# American Institute of Mathematical Sciences

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

## Optimal investment problem with delay under partial information

 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  June 2020 Early access  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.

Citation: Shuaiqi Zhang, Jie Xiong, Xin Zhang. Optimal investment problem with delay under partial information. Mathematical Control and 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. [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. [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. [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. [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. [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. [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. [8] P. Lakner, Utility maximization with partial information, Stochastic Process. Appl., 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3. [9] B. Larssen, Dynamic programming in stochastic control of systems with delay, Stoch. Stoch. Rep., 74 (2002), 651-673.  doi: 10.1080/1045112021000060764. [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. [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. [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. [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. [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. [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. [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. [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. [18] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [8] P. Lakner, Utility maximization with partial information, Stochastic Process. Appl., 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3. [9] B. Larssen, Dynamic programming in stochastic control of systems with delay, Stoch. Stoch. Rep., 74 (2002), 651-673.  doi: 10.1080/1045112021000060764. [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. [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. [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. [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. [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. [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. [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. [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. [18] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics, 18, Oxford University Press, Oxford, 2008. [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. [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. [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.
Comparison of $\hat\mu$
Comparison of $v$
Comparison of $V$
Comparison of $v$
Comparison of $V$
Comparison of $\hat\mu$
Comparison of $v$
Comparison of $V$
Comparison of $V$ and $\bar V$
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