Mean-variance portfolio selection: connections between mean-field systems and forward-backward stochastic systems under initial-terminal cost coupling
Abstract
This paper investigates the connection between mean-field systems and forward-backward stochastic systems arising from the mean-variance portfolio selection problem. By introducing the conditional expectation process of terminal wealth, the mean-field term is represented by a backward state, leading to a class of control problems for forward-backward stochastic systems with coupled initial-terminal values. The coupling between the terminal value of the forward state and the initial value of the backward state in the cost functional prevents classical methods for forward-backward stochastic optimal control from being directly applicable. To overcome this difficulty, we establish a stochastic maximum principle for control problems with running costs and coupled initial-terminal values, and provide sufficient conditions for optimality. Furthermore, we investigate a class of generalized stochastic linear-quadratic control problems with initial-terminal coupling, allowing the control weight in the running cost to be indefinite, and derive the optimal feedback representation via a Riccati equation. The classical mean-variance portfolio selection model is shown to be a special case, for which the corresponding efficient investment strategy and efficient frontier are obtained explicitly, and numerical simulations are presented to illustrate the results and examine parameter sensitivity and its financial implications.
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BibTeX
@misc{li2026meanvariance,
title = {{Mean-variance portfolio selection: connections between mean-field systems and forward-backward stochastic systems under initial-terminal cost coupling}},
author = {Xun Li and Hongyu Shi and Zhen Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02485},
eprint = {2610.02485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02485},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mean-variance portfolio selection: connections between mean-field systems and forward-backward stochastic systems under initial-terminal cost coupling
AU - Xun Li
AU - Hongyu Shi
AU - Zhen Wu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02485
UR - https://arxiv.org/abs/2610.02485
AB - This paper investigates the connection between mean-field systems and forward-backward stochastic systems arising from the mean-variance portfolio selection problem. By introducing the conditional expectation process of terminal wealth, the mean-field term is represented by a backward state, leading to a class of control problems for forward-backward stochastic systems with coupled initial-terminal values. The coupling between the terminal value of the forward state and the initial value of the backward state in the cost functional prevents classical methods for forward-backward stochastic optimal control from being directly applicable. To overcome this difficulty, we establish a stochastic maximum principle for control problems with running costs and coupled initial-terminal values, and provide sufficient conditions for optimality. Furthermore, we investigate a class of generalized stochastic linear-quadratic control problems with initial-terminal coupling, allowing the control weight in the running cost to be indefinite, and derive the optimal feedback representation via a Riccati equation. The classical mean-variance portfolio selection model is shown to be a special case, for which the corresponding efficient investment strategy and efficient frontier are obtained explicitly, and numerical simulations are presented to illustrate the results and examine parameter sensitivity and its financial implications.
ER -