Path independence of additive functionals for stochastic differential equations under G-framework

Abstract

The path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.

Path independence of the additive functionals for stochastic differential equations driven by G-Lévy processes

“…Next, we treat sufficiency. By the Itô formula for G-Itô-Lévy processes to V (t, Y t ), we have (12). And then one can apply (10) to (12) to get (11).…”

“…By the Itô formula for G-Itô-Lévy processes to V (t, Y t ), we have (12). And then one can apply (10) to (12) to get (11). That is, F s,t is path independent in the sense of (7).…”

“…Next, we say our motivations. First, we mention that Ren-Yang [12] proved path independence of additive functionals for SDEs driven by G-Brownian motions. Since these equations can not satisfy the actual demand very well, to extend them becomes one of our motivations.…”

Path independence of the additive functionals for McKean–Vlasov stochastic differential equations with jumps

“…We aim to characterise the path-independence of additive functionals of McKean-Vlasov stochastic differential equations with jumps by certain partial integro-differential equations involving L-derivatives with respect to probability measures, following our previous work [14,15] where therein stochastic differential equations with jumps in finite and infinite dimensions were studied, respectively. Let us also mention further interesting work [17,10], where characterisation theorems for the path independence of additive functionals of stochastic differential equations driven by G-Brownian motion as well as for stochastic differential equations driven by Brownian motion with non-Markovian coefficients (i.e. random coefficients) are established, respectively.…”

On path independence of Girsanov transformation for stochastic differential equations

“…定理, 然后把主要结果推广到其他模型, 包括带跳情形 [9] 、半线性随机偏微分方程 [8,10] 和分布依赖情 形 [15] . 受篇幅所限, 除了非退化随机微分方程外, 本文对于所介绍的这些推广结果不予证明, 有兴趣的 读者可参考相关文献, 并延伸阅读文献 [11][12][13][14] 关于退化的分布依赖的带跳随机微分方程的研究、文 献 [16] 关于非线性期望下的随机微分方程的研究, 以及文献 [17] 关于倒向随机微分方程的研究.…”