A $C^1$-Itô’s Formula for Flows of Semimartingale Distributions

IF 1.6 2区数学 Q2 MATHEMATICS, APPLIED

Applied Mathematics and Optimization Pub Date : 2024-07-27 DOI:10.1007/s00245-024-10165-y

Bruno Bouchard, Xiaolu Tan, Jixin Wang

引用次数: 0

Abstract

We provide an Itô’s formula for $C^1$-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the $C^1$-Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires $C^1$-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.

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半马丁分布流动的 C^1$$-Itô 公式

我们提供了连续半马汀式的条件边际分布流的$C^1$-函数的伊托公式。它基于弱狄利克特过程的概念，并将 Gozzi 和 Russo (Stoch Process Appl 116(11):1563-1583, 2006) 中的$C^1$-Itô's 公式扩展到这一上下文。作为第一个应用，我们研究了一类麦金-弗拉索夫最优控制问题，并建立了一个验证定理，该定理只要求其值函数具有 $C^1$-regularity 性，这等同于相关 HJB 主方程的（粘性）解。它与一个新颖的对偶性结果相辅相成。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Mathematics and Optimization 数学-应用数学

CiteScore

3.30

自引率

5.60%

发文量

103

审稿时长

>12 weeks

期刊介绍： The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.

A \(C^1\)-Itô’s Formula for Flows of Semimartingale Distributions

Abstract