American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021137
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A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory

 1 Center for Advanced Statistics and Econometrics Research, School of Mathematical Sciences, Soochow University, 1 Shi-Zi Street, Suzhou 215000, China 2 School of Mathematics, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China

* Corresponding author: Lei Hu

Received  January 2021 Revised  April 2021 Early access August 2021

Fund Project: The work is supported by NSF grant No.11871435

Portfolio optimization problem with memory effect, namely, taking the past performance of portfolio into account, which is motivated by the fact that the decisions of the investors are influenced by the historic performance of the portfolio, has recently been of interest to researchers. Due to the memory effect, this type of problem is challenging. The main purpose of this paper is to propose a new approach to solve the Hamilton-Jacobi-Bellman (HJB) equation for the stochastic portfolio optimization model, which is more simple and generalized. We study the portfolio management problem under single investor and multi-investor framework. We establish certain conditions under which the problem can be reduced to the classical stochastic control problem and give some examples to show the advantages of our approach.

Citation: Xuhui Wang, Lei Hu. A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021137
References:
 [1] H. Bauer and U. Rieder, Stochastic control problems with delay, Math. Methods Oper. Res., 62 (2005), 411-427.  doi: 10.1007/s00186-005-0042-4. [2] 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. [3] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics Stochastics Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [4] D. Lacker and T. Zariphopoulou, Mean field and n-agent games for optimal investment under relative performance criteria, Math. Financ., 29 (2019), 1003-1038.  doi: 10.1111/mafi.12206. [5] B. Larssen and N. H. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?, Stochastic Anal. Appl., 21 (2003), 643-671.  doi: 10.1081/SAP-120020430. [6] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat., 51 (1969), 247-257.  doi: 10.2307/1926560. [7] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [8] S. E. A. Mohammed, Stochastic Functional Differential Equations, vol. 99 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. [9] B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, 64–79. [10] N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, vol. 29 of Fields Institute Monographs, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013, With Chapter 13 by Angès Tourin. doi: 10.1007/978-1-4614-4286-8.

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References:
 [1] H. Bauer and U. Rieder, Stochastic control problems with delay, Math. Methods Oper. Res., 62 (2005), 411-427.  doi: 10.1007/s00186-005-0042-4. [2] 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. [3] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics Stochastics Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [4] D. Lacker and T. Zariphopoulou, Mean field and n-agent games for optimal investment under relative performance criteria, Math. Financ., 29 (2019), 1003-1038.  doi: 10.1111/mafi.12206. [5] B. Larssen and N. H. Risebro, When are HJB-equations in stochastic control of delay systems finite dimensional?, Stochastic Anal. Appl., 21 (2003), 643-671.  doi: 10.1081/SAP-120020430. [6] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat., 51 (1969), 247-257.  doi: 10.2307/1926560. [7] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X. [8] S. E. A. Mohammed, Stochastic Functional Differential Equations, vol. 99 of Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1984. [9] B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, in Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, 64–79. [10] N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE, vol. 29 of Fields Institute Monographs, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013, With Chapter 13 by Angès Tourin. doi: 10.1007/978-1-4614-4286-8.
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