随机微分博弈和带列维跳跃的统一前后耦合随机偏微分方程

IF 2.3 3区 数学 Q1 MATHEMATICS Mathematics Pub Date : 2024-09-16 DOI:10.3390/math12182891
Wanyang Dai
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引用次数: 0

摘要

我们建立了随机微分博弈(SDGs)与具有不连续莱维跳跃的统一前后耦合随机偏微分方程(SPDE)之间的关系。SDGs 有 q 个参与者,由一般维度的向量 Lévy 过程驱动。通过建立向量形式的伊托-文采公式和统一 SPDE 的 4 元组向量场解,我们得到了帕累托最优纳什均衡政策过程或非零和或零和意义上的 SDG 鞍点政策过程。统一的 SPDE 既有通维向量形式,也有前后耦合方式。其漂移、扩散和跃迁系数中的偏微分算子是域上的时变参数和位置参数。由于统一 SPDE 具有一般非线性和一般高阶,我们将最近的研究从现有的布朗运动(BM)驱动的后向情况扩展到一般的莱维驱动的前向后向耦合情况。在此过程中,我们构建了一个新的拓扑空间,以支持证明统一 SPDE 的适应解的存在性和唯一性,该解在 4 元组强意义上。拓扑空间的构建是通过在一组一般局部条件下构建一组与一组指数{γ1,γ2,...}相关联的拓扑空间,这与单指数情况下的构建有显著不同。此外,由于前向 SPDE 的耦合和不连续 Lévy 跳变的参与,我们的研究也与 BM 驱动的后向情况有显著不同。前向 SPDE 与后向 SPDE 之间的耦合基本上对应于当前生成式人工智能热扩散变压器模型中噪声编码与噪声解码之间的相互作用。
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Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps
We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents {γ1,γ2,…} under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.
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来源期刊
Mathematics
Mathematics Mathematics-General Mathematics
CiteScore
4.00
自引率
16.70%
发文量
4032
审稿时长
21.9 days
期刊介绍: Mathematics (ISSN 2227-7390) is an international, open access journal which provides an advanced forum for studies related to mathematical sciences. It devotes exclusively to the publication of high-quality reviews, regular research papers and short communications in all areas of pure and applied mathematics. Mathematics also publishes timely and thorough survey articles on current trends, new theoretical techniques, novel ideas and new mathematical tools in different branches of mathematics.
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