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Dynamic Economic Problems with Regime Switches pp 287–309Cite as

A Regime-Switching Model with Applications to Finance: Markovian and Non-Markovian Cases

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Part of the book series: Dynamic Modeling and Econometrics in Economics and Finance ((DMEF,volume 25))

Abstract

A Markov regime-switching model may capture abrupt changes in the financial market efficiently, which are generated by inner or outer effects in an economy. These systems are governed by both continuous and discrete time dynamics, for which they are also called hybrid systems and have many applications in science and technology. In this work, we present a survey of financial applications under a specific semimartingale result of Markov chains. First, we present a robust portfolio strategy, which is also a zero-sum stochastic differential game. Second, optimal portfolio formulas of two investors’ collaboration are established by a nonzero-sum game. Both of these applications are solved by the Dynamic Programming Principle (DPP) approach in stochastic games. Our third result is an optimal consumption problem of a cash flow with regimes and time delay. This last financial problem is represented by the results of the necessary and sufficient stochastic Maximum Principle (MP).

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References

  1. Bensoussan, A., & Menaldi, J. L. (2003). Stochastic hybrid control. Journal of Mathematical Analysis and Applications, 249, 261–288.

    Article  Google Scholar 

  2. Bensoussan, A., Siu, C. C., Yamd, S. C. P., & Yange, H. (2014). A class of nonzero-sum stochastic differential investment and reinsurance games. Automatica, 50(6), 2025–2037.

    Article  Google Scholar 

  3. Cadenillas, A., & Karatzas, I. (1995). The stochastic maximum principle for linear, convex optimal control with random coefficients. SIAM Journal of Control and Optimization, 33(2), 590–624.

    Article  Google Scholar 

  4. Cohen, S. N., & Elliott, R. J. (2010). Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions. Annals of Applied Probability, 20(1), 267–311.

    Article  Google Scholar 

  5. Crepey, S. (2010). About the pricing equations in finance. Paris-Princeton Lectures on Mathematical Finance, 63–203.

    Google Scholar 

  6. Crepey, S., & Matoussi, A. (2008). Reflected and doubly reflected BSDEs with jumps: A priori estimate and comparison. The Annals of Applied Probability, 18(5), 2041–2069.

    Article  Google Scholar 

  7. Deng, C., Zeng, X., & Zhu, H. (2018). Nonzero-sum stochastic differential reinsurance and investment games with default risk. European Journal of Operational Research, 264(3), 1144–1158.

    Article  Google Scholar 

  8. El-Karoui, N., & Hamadene, S. (2003). BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Processes and their Applications, 107(1), 145–169.

    Article  Google Scholar 

  9. El-Karoui, N., Peng, S., & Quenez, M. C. (1997). Backward stochastic differential equations. Mathematical Finance, 7(1), 1–71.

    Article  Google Scholar 

  10. Elliott, R. J., Aggoun, L., & Moore, J. B. (1995). Hidden Markov models: Estimation and control. New York: Springer.

    Google Scholar 

  11. Elliott, R. J., Chan, L., & Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance, 1(4), 423–432.

    Article  Google Scholar 

  12. Elliott, R. J., & Siu, T. K. (2008). On risk minimizing portfolios under a Markovian regime-switching Black-Sholes economy. Annals of Operations Research, 176(1), 271–291.

    Article  Google Scholar 

  13. Elliott, R. J., & Siu, T. K. (2009). Robust optimal portfolio choice under Markovian regime-switching model. Methodology and Computing in Applied Probability, 11(2), 145–157.

    Article  Google Scholar 

  14. Elsanosi, I., Øksendal, B., & Sulem, A. (2000). Some solvable stochastic control problems with delay. Stochastics and Stochastic Reports, 71, 69–89.

    Article  Google Scholar 

  15. Federico, S. (2011). A stochastic control problem with delay arising in a pension fund model. Finance and Stochastics, 15(3), 421–459.

    Article  Google Scholar 

  16. Federico, S., Goldys, B., & Gozzi, F. (2010). HJB equations for the optimal control of differential equations with delays and state constraints, i: Regularity of viscosity solutions. SIAM Journal on Control and Optimization, 48(8), 4910–4937.

    Article  Google Scholar 

  17. Federico, S., Goldys, B., & Gozzi, F. (2011). HJB equations for the optimal control of differential equations with delays and state constraints, ii: Verification and optimal feedbacks. SIAM Journal on Control and Optimization, 49(6), 2378–2414.

    Article  Google Scholar 

  18. Ghosh, M. K., & Bagchi, A. (2003). Modeling stochastic hybrid systems. In 21st IFIP TC7 Conference on System Modelling and Optimization.

    Google Scholar 

  19. Goldfeld, S. M., & Quandt, R. E. (1973). A Markov model for switching regressions. Journal of Econometrics, 1(1), 3–16.

    Article  Google Scholar 

  20. Hamadene, S. (1998). Backward-forward SDE’s and stochastic differential games. Stochastic Processes and their Applications, 77(1), 1–15.

    Article  Google Scholar 

  21. Hamadene, S., & Hassani, M. (2006). BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game. http://www.math.washington.edu/\(> ejpecp/\)11(5), 121–145.

  22. Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and business cycle. Econometrica, 57(2), 357–384.

    Article  Google Scholar 

  23. Hu, J., Lygeros, J., & Sastry, S. (2000). Towards a theory of stochastic hybrid systems. In N. Lynch & B. H. Krogh (Eds.), Hybrid systems: Computation and control, 1790 in LNCS pp. 160–173.

    Google Scholar 

  24. Huang, W., Lehalle, C.-A., & Rosenbaum, M. (2016). How to predict the consequences of a tick value change? Evidence from the Tokyo stock exchange pilot program. Market Microstructure and Liquidity 2(03n04), 1750001.

    Google Scholar 

  25. Itô, K., & Nisio, M. (1964). On stationary solutions of a stochastic differential equation. Journal of Mathematics, 4(1), 1–75.

    Google Scholar 

  26. Karatzas, I., & Li, Q. (2011). BSDE approach to nonzero-sum stochastic differential games of control and stopping. In Stochastic processes, finance and control: A festschrift in honor of Robert J. Elliott. World Scientific Publishing Co. Pte. Ltd.

    Google Scholar 

  27. Korn, R., & Menkens, O. (2005). Worst-case scenario portfolio optimization: A new stochastic control approach. Mathematical Methods of Operations Research, 62(1), 123–140.

    Article  Google Scholar 

  28. Kropat, E., Turkay, M., & Weber, G.-W. (2020). Fuzzy analytics and stochastic methods in neurosciences. IEEE Transactions on Fuzzy Systems (IEEE TFS), Special issue (to appear), 28(1), 1–121.

    Google Scholar 

  29. Kushner, H. J. (1968). On the stability of processes defined by stochastic differential difference equations. Journal of Differential Equations, 4(3), 424–443.

    Article  Google Scholar 

  30. Laruelle, S., Rosenbaum, M., & Savku, E. (2020). Assessing MiFID 2 regulation on tick sizes: A transaction costs analysis viewpoint. Market Microstructure and Liquidity, 5(1), 2050003. World Scientific.

    Google Scholar 

  31. Laruelle, S., Rosenbaum, M., & Savku, E. (2018). Assessing MiFID 2 regulation on tick sizes: A transaction costs analysis viewpoint. Available at SSRN, 3256453.

    Google Scholar 

  32. Li, C.-Y., Chen, S.-N., & Lin, S.-K. (2016). Pricing derivatives with modeling co2 emission allowance using a regime-switching jump diffusion model: With regime-switching risk premium. The European Journal of Finance, 22(10), 887–908.

    Article  Google Scholar 

  33. Lin, X., Zhang, C., & Siu, T. K. (2012). Stochastic differential portfolio games for an insurer in a jump-diffusion risk process. Mathematical Methods of Operations Research, 75(1), 83–100.

    Google Scholar 

  34. Lv, S., Tao, R., & Wu, Z. (2016). Maximum principle for optimal control of anticipated forward-backward stochastic differential delayed systems with regime-switching. Optimal Control Applications and Methods, 37(1), 154–175.

    Article  Google Scholar 

  35. Ma, C., Wu, H., & Lin, X. (2015). Nonzero-sum stochastic differential portfolio games under a Markovian regime switching model. Hindawi Publishing Corporation Mathematical Problems in Engineering.

    Google Scholar 

  36. Mataramvura, S., & Øksendal, B. (2008). Risk minimizing portfolios and HJBI equations for stochastic differential games. Stochastics: An International Journal of Probability and Stochastic Processes, 80(4), 317–337.

    Google Scholar 

  37. Menoukeu-Pamen, O., & Momeya, R. H. (2017). A maximum principle for Markov regime-switching forward-backward stochastic differential games and applications. Mathematical Methods of Operations Research, 85(3), 349–388.

    Google Scholar 

  38. Meyer-Brandis, T., Øksendal, B., & Zhou, X. Y. (2012). A mean-field stochastic maximum principle via Malliavin calculus. Stochastics: An International Journal of Probability and Stochastic Processes 84, 5–6 (201), 643–666.

    Google Scholar 

  39. Mohammed, S. E. A. (1984). Stochastic functional differential equations. Pitman.

    Google Scholar 

  40. Øksendal, B., & Sulem, A. (2001). A maximum principle for optimal control of stochastic systems with delay, with applications to finance. Optimal Control and Partial Differential Equations (01 2000).

    Google Scholar 

  41. Øksendal, B., & Sulem, A. (2011). A maximum principle for optimal control of stochastic systems with delay, with application to finance. Amsterdam: IOS press.

    Google Scholar 

  42. Øksendal, B., Sulem, A., & Zhang, T. (2011). Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Advances in Applied Probability, 43(2), 572–596.

    Article  Google Scholar 

  43. Pamen, O. M. (2015). Optimal control for stochastic delay systems under model uncertainty: A stochastic differential game approach. Journal of Optimization Theory and Applications, 167(3), 998–1031.

    Article  Google Scholar 

  44. Pamen, O. M. (2017). Maximum principles of Markov regime-switching forward-backward stochastic differential equations with jumps and partial information. Journal of Optimization Theory and Applications, 175(2), 373–410.

    Article  Google Scholar 

  45. Pang, T., & Hussain, A. (2016). An infinite time horizon portfolio optimization model with delays. Mathematical Control & Related Fields, 6(4), 629–651.

    Article  Google Scholar 

  46. Pardoux, E., & Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems and Control Letters, 14(1), 55–61.

    Article  Google Scholar 

  47. Peng, S., & Yang, Z. (2009). Anticipated backward stochastic differential equations. The Annals of Probability, 37(3), 877–902.

    Article  Google Scholar 

  48. Quandt, R. E. (1958). The estimation of the parameters of a linear regression system obeying two seperate regimes. Journal of the American Association, 53(284), 873–880.

    Article  Google Scholar 

  49. Savku, E. (2017). Advances in optimal control of Markov regime-switching models with applications in finance and economics. Ph.D. thesis, IAM, METU, Turkey.

    Google Scholar 

  50. Savku, E., & Weber, G.-W. (2020). Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market. Annals of Operations Research, 1–26. Springer.

    Google Scholar 

  51. Savku, E., & Weber, G.-W. (2018). A stochastic maximum principle for a markov regime-switching jump-diffusion model with delay and an application to finance. Journal of Optimization Theory and Applications, 179(2), 696–721.

    Article  Google Scholar 

  52. Seifried, F. T. (2010). Optimal investment for worst-case crash scenarios: A martingale approach. Mathematics of Operations Research, 35(3), 559–579.

    Article  Google Scholar 

  53. Shen, Y., Meng, Q., & Shi, P. (2014). Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance. Automatica, 50(5), 1565–1579.

    Article  Google Scholar 

  54. Shen, Y., & Siu, T. K. (2013). The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem. Nonlinear Analysis, 86(2), 58–73.

    Article  Google Scholar 

  55. Shen, Y., & Siu, T. K. (2013). Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model. Insurance: Mathematics and Economics 53(3), 757–768.

    Google Scholar 

  56. Tu, S., & Hao, W. (2014). Anticipated backward stochastic differential equations with jumps under the non-Lipschitz condition. Statistics and Probabilty Letters, 92, 215–225.

    Article  Google Scholar 

  57. Yiu, K.-F. C., Liu, J., Siu, T. K., & Ching, W.-K. (2010). Optimal portfolios with regime switching and value-at-risk constraint. Automatica, 46(6), 979–989.

    Article  Google Scholar 

  58. Zhang, Q. (2001). Stock trading: An optimal selling rule. Siam Journal on Control and Optimization, 40(1), 64–87.

    Article  Google Scholar 

  59. Zhang, Q., & Yin, G. (2004). Nearly-optimal asset allocation in hybrid stock investment. Journal of Optimization Theory and Applications, 121(2), 121–419.

    Article  Google Scholar 

  60. Zhang, X., Elliott, R. J., & Siu, T. K. (2012). A stochastic maximum principle for a Markov regime-switching jump-diffusion model and an application to finance. SIAM Journal on Control and Optimization, 50(2), 964–990.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Ecole Polytechnique, CMAP, 91128, Palaiseau, France

    E. Savku

  2. Middle East Technical University, IAM, Dumlupinar Bulvari, Universiteler Mahallesi, No:1, 06800, Cankaya, Ankara, Turkey

    E. Savku & G.-W. Weber

  3. Faculty of Management Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965, Poznan, Poland

    G.-W. Weber

Authors
  1. E. Savku
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  2. G.-W. Weber
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Correspondence to E. Savku .

Editor information

Editors and Affiliations

  1. Inst. of Stochastics & Math. Economics, Vienna University of Technology, Wien, Austria

    Josef L. Haunschmied

  2. Institute of Stochastics and Mathematical Economics, Vienna University of Technology, Wien, Austria

    Raimund M. Kovacevic

  3. Department of Economics, The New School for Social Research, New York, NY, USA

    Willi Semmler

  4. Institute of Stochastics and Mathematical Economics, Vienna University of Technology, Wien, Austria

    Vladimir M. Veliov

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Savku, E., Weber, GW. (2021). A Regime-Switching Model with Applications to Finance: Markovian and Non-Markovian Cases. In: Haunschmied, J.L., Kovacevic, R.M., Semmler, W., Veliov, V.M. (eds) Dynamic Economic Problems with Regime Switches. Dynamic Modeling and Econometrics in Economics and Finance, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-54576-5_13

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