Embedding laws in diffusions by functions of time
Embedding laws in diffusions by functions of time
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通过时间函数将定律嵌入扩散中
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发表时间：
2012
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通讯作者：
Goran Peskir
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作者：
Alexander M. G. Cox;Goran Peskir
我们提出了一个建设性的概率证明，证明如果 B=(Bt)t≥0 是从 0 开始的标准布朗运动，μ 是 R 上给定的概率测度，使得 μ({0})=0，则存在唯一的左连续递增函数 b:(0,∞)→R∪{+∞} 和唯一的左连续递减函数 c:(0,∞)→R∪{−∞} 使得 B 停止于τb,c=inf{t>0|Bt≥b(t) 或 Bt≤c(t)} 具有 μ 定律。证明方法依赖于 Helly 选择定理产生的弱收敛论证，并利用了在嵌入定理背景下显得新颖的 Levy 度量。我们证明 τb,c 在 Monroe 意义上是最小的，因此停止过程 Bτb,c=(Bt∧τb,c)t≥0 满足以 μ 表示的自然一致可积条件。我们还表明，在将 μ 嵌入到 B 中的所有停止时间中，τb,c 具有最小的截断期望。主要结果从标准布朗运动扩展到实线上的所有循环扩散过程。
We present a constructive probabilistic proof of the fact that if B=(Bt)t≥0 is standard Brownian motion started at 0, and μ is a given probability measure on R such that μ({0})=0, then there exists a unique left-continuous increasing function b:(0,∞)→R∪{+∞} and a unique left-continuous decreasing function c:(0,∞)→R∪{−∞} such that B stopped at τb,c=inf{t>0|Bt≥b(t) or Bt≤c(t)} has the law μ. The method of proof relies upon weak convergence arguments arising from Helly’s selection theorem and makes use of the Levy metric which appears to be novel in the context of embedding theorems. We show that τb,c is minimal in the sense of Monroe so that the stopped process Bτb,c=(Bt∧τb,c)t≥0 satisfies natural uniform integrability conditions expressed in terms of μ. We also show that τb,c has the smallest truncated expectation among all stopping times that embed μ into B. The main results extend from standard Brownian motion to all recurrent diffusion processes on the real line.