Stochastic Processes · 免费预览
鞅的应用
可选停止定理只需几行便可刻画随机游走的退出分布和退出时间；对称随机游走在选定水平退出的概率为距离之比，其持续时间则由一个乘积给出；瓦尔德等式给出停止和式的期望；指数鞅既可用于界定破产界限，也可推出分支过程的灭绝概率。
1. The recipe, invent a martingale, stop it
The recipe, invent a martingale, stop it
2. Exit probabilities for the symmetric walk
Exit probabilities for the symmetric walk
3. How long the game lasts
How long the game lasts
4. Ruin bounds and extinction recovered
Ruin bounds and extinction recovered
5. The recipe and its fine print
What condition lets you conclude E M T equals E M 0 for a stopping time T?
6. Fair exits split by distance
A symmetric walk starts at x between walls a and b. What is the probability of exiting through b?
7. The duration parabola
What shape does the expected exit time E sub x T take as a function of the starting point x between a and b?
8. Wald's equation
What does Wald's equation say for a walk with step mean mu and a stopping time T of finite expectation?
9. Pricing a reserve cushion
In Cramer's estimate, what does the ruin bound e to the minus 2 mu S 0 over sigma squared say about reserves?
10. Two chapters shake hands
How does the martingale chapter recover the branching process extinction probability?
11. Exit through the near wall
A symmetric walk starts at 0 with walls at minus 4 and 2. Compute the probability of exiting through 2 and write the answer as a fraction or decimal.
12. The longest game in the interval
For the same walls at minus 4 and 2, the expected exit time is the product of the distances from x to the walls. Compute this expectation starting from the midpoint x equal to minus 1, and write the number of rounds.
13. An insurer's ruin bound
An insurance company has reserves 10, yearly profit mean mu equal 1 and variance sigma squared equal 4. Compute Cramer's bound on the ruin probability, e to the minus 2 mu S 0 over sigma squared, and write the answer as a decimal.