Stochastic differential games on robust optimal asset-liability management with delay under the CEV model

Xueping Luo; Qing Zhou

doi:10.3934/jimo.2024148

Article Contents

2025, Volume 21, Issue 3: 1771-1796. Doi: 10.3934/jimo.2024148

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Stochastic differential games on robust optimal asset-liability management with delay under the CEV model

Xueping Luo^{1, 2,} and
Qing Zhou^{1, 2, ,}

1.
School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China
2.
Key Laboratory of Mathematics and Information Networks, Beijing, University of Posts and Telecommunications, Ministry of Education, Beijing, 100876, China

^*Corresponding author: Qing Zhou
^*Corresponding author: Qing Zhou

Received: March 2024

Revised: September 2024

Early access: October 2024

Published: March 2025

This work is supported by the National Key R&D Program of China [No. 2023YFA1009204] and the Fundamental Research Funds for the Central Universities [No. 2023ZCJH02].

Abstract / Introduction Full Text(HTML) Figure(5) / Table(2) Related Papers Cited by

Abstract

This paper studies the stochastic differential game problem of robust optimal asset-liability management between two ambiguity-averse investors with delay under the constant elasticity of variance (CEV) model, aiming to optimize resource allocation, enhance profitability, manage risks and provide guidance. In our model, each investor has access to a risk-free asset and a risky asset whose price process follows the CEV model in a financial market. Their competitive relationship is characterized by the non-zero-sum stochastic differential game where they consider the relative performance measured by the difference in terminal wealth. Each investor aims to maximize the utility of the combination of his terminal wealth and the relative performance with delay. For cases of the power utility and the mean-variance utility, this paper utilizes robust optimal control theory and dynamic programming principle to obtain explicit expressions of the investment strategies and value functions. Finally, we provide theoretical guidance for asset-liability management (ALM) by illustrating the impact of model parameters on the optimal investment strategy under the expected power utility framework through several numerical examples.

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Mathematics Subject Classification: Primary: 91A15, 91B70; Secondary: 91G10, 93E20.

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Figure 1. Effect of $ \eta $ on $ \pi_k^*(t) $, for $ k = 1, 2 $

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Figure 2. Effect of $ \lambda_k $ on $ \pi_k^*(t) $, for $ k = 1, 2 $

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Figure 3. Effect of $ \sigma_s $ and $ \mu_s $ on $ \pi_k^*(t) $, for $ k = 1, 2 $

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Figure 4. Effect of $ \rho_{k, i} $ and $ \lambda_k $ on $ \pi_k^*(t) $, for $ k = 1, 2 $

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Figure 5. Effect of $ \eta_{k, 1} $ and $ \eta_{k, 2} $ on $ \pi_k^*(t) $, for $ k = 1, 2 $

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Table 1. The common parameters values

$r$	$ \mu_s$	$ \sigma_s$	$ \beta$	s	t	T	$ \eta$
0.05	0.1	0.4	1	1	0	10	0.4

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Table 2. The parameter values of investors

investor 1		investor 2
$ \eta_{1, 1} $	0.04	$ \eta_{2, 1} $	0.03
$ \delta_1 $	0.05	$ \delta_2 $	0.03
$ \lambda_1 $	0.3	$ \lambda_2 $	0.7
$ \gamma_1 $	0.2	$ \gamma_2 $	0.3
$ h_1 $	2	$ h_2 $	3
$ \rho_1 $	0.5	$ \rho_2 $	0.4
$ \omega_1 $	0.3	$ \omega_2 $	0.1

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