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Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 ISSN Print: 2576-0645 CODEN: JAMCEZ
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ArticleOpen Access http://dx.doi.org/10.26855/jamc.2025.09.002

Stochastic Analysis of Local Risk Minimization Strategies with Multiple Assets, Including Jump Processes

W. Nangolo1,*, R. Gnitchogna2

1Department of Computing, Mathematical & Statistical Science, University of Namibia, Private Bag 13301, Windhoek, Namibia. 

2Department of Mathematics Statistics and Actuarial Sciences, Namibia University of Science and Technology, Private Bag 13388, Windhoek, Namibia.

*Corresponding author:W. Nangolo

Published: October 15, 2025

Abstract

This study quantifies risk and mitigates hedging ambiguity in incomplete multi-asset financial markets using Local Risk Minimization (LRM) strategies. In such markets, the absence of unique asset price distributions requires robust methodologies to assess residual risk. This work focuses on the difficulties that arise from the multiplicity of equivalent martingale measures, which creates a set of arbitrage-free valuations rather than a single price. The study emphasizes the need for risk management tools that adhere to no-arbitrage principles, are computationally efficient, and can reliably estimate residual risk. Specifically, extend a single-dimensional risk model to a multi-asset framework by employing Local Risk Minimization (LRM) strategies. This approach is used to develop an uncertainty quantification model for incomplete multi-asset markets that explicitly includes stochastic jump processes. This allows for a more comprehensive analysis of hedging and managing the unhedgeable risks that arise from market frictions and sudden price jumps, ultimately providing a more robust methodology for handling financial derivatives. This is formally connected to the Föllomer-Schweizer (FS) decomposition, which expresses a contingent claim as an initial endowment, a hedged component, and an unhedgeable martingale, thus defining the optimal trading strategy. The findings show that LRM significantly reduces the diffusion risk, but a quantifiable residual jump risk persists, underscoring the need for a framework that explicitly manages these sudden market movements in derivative instruments.

Keyword

Uncertainty quantification; Locally Risk-Minimizing (LRM) strategies; Poisson process; Stochastic differential equation; Jump process; Multiple assets

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How to cite this paper

Stochastic Analysis of Local Risk Minimization Strategies with Multiple Assets, Including Jump Processes

How to cite this paper: W. Nangolo, R. Gnitchogna. (2025) Stochastic Analysis of Local Risk Minimization Strategies with Multiple Assets, Including Jump Processes. Journal of Applied Mathematics and Computation9(3), 155-174.

DOI: http://dx.doi.org/10.26855/jamc.2025.09.002