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June  2020, 10(2): 365-378. doi: 10.3934/mcrf.2020001

Optimal investment problem with delay under partial information

Shuaiqi Zhang 1,, , Jie Xiong 2, and  Xin Zhang 3,

1. 

School of Mathematics, China University of Mining and Technology, Jiangsu 221116, China
2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
3. 

School of Mathematics, Southeast University, Nanjing, China

* Corresponding author: Shuaiqi Zhang

Received  July 2018 Revised  December 2018 Published  November 2019

In this paper, we investigate the optimal investment problem in the presence of delay under partial information. We assume that the financial market consists of one risk free asset (bond) and one risky asset (stock) and only the price of the risky asset can be observed from the financial market. The objective of the investor is to maximize the expected utility of the terminal wealth and average of the path segment. By using the filtering theory, we establish the separation principle and reduce the problem to the complete information case. Explicit expressions for the value function and the corresponding optimal strategy are obtained by solving the corresponding Hamilton-Jacobi-Bellman equation. Furthermore, we study the sensitivity of the optimal investment strategy on the model parameters in a numerical section and both of the full and partial information schemes are simulated and compared.

Keywords: Partial information, investment, delay.
Mathematics Subject Classification: Primary: 49L20, 37N40; Secondary: 93E11.
Citation: Shuaiqi Zhang, Jie Xiong, Xin Zhang. Optimal investment problem with delay under partial information. Mathematical Control & Related Fields, 2020, 10 (2) : 365-378. doi: 10.3934/mcrf.2020001
References:
[1]

C. X. A and Z. F. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.  Google Scholar

[2]

L. H. Bai and J. Y. Guo, Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach, Front. Math. China, 2 (2007), 527-537.  doi: 10.1007/s11464-007-0032-3.  Google Scholar

[3]

T. Björk, M. H. Davis and C. Landén, Optimal investment under partial information, Math. Methods Oper. Res., 71 (2010), 371-399.  doi: 10.1007/s00186-010-0301-x.  Google Scholar

[4]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Math. Oper. Res., 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508.  Google Scholar

[6]

I. Elsanousi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259.  Google Scholar

[7]

H. Hata and S. J. Sheu, An optimal consumption and investment problem with partial information, Adv. in Appl. Probab., 50 (2018), 131-153.  doi: 10.1017/apr.2018.7.  Google Scholar

[8]

P. Lakner, Utility maximization with partial information, Stochastic Process. Appl., 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[9]

B. Larssen, Dynamic programming in stochastic control of systems with delay, Stoch. Stoch. Rep., 74 (2002), 651-673.  doi: 10.1080/1045112021000060764.  Google Scholar

[10]

Z. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance Math. Econom., 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008.  Google Scholar

[11]

B. Øksendal, A. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. in Appl. Probab., 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.  Google Scholar

[12]

T. Pang and A. Hussain, A stochastic portfolio optimization model with complete memory, Stoch. Anal. Appl., 35 (2017), 742-766.  doi: 10.1080/07362994.2017.1299629.  Google Scholar

[13]

T. Pang and A. Hussain, An infinite time horizon portfolio optimization model with delays, Math. Control Relat. Fields, 6 (2016), 629-651.  doi: 10.3934/mcrf.2016018.  Google Scholar

[14]

T. Pang and A. Hussain, An application of functional Ito's formula to stochastic portfolio optimization with bounded memory, Proceedings of the SIAM Conference on Control and Its Applications, Paris, France, 2015,159–166. doi: 10.1137/1.9781611974072.23.  Google Scholar

[15]

X. C. Peng and Y. J. Hu, Optimal proportional reinsurance and investment under partial information, Insurance Math. Econom., 53 (2013), 416-428.  doi: 10.1016/j.insmatheco.2013.07.004.  Google Scholar

[16]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 565-601. doi: 10.1016/B978-0-12-775850-3.50017-0.  Google Scholar

[17]

Y. Shen and Y. Zeng, Optimal investment re-insurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004.  Google Scholar

[18] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.   Google Scholar
[19]

J. Xiong and X. Y. Zhou, Mean-variance portfolio selection under partial information, SIAM J. Control Optim., 46 (2007), 156-175.  doi: 10.1137/050641132.  Google Scholar

[20]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[21]

Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

show all references

References:
[1]

C. X. A and Z. F. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005.  Google Scholar

[2]

L. H. Bai and J. Y. Guo, Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach, Front. Math. China, 2 (2007), 527-537.  doi: 10.1007/s11464-007-0032-3.  Google Scholar

[3]

T. Björk, M. H. Davis and C. Landén, Optimal investment under partial information, Math. Methods Oper. Res., 71 (2010), 371-399.  doi: 10.1007/s00186-010-0301-x.  Google Scholar

[4]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[5]

M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Math. Oper. Res., 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508.  Google Scholar

[6]

I. Elsanousi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics and Stochastic Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259.  Google Scholar

[7]

H. Hata and S. J. Sheu, An optimal consumption and investment problem with partial information, Adv. in Appl. Probab., 50 (2018), 131-153.  doi: 10.1017/apr.2018.7.  Google Scholar

[8]

P. Lakner, Utility maximization with partial information, Stochastic Process. Appl., 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar

[9]

B. Larssen, Dynamic programming in stochastic control of systems with delay, Stoch. Stoch. Rep., 74 (2002), 651-673.  doi: 10.1080/1045112021000060764.  Google Scholar

[10]

Z. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance Math. Econom., 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008.  Google Scholar

[11]

B. Øksendal, A. Sulem and T. Zhang, Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations, Adv. in Appl. Probab., 43 (2011), 572-596.  doi: 10.1239/aap/1308662493.  Google Scholar

[12]

T. Pang and A. Hussain, A stochastic portfolio optimization model with complete memory, Stoch. Anal. Appl., 35 (2017), 742-766.  doi: 10.1080/07362994.2017.1299629.  Google Scholar

[13]

T. Pang and A. Hussain, An infinite time horizon portfolio optimization model with delays, Math. Control Relat. Fields, 6 (2016), 629-651.  doi: 10.3934/mcrf.2016018.  Google Scholar

[14]

T. Pang and A. Hussain, An application of functional Ito's formula to stochastic portfolio optimization with bounded memory, Proceedings of the SIAM Conference on Control and Its Applications, Paris, France, 2015,159–166. doi: 10.1137/1.9781611974072.23.  Google Scholar

[15]

X. C. Peng and Y. J. Hu, Optimal proportional reinsurance and investment under partial information, Insurance Math. Econom., 53 (2013), 416-428.  doi: 10.1016/j.insmatheco.2013.07.004.  Google Scholar

[16]

J. Serrin, Gradient estimates for solutions of nonlinear elliptic and parabolic equations, in Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, 565-601. doi: 10.1016/B978-0-12-775850-3.50017-0.  Google Scholar

[17]

Y. Shen and Y. Zeng, Optimal investment re-insurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004.  Google Scholar

[18] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008.   Google Scholar
[19]

J. Xiong and X. Y. Zhou, Mean-variance portfolio selection under partial information, SIAM J. Control Optim., 46 (2007), 156-175.  doi: 10.1137/050641132.  Google Scholar

[20]

H. L. Yang and L. H. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[21]

Y. Zeng and Z. F. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001.  Google Scholar

Figure 1.  Comparison of $ \hat\mu $
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Figure 2.  Comparison of $ v $
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Figure 3.  Comparison of $ V $
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Figure 4.  Comparison of $ v $
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Figure 5.  Comparison of $ V $
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Figure 6.  Comparison of $ \hat\mu $
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Figure 7.  Comparison of $ v $
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Figure 8.  Comparison of $ V $
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Figure 9.  Comparison of $ V $ and $ \bar V $
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