Pathwise differentiability of reflected diffusions in convex polyhedral domains
Pathwise differentiability of reflected diffusions in convex polyhedral domains
复制标题
凸多面体域中反射扩散的路径可微性
DOI：
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发表时间：
2017
影响因子：
1.5
通讯作者：
K. Ramanan
中科院分区：
文献类型：
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作者：
David Lipshutz;K. Ramanan
凸多面体域的反射扩散有多种应用，包括相互作用粒子系统、排队网络、生化反应网络和数学金融。在适当的数据条件下，我们建立了这种反射扩散关于其定义参数的路径可微性，即它的初始条件，漂移和扩散系数，以及沿域边界的反射（斜）方向。我们将反射扩散的路径导数的右连续正则化描述为具有跳跃的受限线性随机微分方程的路径唯一解，该方程的漂移和扩散系数、反射的域和方向取决于反射扩散的状态。这一结果的证明依赖于相关的（扩展的）Skorokhod反射映射的方向导数的性质以及它们在所谓的导数问题中的表征，并且还涉及建立多面体域边界非光滑部分的反射扩散的某些路径性质，这可能是独立的兴趣。作为推论，我们得到了反射扩散的泛函期望导数的概率表示，这对反射扩散的灵敏度分析是有用的。
Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data, we establish pathwise differentiability of such a reflected diffusion with respect to its defining parameters --- namely, its initial condition, drift and diffusion coefficients, and (oblique) directions of reflection along the boundary of the domain. We characterize the right-continuous regularization of a pathwise derivative of the reflected diffusion as the pathwise unique solution to a constrained linear stochastic differential equation with jumps whose drift and diffusion coefficients, domain and directions of reflection depend on the state of the reflected diffusion. The proof of this result relies on properties of directional derivatives of the associated (extended) Skorokhod reflection map and their characterization in terms of a so-called derivative problem, and also involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be of independent interest. As a corollary, we obtain a probabilistic representation for derivatives of expectations of functionals of reflected diffusions, which is useful for sensitivity analysis of reflected diffusions.