http://dx.doi.org/10.1016/j.insmatheco.2009.04.001

[13]   French, W.E., Schwert, G.W. and Stambaugh, R.F. (1987) Expected Stock Returns and Volatility. Journal of Financial Economics, 19, 3-29.
http://dx.doi.org/10.1016/0304-405X(87)90026-2

[14]   Pham, H. (2002) Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints. Applied Mathematics and Optimization, 46, 55-78.
http://dx.doi.org/10.1007/s00245-002-0735-5

[15]   Zariphopoulou, T. (1999) Optimal Investment and Consumption Models with Nonlinear Stock Dynamics. Mathematical Methods of Operations Research, 50, 271-296.
http://dx.doi.org/10.1007/s001860050098

[16]   Zariphopoulou, T. (2001) A Solution Approach to Valuation with Unhedgeable Risks. Finance and Stochastics, 5, 61-82.
http://dx.doi.org/10.1007/PL00000040

 
 
Top