Stochastic Processes · 免费预览
鞅的示例
鞅是一种财富过程，其每一轮的条件利润均为零，这是公平游戏的标志。示例包括平移后的随机游走、平方随机游走减去其方差、资产价格的乘积模型、指数鞅，以及赌徒破产似然比。
1. Fair games formalized
Fair games formalized
2. The additive gallery
The additive gallery
3. Products and exponentials
Products and exponentials
4. Martingales from Markov chains
Martingales from Markov chains
5. The defining property
What makes a process M n a martingale with respect to the outcomes X n?
6. Super versus sub
Which phrase correctly pairs the inequality with the game's character?
7. Deleting the drift
Why is S n minus n mu a martingale for a random walk with step mean mu?
8. Squares minus their variance
What is the martingale associated with a mean-zero random walk of step variance sigma squared?
9. The product model of prices
Why does finance prefer the multiplicative martingale M n equals M 0 times X 1 through X n?
10. Exponential cancellation
Why divide e to the theta S n by phi of theta to the n, where phi is the moment generating value at theta?
11. Fair steps have unit variance
A random walk has steps of plus or minus 1 with equal probability. Compute the step variance, the correction term for the squared-walk martingale, and write the number.
12. The ruin likelihood ratio
In asymmetric gambler's ruin with p equal to 0.4 for an up-step, the martingale is the ratio q over p raised to S n. Compute this ratio and write the answer as a decimal.
13. The exponential martingale's growth factor
Steps are plus or minus 1 with equal probability, so the moment generating value at theta is one half times e to the theta plus one half times e to the minus theta, the hyperbolic cosine. Compute this value at theta equal to the natural log of 2, and write the answer as a fraction or decimal.