A**u
发帖数: 2458 | 1 2, fair coin
1) flip 100 times, if H you win 2$, if T you lose 1$, what is the
E[gain]?
2) assume you start with 50$. you will stop either hit 0 or after 100
times
flip. calculate E[gain]
3) You can flip the coin 100 times, but instead of having a fixed stake,
you
can freely choose the stake for each flip. Just before the flip, you
start
with 100$. After each flip, if is comes up with H, you win twice your
stake
(and your stake is returned). If it comes up with T, you lose your
stake. i.
e. if you start with x and select a stake of s, then after the flip you
will
either have x-s or x+2s. you can never make your stake larger than your
balance. Q: what is E[gain] and what is E[log(gain+100)]?
这是题目.
3题目不清楚，就算了
问题在2, 怎么手算出 hit 0 的概率 | s********r
发帖数: 529 | 2 第二道题目可以构建一个martingale来做，我说一下我的思路，不知道对不对，大牛指
正：
f(x)=k^x,那么E_t[f(X_{t+1})]=k^{X_t+2}*0.5+k^{X_t-1}*0.5=f(X_t)=k^{X_t},由此
可得k=(\sqrt{5}-1)/2
那么另\tau={inf t: X_{\tau}=0 or 100}，则f(\tau)也是一个martingale，f(k^{100
})*p+f(k^{0})*(1-p)=f(k^{50})可以求出去到100和0的概率，就gain的期望也就不难了
这个当中的数字好像很诡异，可能我的想法有问题了，不过就当抛砖引玉，望牛人指点
则个
times
you
stake
i.
will
【在 A**u 的大作中提到】
: 2, fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the
: E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100
: times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake,
: you
: can freely choose the stake for each flip. Just before the flip, you
: start
| A**u
发帖数: 2458 | 3 对的
多谢了
100
难了
【在 s********r 的大作中提到】
: 第二道题目可以构建一个martingale来做，我说一下我的思路，不知道对不对，大牛指
: 正：
: f(x)=k^x,那么E_t[f(X_{t+1})]=k^{X_t+2}*0.5+k^{X_t-1}*0.5=f(X_t)=k^{X_t},由此
: 可得k=(\sqrt{5}-1)/2
: 那么另\tau={inf t: X_{\tau}=0 or 100}，则f(\tau)也是一个martingale，f(k^{100
: })*p+f(k^{0})*(1-p)=f(k^{50})可以求出去到100和0的概率，就gain的期望也就不难了
: 这个当中的数字好像很诡异，可能我的想法有问题了，不过就当抛砖引玉，望牛人指点
: 则个
:
: times
| w******e
发帖数: 199 | 4 但并不是说走到0或100停，而是说走到0，或100次以后停。。。
100
难了
【在 s********r 的大作中提到】
: 第二道题目可以构建一个martingale来做，我说一下我的思路，不知道对不对，大牛指
: 正：
: f(x)=k^x,那么E_t[f(X_{t+1})]=k^{X_t+2}*0.5+k^{X_t-1}*0.5=f(X_t)=k^{X_t},由此
: 可得k=(\sqrt{5}-1)/2
: 那么另\tau={inf t: X_{\tau}=0 or 100}，则f(\tau)也是一个martingale，f(k^{100
: })*p+f(k^{0})*(1-p)=f(k^{50})可以求出去到100和0的概率，就gain的期望也就不难了
: 这个当中的数字好像很诡异，可能我的想法有问题了，不过就当抛砖引玉，望牛人指点
: 则个
:
: times
| k*****y
发帖数: 744 | 5 前两天刚在版上翻到过，好像有大牛说的可以证明100步内hit 0的概率很小，所以基本
上是等于$100。大概可以这么想：
假设原题的gain是X_t，相应的fair play过程是Y_t(win $1, lose $1)。对相同的路径
X_t >= Y_t。记相应的到0的first hitting time分别是\tau_X和\tau_Y
所以P( \tau_X <= 100)
<= P(\tau_Y <= 100)
= 2 P(Y_100 <= 0) --------------(reflection principle)
而Y_100约等于normal分布N(50, 10)，所以在５倍的standard deviation以外的概率基
本为０。
另外如果要精确求的话，可以参考shreve讲义上的方法，不难算出\tau_X的moment
generating function（不过要invert一个3次函数）。但是要求出前100项的和，貌似
相当messy的样子。
【在 A**u 的大作中提到】
: 2, fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the
: E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100
: times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake,
: you
: can freely choose the stake for each flip. Just before the flip, you
: start
| R**T
发帖数: 784 | 6 这个第二题我觉得算出hit 0的概率也没用啊，
你得知道这些hit 0的path如果没有被knock out他们的pay off是多少
【在 A**u 的大作中提到】
: 2, fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the
: E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100
: times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake,
: you
: can freely choose the stake for each flip. Just before the flip, you
: start
| L**********u
发帖数: 194 | 7 This problem can be solved by Wald's Equality.
Let X_i= either 2 or -1 with probability 1/2 respectively. N be the stopping
rule.
The gain can be written as
S=50+X_1+X_2+.....X_N.
By Wald's equality
E[S]=50+E[X_i]E[N]=1/2E[N].
and
E[N]=50/2^{50}+53/2^{53}+56/2^{56}+...+100*(1-1/2^{50}-1/2^{53}-1/2^{56}-...)
If you just want an approximation, then you can say E[S]=100 since the
probability
1-1/2^{50}-1/2^{53}-1/2^{56}-... is almost 1. i.e. E[N] is almost 100
【在 A**u 的大作中提到】
: 2, fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the
: E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100
: times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake,
: you
: can freely choose the stake for each flip. Just before the flip, you
: start
| L**********u
发帖数: 194 | 8 I have missed some terms in calculation of E[N].
the term for one head case, I should multiply it by a number k_1 that is the
number of solutions of the following equation x_1+x_2=52, where 0<=x_1<=49,
0<=x_2<=52.i,e. the 53/2^{53} term become 53*k_1/2^{53}.
The number of solutions of the above equation can be found by fundamental
combination technique.
so for the other terms.
sorry for the mistake. |
|
