Characterising the path-independent property of the Girsanov density for degenerated stochastic differential equations

Abstract

In this paper, we derive a characterisation theorem for the path-independent property of the density of the Girsanov transformation for degenerated stochastic differential equations (SDEs), extending the characterisation theorem of [13] for the nondegenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalises the corresponding characterisation theorem established in [10].

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

“…In Section 2, following the line of [15,22], we present a decomposition theorem for multidimensional G-semimartingales. In Section 3, we characterize the path independence of A f,g s,t using nonlinear PDEs, so that main results in [16,23,24,26] are extended to the present nonlinear expectation setting. Finally, in Section 4, we provide an example to illustrate the main result for α = 0 as mentioned in Remark 1.1.…”

Stochastic differential equations with singular coefficients on the straight line

“…For more details in this direction, we refer to [10][11][12][13]. For some extensions and applications, we refer to [14][15][16][17][18] and the references cited therein.…”

On path independence of Girsanov transformation for stochastic differential equations

“…文献 [6] 还将该结果推广到了 Riemann 流形情形. 此外, 对于形如 b(t, x) = σ(t, x)γ(t, x) 的漂移 项, 文献 [14] 把该结果推广到了退化 (亚椭圆) 情形, 见本文第 5 节关于更一般 (分布依赖、可退化) 模型的讨论.…”

“…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.…”