Stochastic differential equations with a convex constraint
Stochastic differential equations with a convex constraint
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DOI：
10.1080/17442509508833992
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发表时间：
1995-05
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影响因子：
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通讯作者：
A. Storm
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文献类型：
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作者：
A. Storm
本文利用凸分析的技巧建立了Hilbert空间中随机常微分方程约束强解的存在唯一性。解由一个半鞅u和一个有界变分η组成，其中u必须在给定凸函数φ的定义域内，且η满足变分不等式a.s.粗略地说，η在某种意义上是奇异的，并且表示维持对u的约束所需的推动量。对φ的限制是它是下连续的，并且有一个非空内部的区域。如果是一个开凸集，φ是的指示符，则u被反射为保持在，η是非绝对连续的，并且仅当u在的边界处时才变化。因此，我们恢复以前的结果在有限维反射扩散过程。如果φ有一个开域并且是Gateaux可微的，则η成为绝对连续过程。在这种情况下...
This work uses techniques from convex analysis to establish the existence and uniqueness of constrained strong solutions to stochastic ordinary differential equations in Hilbert space. The solution consists of a semimartingale u with a bounded variation component η, in which u must stay within the domain of a given convex function φ, and η satisfies the variational inequality a.s. for all suitable test functions v. Roughly speaking η is singular in some sense and represents the amount of pushing needed to maintain the constraint on u. The restrictions on φ are that it be lower semicontinuous and have a domain with non-empty interior. If is an open convex set and φ is the indicator of then u is reflected to stay in and η is non absolutely continuous and varies only when u is at the boundary of . We thus recover previous results on reflected diffusion processes in finite dimensions. If φ has an open domain and is Gateaux differentiable than η becomes the absolutely continuous process . In this situation is ...