Weak convergence of utility-risk portfolios

Kwok Chuen Wong; Sheung Chi Phillip Yam; Hailiang Yang; Harry Zheng

doi:10.3934/mcrf.2025049

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2025. Doi: 10.3934/mcrf.2025049

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Weak convergence of utility-risk portfolios

1.
School of Mathematical Sciences, Dublin City University, Dublin, Ireland
2.
Department of Statistics and Data Science, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR
3.
Department of Financial and Actuarial Mathematics, Xi'an Jiaotong-Liverpool University, Suzhou, China
4.
Department of Mathematics, Imperial College London, London, United Kingdom

^*Corresponding author: Sheung Chi Phillip Yam
^*Corresponding author: Sheung Chi Phillip Yam

In Celebration of Alain Bensoussan on the Occasion of his 85^th Birthday

Received: July 1, 2025

Revised: August 1, 2025

Early access: September 3, 2025

Published online: September 3, 2025

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Abstract

In this article, we establish the validity of the Donsker invariance principle for weak convergence of the optimal utility-downside-risk portfolio payoff. Thus, the optimal solution is robust to changes in the market model via the pricing kernel. The convergence result is valid for all feasible combinations of utility functions satisfying the Inada conditions, convex downside deviation risk measures, and pricing kernels satisfying integrability conditions. Thus, our results provide a unified framework to understand whether utility-downside-risk solutions are stable under mild misspecifications of asset price processes. As another application, we further obtain a result that the optimal utility-risk values and policies driven by discrete-time binomial models converge weakly to those of the limiting continuous-time Black-Scholes model. We demonstrate the convergence result using numerical examples and find that the optimal utility-risk value converges as the time interval of the binomial model gets smaller.

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Mathematics Subject Classification: Primary: 91G10, 93E20; Secondary: 90C46.

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Table 1. Convergence of optimal solutions under the case of utility-semivariance

$N$	$Y^{(N)}$	$M^{(N)}$	$R^{(N)}$	$\pi^{(N)}(0) $	$U^{(N)}$	$SV^{(N)}$	$J^{(N)} $	Change
25	1.2547	3.9077	1.5571	2.3042	3.4072	2.2446	3.182717
50	1.2561	3.8816	1.5456	2.2517	3.3959	2.2093	3.174965	-0.007752
100	1.2570	3.8666	1.5428	2.2262	3.3902	2.1905	3.171126	-0.003839
200	1.2573	3.8607	1.5394	2.2135	3.3874	2.1823	3.169219	-0.001908
400	1.2575	3.8573	1.5383	2.2073	3.3861	2.1779	3.168274	-0.000945
800	1.2575	3.8557	1.5375	2.2041	3.3854	2.1758	3.167795	-0.000479
1600	1.2576	3.8550	1.5371	2.2025	3.3850	2.1748	3.167557	-0.000238
3200	1.2576	3.8546	1.5369	2.2018	3.3849	2.1742	3.167439	-0.000118
6400	1.2576	3.8544	1.5368	2.2014	3.3848	2.1740	3.167379	-0.000060

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