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Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market

Abstract

We apply dynamic programming principle to discuss two optimal investment problems by using zero-sum and nonzero-sum stochastic game approaches in a continuous-time Markov regime-switching environment within the frame work of behavioral finance. We represent different states of an economy and, consequently, investors’ floating levels of psychological reactions by a D-state Markov chain. The first application is a zero-sum game between an investor and the market, and the second one formulates a nonzero-sum stochastic differential portfolio game as the sensitivity of two investors’ terminal gains. We derive regime-switching Hamilton–Jacobi–Bellman–Isaacs equations and obtain explicit optimal portfolio strategies with Feynman–Kac representations of value functions. We illustrate our results in a two-state special case and observe the impact of regime switches by comparative results.

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References

  1. 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. https://doi.org/10.1016/j.automatica.2014.05.033.

    Article  Google Scholar 

  2. 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 

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

  4. 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 

  5. De Bondt, W. F., Muradoglu, Y. G., Shefrin, H., & Staikouras, S. K. (2008). Behavioral finance: Quo vadis? Journal of Applied Finance (Formerly Financial Practice and Education), 18(2).

  6. 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. https://doi.org/10.1016/S0304-4149(03)00059-0.

    Article  Google Scholar 

  7. 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. https://doi.org/10.1007/s10479-008-0448-5.

    Article  Google Scholar 

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

    Google Scholar 

  9. Gökgöz, N., Öktem, H., & Weber, G. W. (2020). Modeling of tumor-immune nonlinear stochastic dynamics with hybrid systems with memory approach. Results in Nonlinear Analysis, 3(1), 24–34.

    Google Scholar 

  10. 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.

    Article  Google Scholar 

  11. Isaacs, R. (1965). Differential games: a mathematical theory with applications to warfare and pursuit., Control and optimization New York: Wiley.

    Google Scholar 

  12. Karoui, N. E., & M’Rad, M. (2010). Stochastic utilities with a given optimal portfolio: approach by stochastic flows. arXiv preprint arXiv:10045192.

  13. Korn, R., Melnyk, Y., & Seifried, F. T. (2017). Stochastic impulse control with regime-switching dynamics. European Journal of Operations Research, 260(3), 1024–1042. https://doi.org/10.1016/j.ejor.2016.12.029.

    Article  Google Scholar 

  14. Kourtidis, D., Šević, Ž., & Chatzoglou, P. (2011). Investors’ trading activity: a behavioural perspective and empirical results. The Journal of Socio-Economics, 40(5), 548–557.

    Article  Google Scholar 

  15. Kuhnen, C. M., & Knutson, B. (2011). The influence of affect on beliefs, preferences, and financial decisions. Journal of Financial and Quantitative Analysis, 46(3), 605–626.

    Article  Google Scholar 

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

  17. 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. https://doi.org/10.1080/1351847X.2015.1050526.

    Article  Google Scholar 

  18. Loewenstein, G. (2000). Emotions in economic theory and economic behavior. American Economic Review, 90(2), 426–432.

    Article  Google Scholar 

  19. 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. https://doi.org/10.1002/oca.2160.

    Article  Google Scholar 

  20. 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,. https://doi.org/10.1155/2015/738181.

    Article  Google Scholar 

  21. Mao, X., & Yuan, C. (2006). Stochastic differential equations with Markovian switching. London: Imperial College Press.

    Book  Google Scholar 

  22. Masood, O., Aktan, B., Gavurova, B., Fakhry, B., Tvaronaviciene, M., & Martinkute-Kauliene, R. (2017). The impact of regime-switching behaviour of price volatility on efficiency of the US sovereign debt market. Economic Research-Ekonomska Istrazivanja, 30(1), 1865–1881.

    Article  Google Scholar 

  23. 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. https://doi.org/10.1080/17442500701655408.

    Article  Google Scholar 

  24. 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. https://doi.org/10.1007/s00186-017-0574-4.

    Article  Google Scholar 

  25. Øksendal, B. (2003). Stochastic differential equations: an introduction with applications. Germany: Springer.

    Book  Google Scholar 

  26. Øksendal, B., & Sulem, A. (2007). Applied stochastic control of jump diffusions. Berlin: Springer.

    Book  Google Scholar 

  27. Papantaleon, A. (2000). An introduction to Lévy processes with application in finance. Freiburg: University of Freiburg.

    Google Scholar 

  28. Peskir, G., & Shorish, J. (2002). Market forces and dynamic asset pricing. Stochastic Analysis and Applications, 20(5), 1027–1082. https://doi.org/10.1081/SAP-120014553.

    Article  Google Scholar 

  29. 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 

  30. Savku, E., Azevedo, N., & Weber, G. (2014). Optimal control of stochastic hybrid models in the framework of regime switches. International Conference on Dynamics (pp. 371–387). Games and Science: Springer.

  31. Shefrin, H., & Statman, M. (1993). Behavioral aspects of the design and marketing of financial products. Financial Management, 1, 123–134.

    Article  Google Scholar 

  32. 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. https://doi.org/10.1016/j.insmatheco.2013.09.016.

    Article  Google Scholar 

  33. Statman, M. (2008). What is behavioral finance? handbook of Finance. Wiley Online Library 2.

  34. Subrahmanyam, A. (2008). Behavioural finance: a review and synthesis. European Financial Management, 14(1), 12–29.

    Google Scholar 

  35. Wu, C. C., Sacchet, M. D., & Knutson, B. (2012). Toward an affective neuroscience account of financial risk taking. Frontiers in Neuroscience, 6, 159.

    Article  Google Scholar 

  36. Yong, J., & Zhou, X. Y. (1999). Stochastic controls: Hamiltonian systems and HJB equations. New York: Springer.

    Book  Google Scholar 

  37. 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. https://doi.org/10.1137/110839357.

    Article  Google Scholar 

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Correspondence to E. Savku.

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This work extends a part of Emel Savku’s PhD thesis and the second author is the advisor of the thesis. All authors contributed to the study conception, design and the second author motivated about neuroscience. The first draft of the manuscript was written by Emel Savku as the principal author and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript

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Appendix

Appendix

Let us show that the nonzero-sum game application illustrated in Sect. 4.1 satisfies the conditions of the verification theorem, Theorem 2.

Please remember that \(\mu _{m}(t):=\mu _{m}(t,\alpha (t))=\left\langle \mu _{m},\alpha (t)\right\rangle =(\mu _{m}^{1},\mu _{m}^{2},\ldots ,\mu _{m}^{D})^{T}\in \mathbb {R}^{D}\), \(\sigma _{m}(t):=\sigma _{m}(t,\alpha (t))=\left\langle \sigma _{m},\alpha (t)\right\rangle =(\sigma _{m}^{1},\sigma _{m}^{2},\ldots ,\sigma _{m}^{D})^{T}\in \mathbb {R}^{D}\), and \(\eta _{m}(t,z):= \eta _{m}(t,\alpha (t),z)=\left\langle \eta _{m},\alpha (t)\right\rangle =(z\eta _{m}^{1},z\eta _{m}^{2},\ldots ,z\eta _{m}^{D})^{T}\in \mathbb {R}^{D}\) denote the appreciation rate, the volatility rate, and the jump size of the mth risky asset for \(m=1,2\), \(t\in [0,T]\).

Hence, we are working on a family of geometric Lévy processes, where we assume

$$\begin{aligned} \nu (\left\{ 0\right\} )=0 \qquad \hbox {and} \qquad \int _{\mathbb {R}}(1\wedge |z|^{2})\nu (dz)<\infty . \end{aligned}$$
(22)

Infinitely many jumps can occur around the origin (i.e., small jumps); however, the mass away from the origin is bounded; i.e., only a finite number of big jumps can occur [for more details see (Papantaleon 2000)]. Then, \(\pi _{1}^{*}\) and \(\pi _{2}^{*}\), given by Eqs. (15) and (16) correspondingly, are constant functions at each state, \(e_{i}, \ i=1,2,\ldots ,D\). Therefore, \(h(t,\alpha (t))\), \(t\in [0,T]\) is a piecewise continuous function, then \(h(t,\alpha (t))\), \(t\in [0,T]\) is uniformly bounded. Consequently,

$$\begin{aligned} k_{m}(t,e_{i})=\gamma _{m}E\biggl [\exp \biggl (\int _{t}^{T}h(s,\alpha (s))ds\biggr )|\alpha (t)=e_{i}\biggr ], \qquad \hbox {for} \quad m=1,2,\ e_{i}\in S, \end{aligned}$$

are well defined. By dominated convergence theorem, \(k_{m}(t,e_{i})\) is continuously differentiable with respect to t for each \(e_{i}\in S\) and \(m=1,2\).

Then, we can easily see that the value functions

$$\begin{aligned} V_{m}(t,x_{1},x_{1},e_{i})=k_{m}(t,e_{i})x_{1}x_{2}, \quad m=1,2, \end{aligned}$$

have continuous partial derivatives with respect to \(x_{1}\) and \(x_{2}\) in any order. Therefore, the value functions are in \(C^{1,2}(G)\cap C(\bar{G})\) for each \(e_{i}\in S\) and for each investor as required to satisfy Theorem 2.

The conditions (i)–(ii) are direct consequences of first order conditions, which have been applied to obtain \(k_{m}(t,\alpha (t))\), \(m=1,2, \ t\in [0,T]\).

(iii) By above explanations, \(\left| k_{m}(t,\alpha (t))\right| \le \gamma _{m}\exp ((T-t)L)\),   \(L>0\). Then, by dominated convergence theorem,

$$\begin{aligned}&\lim _{t\rightarrow T^{-}}\phi _{m}(t,y^{\pi _{1},\pi _{2}},\alpha (t))=\lim _{t\rightarrow T^{-}}k_{m}(t,\alpha (t))x_{1}x_{2} \\&\quad =\lim _{t\rightarrow T^{-}}\gamma _{m}E\biggl [\exp \biggl (\int _{t}^{T}h(s,\alpha (s))ds\biggr )|\alpha (t)=e_{i}\biggr ]x_{1}x_{2} \\&\quad =\gamma _{m}x_{1}x_{2}, \end{aligned}$$

for all \((\pi _{1},\pi _{2})\in \varTheta _{1}\times \varTheta _{2}, \ m=1,2.\)

(iv) If we show that the family \((\phi _{m}(t,Y^{\pi _{1},\pi _{2}}(t),\alpha (t)))_{t\in [0,T]}\) is in \(\mathbb {L}^{2}\) for all \((\pi _{1},\pi _{2})\in \varTheta _{1}\times \varTheta _{2}, \ m=1,2\), then uniform integrability condition will be satisfied.

Our wealth processes are as follows:

$$\begin{aligned} dX_{m}(t)&=X_{m}(t-)\biggl (\pi _{m}(t)\mu _{m}(t,\alpha (t-))+(1-\pi _{m}(t))r(t,\alpha (t-))\biggr )dt+X_{m}(t-) \nonumber \\&\quad \times \pi _{m}(t)\biggl (\sigma _{m}(t,\alpha (t-))dW(t)+\int _{\mathbb {R}_{0}}\eta _{m}(t,\alpha (t-),z)\tilde{N}(dt,dz)\biggr ),\ t\in [0,T], \\&\quad X_{m}(0)=x_{m}>0. \end{aligned}$$

By Itô’s differentiation rule on \(f(s,X_{m}(s),\alpha (s))=\ln (X_{m}(s))\), \(m=1,2, \ s\in [0,T]\) (see Theorem 4.1 by Zhang et al. (2012)), we obtain:

$$\begin{aligned} X_{m}(t)&=x_{m}\exp \biggl \{\int _{0}^{t} \biggl ((\pi _{m}(s)\mu _{m}(s,\alpha (s-))+(1-\pi _{m}) r(s,\alpha (s-)))-\frac{1}{2}\pi _{m}^{2}(s)\\&\quad \times \sigma _{m}(s,\alpha (s-))\biggr )ds +\int _{0}^{t}\int _{\mathbb {R}_{0}} \biggl (\ln (1+\pi _{m}(s)\eta _{m}(s,\alpha (s-),z))\\&\quad -\pi _{m}(s)\eta _{m}(s,\alpha (s-),z)\biggr ) \nu (dz)ds+\int _{0}^{t}\pi _{m}(s)\sigma _{m}(s,\alpha (s-))dW(s)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}} \ln (1+\pi _{m}(s)\eta _{m}(s,\alpha (s-),z))\tilde{N}(ds,dz)\biggr \}. \end{aligned}$$

Then,

$$\begin{aligned}&X_{1}(t)X_{2}(t)=x_{1}x_{2}\exp \biggl \{ \int _{0}^{t}\biggl (\pi _{1}(s)\mu _{1}(s,\alpha (s-))+\pi _{2}(s)\mu _{2}(s,\alpha (s-)) \\&\quad +(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\frac{1}{2}(\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))\\&\quad +\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-)))\biggl )ds+\int _{0}^{t}\biggl (\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-))\biggr )dW(s)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl (\ln (1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))+\ln (1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )\tilde{N}(ds,dz)\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl ( (\ln (1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))+\ln (1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)))\\&\quad -\pi _{1}(s)\eta _{1}(s,\alpha (s-),z)-\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)\biggr )\nu (dz)ds \biggr \}. \end{aligned}$$

Therefore, we can state:

$$\begin{aligned}&E[X_{1}^{2}(t)X_{2}^{2}(t)]=x_{1}x_{2}E\biggl [\exp \biggl \{ \biggl (\int _{0}^{t}2\pi _{1}(s)\mu _{1}(s,\alpha (s-))+2\pi _{2}(s)\mu _{2}(s,\alpha (s-))\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))-\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-))\biggr )ds\\&\quad +\int _{0}^{t}2\biggl (\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-))\biggr )dW(s) \\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\tilde{N}(ds,dz) \\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl (\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2} \\&\quad -2\pi _{1}(s)\eta _{1}(s,\alpha (s-),z)-2\pi _{2}(s)\eta _{2}(s,\alpha (s-),z)\biggr )\nu (dz)ds\biggr \}\biggr ]. \end{aligned}$$

Since \(\pi _{m}\) and \(\sigma _{m}\), for \(m=1,2\), are piecewise constant, then we can find a \(K>0\) such that

$$\begin{aligned}&E\biggl [\exp \biggl (\int _{0}^{t}2(\pi _{1}(s)\sigma _{1}(s,\alpha (s-))+\pi _{2}(s)\sigma _{2}(s,\alpha (s-)))dW(s)\biggr )\biggr ] \\&\quad \le E\biggl [\exp \biggl (\int _{0}^{t}2K\biggr )dW(s)\biggr )\biggr ]=\exp (2K^{2}t)< \infty . \end{aligned}$$

Moreover, if we apply Example 1.6 in Øksendal and Sulem (2007) state by state, we reach

$$\begin{aligned}&E\biggl [\exp \biggl (\int _{0}^{t}\int _{\mathbb {R}_{0}}\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\tilde{N}(ds,dz)\biggr )\biggr ] \\&\quad =\exp \biggl (\int _{0}^{t}\int _{\mathbb {R}_{0}}\biggl \{2(1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))-1\\&\quad -\ln \biggl ((1+\pi _{1}(s)\eta _{1}(s,\alpha (s-),z))(1+\pi _{2}(s)\eta _{2}(s,\alpha (s-),z))\biggr )^{2}\biggr \}\nu (dz)ds\biggr ). \end{aligned}$$

Hence, by Eq. (22) and by the construction of coefficients,

$$\begin{aligned}&E[k_{m}^{2}(t,\alpha (t))X_{1}^{2}(t)X_{2}^{2}(t)]\le \gamma ^{2}_{m}\exp (2(T-t)L))E[X_{1}^{2}(t)X_{2}^{2}(t)] \\&\quad \le \gamma ^{2}_{m}\exp (2(T-t)L+2K^{2}t)x_{1}x_{2}E\biggl [\exp \biggl \{\biggl (\int _{0}^{t}2\pi _{1}(s)\mu _{1}(s,\alpha (s-))+2\pi _{2}(s)\mu _{2}(s,\alpha (s-))\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r(s,\alpha (s-))-\pi _{1}^{2}(s)\sigma _{1}^{2}(s,\alpha (s-))-\pi _{2}^{2}(s)\sigma _{2}^{2}(s,\alpha (s-))\biggr )ds\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}(1+\pi _{1}(s)\pi _{2}(s)\eta _{1}(s,\alpha (s-),z)\eta _{2}(s,\alpha (s-),z))\nu (dz)ds\biggr ]< \infty , \quad m=1,2, \end{aligned}$$

i.e.,

$$\begin{aligned}&E[k_{m}^{2}(t,e_{i})X_{1}^{2}(t)X_{2}^{2}(t)]\le \gamma ^{2}_{m}\exp (2(T-t)L))E[X_{1}^{2}(t)X_{2}^{2}(t)] \\&\quad \le \gamma ^{2}_{m}\exp (2(T-t)L+2K^{2}t)x_{1}x_{2}E\biggl [\exp \biggl \{\biggl (\int _{0}^{t}2\pi _{1}(s)\mu ^{i}_{1}+2\pi _{2}(s)\mu ^{i}_{2}\\&\quad +2(2-\pi _{1}(s)-\pi _{2}(s))r^{i}-\pi _{1}^{2}(s)(\sigma ^{i}_{1})^{2}-\pi _{2}^{2}(s)(\sigma ^{i}_{2})^{2}\biggr )ds\\&\quad +\int _{0}^{t}\int _{\mathbb {R}_{0}}(1+\pi _{1}(s)\pi _{2}(s)\eta ^{i}_{1}\eta ^{i}_{2}z^{2})\nu (dz)ds\biggr ]< \infty , \quad m=1,2, \quad i=1,2,\ldots ,D. \end{aligned}$$

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Savku, E., Weber, GW. Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market. Ann Oper Res (2020). https://doi.org/10.1007/s10479-020-03768-5

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Keywords

  • Control
  • Stochastic processes
  • Behavioral finance
  • Game theory
  • Dynamic programming