# American Institute of Mathematical Sciences

April  2013, 9(2): 487-504. doi: 10.3934/jimo.2013.9.487

## Optimal portfolio in a continuous-time self-exciting threshold model

 1 China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China 2 Department of Actuarial Mathematics and Statistics, and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom 3 Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom 4 Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

Received  February 2011 Revised  October 2012 Published  February 2013

This paper discusses an optimal portfolio selection problem in a continuous-time economy, where the price dynamics of a risky asset are governed by a continuous-time self-exciting threshold model. This model provides a way to describe the effect of regime switching on price dynamics via the self-exciting threshold principle. Its main advantage is to incorporate the regime switching effect without introducing an additional source of uncertainty. A martingale approach is used to discuss the problem. Analytical solutions are derived in some special cases. Numerical examples are given to illustrate the regime-switching effect described by the proposed model.
Citation: Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487
##### References:
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##### References:
 [1] R. J. Elliott and P. E. Kopp, "Mathematics of Financial Markets," $2^{nd}$ edition, Springer Finance, Springer-Verlag, New York, 2005.  Google Scholar [2] S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Journal of Econometrics, 1 (1973), 3-16. Google Scholar [3] J. D. Hamilton, A new approach to the economic analysis of non-stationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.  Google Scholar [4] B. G. Jang, H. K. Koo and M. Loewenstein, Liquidity premia and transaction costs, Journal of Finance, 62 (2007), 2329-2366. Google Scholar [5] I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Applications of Mathematics (New York), 39, Springer-Verlag, New York, 1998.  Google Scholar [6] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1965), 12-37. Google Scholar [7] H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. Google Scholar [8] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economics and Statistics, 51 (1969), 247-257. Google Scholar [9] R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.  Google Scholar [10] J. Mossin, Equilibrium in a capital asset market, Econometrica, 34 (1966), 768-783. Google Scholar [11] R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Association, 53 (1958), 873-880.  Google Scholar [12] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246. Google Scholar [13] W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. Google Scholar [14] H. Tong, On a threshold model, in "Pattern Recognition and Signal Processing," Sijthoff and Noordhoff, Amsterdam, (1978), 575-586. Google Scholar [15] H. Tong and K. S. Lim, Threshold autoregression, limit cycles and cyclical data (with discussion), Journal of Royal Statistical Society - B, 42 (1980), 245-292. Google Scholar [16] H. Tong, "Threshold Models in Nonlinear Time Series Analysis," Lecture Notes in Statistics, 21, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4684-7888-4.  Google Scholar [17] J. L. Treynor, Toward a theory of market value of risky assets,, unpublished manuscript., ().   Google Scholar [18] G. Yin and X. Zhou, Markowitz's mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits, IEEE Transactions on Automatic Control, 49 (2004), 349-360. doi: 10.1109/TAC.2004.824479.  Google Scholar [19] X. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.  Google Scholar
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