{"ID":2879687,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.15241","arxiv_id":"2508.15241","title":"Differential Stochastic Variational Inequalities with Parametric Optimization","abstract":"The differential stochastic variational inequality with parametric convex optimization (DSVI-O) is an ordinary differential equation whose right-hand side involves a stochastic variational inequality and solutions of several dynamic and random parametric convex optimization problems. We consider that the distribution of the random variable is time-dependent and assume that the involved functions are continuous and the expectation is well-defined. We show that the DSVI-O has a weak solution with integrable and measurable solutions of the parametric optimization problems. Moreover, we propose a discrete scheme of DSVI-O by using a time-stepping approximation and the sample average approximation and prove the convergence of the discrete scheme. We illustrate our theoretical results of DSVI-O with applications in an embodied intelligence system for the elderly health by synthetic health care data generated by Multimodal Large Language Models.","short_abstract":"The differential stochastic variational inequality with parametric convex optimization (DSVI-O) is an ordinary differential equation whose right-hand side involves a stochastic variational inequality and solutions of several dynamic and random parametric convex optimization problems. We consider that the distribution o...","url_abs":"https://arxiv.org/abs/2508.15241","url_pdf":"https://arxiv.org/pdf/2508.15241v2","authors":"[\"Xiaojun Chen\",\"Jian Guo\",\"Guan Wang\"]","published":"2025-08-21T05:03:47Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.DS\"]","methods":"[\"Language Model\"]","has_code":false}