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叠加于漂移之上的简单随机游动的概率计算
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叠加于漂移之上的简单随机游动的概率计算
1.
叠加于漂移之上的简单随机游动的概率 (2016 年 4
月 25 日) 刘峻峰 摘要:简单随机游动叠加于漂移之后便构成一类复合随机序列。我们 提出一种高效的方法以计算当此序列被限定于某上下界函数之间时的 概率。 关键词:简单随机游动, 漂移, 边界,概率,杨辉三角。 1.简单随机游动及其二维表示: 离散时间域上独立同分布伯努利随机变量(每定义时刻上以概率𝑝(1 − 𝑝) 进行固定步长(𝛥)正(反)向移动)之累积和构成一维简单随机游动,其当前 时刻(𝑡𝑖)位置𝑅(𝑡𝑖) = ∑ (±𝛥𝑖 𝑗=1 ), (𝑖 = 1, … )。对于𝑅(𝑡𝑖),我们以二维象 限 (始于左下角原点(0,0)) 中“向上(纵(𝑦)坐标)”或“向右(横(𝑥)坐标)” 移动 (步长为𝛥) 来对应伯努利随机变量列中的正向或反向移动: 𝑅(𝑡𝑖) = ∑ ∆正移 − ∑ ∆反移 = 𝑦(𝑡𝑖) − 𝑥(𝑡𝑖)。此二维表示路径享有“左下-右上” 单调性以实现即时物理显示累积正(𝑦)/反(𝑥)向移动步数。累积移动步数 (𝑖)对应当前节点处反对角线(依时右移 𝑦 + 𝑥 = 0),累积位移则为相对 基准线𝑦 − 𝑥 = 0的纵向偏离。
2.
如将此图右转45° , 并且将路径步长看作为√2𝛥, 则我们仍在垂直方向上 依次观察到步长为𝛥的简单随机游动。 2.叠加漂移后形成的复合随机序列及其受限域
` 某些离散时间序列模型将一确定(已知或未知)漂移函数 (𝑓(𝑡)) 叠加 于前述的简单随机游动之上而形成一复合随机序列: S(𝑡𝑖) = 𝑅(𝑡𝑖) + 𝑓(𝑡𝑖) 给定漂移函数(𝑓(𝑡))后,我们时常关心某些路径事件的概率。 例如, 序 列 {S(𝑡𝑖)}在累积时间段上不低于某阈值(𝐶𝐿)且不高于某阈值(𝐶 𝑈), 即 Pr(𝑅 path: ⋂ {𝐶𝐿 ≤ 𝑅(𝑡𝑖) + 𝑓(𝑡𝑖) ≤ 𝐶 𝑈}𝑛 𝑖=1 ) = Pr (𝑅 path: ⋂ {𝐶𝐿 − 𝑓(𝑡𝑖) ≤ 𝑅(𝑡𝑖) ≤ 𝐶 𝑈 − 𝑓(𝑡𝑖)} 𝑛 𝑖=1 )
3.
= Pr (𝑅
path: ⋂ {𝐶𝐿 − 𝑓(𝑡𝑖) ≤ 𝑦(𝑡𝑖) − 𝑥(𝑡𝑖) ≤ 𝐶 𝑈 − 𝑓(𝑡𝑖)} 𝑛 𝑖=1 ) = Pr (𝑅 path: ⋂ {𝐶𝐿 − 𝑓(𝑡𝑖) ≤ 𝑦(𝑡𝑖) − (𝑖∆ − 𝑦(𝑡𝑖)) ≤ 𝐶 𝑈 − 𝑓(𝑡𝑖)} 𝑛 𝑖=1 ) = Pr (𝑅 path: ⋂ { 𝐶𝐿 − 𝑓(𝑡𝑖) 2∆ + 𝑖 2 ≤ 𝑦向累积步数(总移动步数𝑖) ≤ 𝐶 𝑈 − 𝑓(𝑡𝑖) 2∆ + 𝑖 2 } 𝑛 𝑖=1 ) 𝐶𝐿 = −∞ 或𝐶 𝑈 = +∞即单边域情形。 3.计算界内样本路径集合及数目 我们演示程序式上下实际边界计算方法。在给定漂移函数𝑓(𝑡)之后,由 每条积时线(𝑡𝑖)上样本路径应满足之界定条件: {𝐶𝐿 ≤ 𝑅(𝑡𝑖) + 𝑓(𝑡𝑖) ≤ 𝐶 𝑈},我们为此线上每一节点标记为“界内”或“界外”,从而标出于 此孤立时间点样本路径应满足的边界(例如,累积上移步数(𝑦)不小于某 一值且不大于另一值)。然后我们编写递归程序将此粗略二维路径边界 精化(由左下至右上)为某条存在的单调右上走势的样本路径以作为 实际边界。实际边界之内的所有样本路径便构成我们所关心的界定内 样本路径集合。(细节待增)
4.
4.计算界定内样本路径集合概率 规则几何连接特性以及单调右上走势助我们甄别边界之内的所有样本 路径, 接着运用“右-上”走势的“杨辉三角”递归式组合相加获取至各 节点路径数目从而精确计算概率。对于某模型配置,我们有如下数值 结果。
5.
5.结论 尽管所涉及的漂移函数(𝑓(𝑡))和上下界函数({𝐶𝐿, 𝐶 𝑈})可以一般化,以上 讨论仅限于简单(等步长)随机游动模型。可能的应用包括计量经济 学上的几何对数序列模型以描述相对变化率的时序变化。 6.参考文献 待增。 7.R
程序 待增。
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