T*****w
发帖数: 802 | 1 好像也是以前MS的面試題了
Given an example of a martingale that is not a Markov process and an example
of a Markov process that is not a martingale
我在mathematically 能看出兩個的區別，但是班上大牛能不能給個清楚的解釋, 為什
麼兩個其實沒有關系了。 我怎麼覺得從定義上理解，Martingale 是Markov Process的一個
特例呢？ 去構造example的思路是什麼？
Markov Process：最重要的特點就是future is independent with history before time s!
Martingale 最重要的特點就是 driftless, ， 但是 it may depend on the history Ft?
我比較在和漿糊。。。
多謝～ |
G********d
发帖数: 10250 | 2 exp(B_t) is markov but not martingale
how about B_t+B_{t+1} for the other one
i am not sure
example
【在 T*****w 的大作中提到】
: 好像也是以前MS的面試題了
: Given an example of a martingale that is not a Markov process and an example
: of a Markov process that is not a martingale
: 我在mathematically 能看出兩個的區別，但是班上大牛能不能給個清楚的解釋, 為什
: 麼兩個其實沒有關系了。 我怎麼覺得從定義上理解，Martingale 是Markov Process的一個
: 特例呢？ 去構造example的思路是什麼？
: Markov Process：最重要的特點就是future is independent with history before time s!
: Martingale 最重要的特點就是 driftless, ， 但是 it may depend on the history Ft?
: 我比較在和漿糊。。。
: 多謝～
|
z****g
发帖数: 1978 | 3 markov process has STATE, which means it is basically for describing
discrete process.
Also, from the definition, martingale's mean is only dependent on current
time stamp. However, a markov chain may not have a STABLE distribution, or
the mean of the state may fluctuate periodically within in a range. |
k******2
发帖数: 111 | 4
continuous markov 也是为了描述discrete？
【在 z****g 的大作中提到】
: markov process has STATE, which means it is basically for describing
: discrete process.
: Also, from the definition, martingale's mean is only dependent on current
: time stamp. However, a markov chain may not have a STABLE distribution, or
: the mean of the state may fluctuate periodically within in a range.
|
z****g
发帖数: 1978 | 5 OK, if say general markov process, I am wrong with this.
However, I think the definition of markov process is clear enough to say
that markov chain has finite memory on the path, so that any length of
delay can be embedded into a vector as lag-one markov process. So any
martingale with infinite memory is not a markov process.
【在 k******2 的大作中提到】
:
: continuous markov 也是为了描述discrete？
|
J**********g
发帖数: 213 | |
B****n
发帖数: 11290 | 7 Markov process是說下一刻的條件分布(conditional on the past history) 只和上一
時刻的值有關
Martingale 只是一個公賭的性質 考慮下一刻的條件期望值 (conditional on the
past history) 只和上一刻的值有關 而不考慮整個distribution 也就是說下一刻
conditional distribution 可以和所有之前的history都有關係
所以很清楚的是 要找一個martingale的例子 和之前整個history有關 而不是只和上一
刻有關 這種例子很容易找 隨便一個martigale Xn 給一個stoping rule 則X_{Tn}大都
可以滿足所要求的
比方說Xn=C1+C2+...+Cn, Ci iid P(Ci=-1)=P(Ci=1)=1/2
T=1 if C1=1 T=infinity otherwise
Tn=min{T,n}
X_{Tn}是一個Martingle 但是一定不是Marcov process 因為它的conditonal
distribution和X1有關係
example
的一個
time s!
history Ft?
【在 T*****w 的大作中提到】
: 好像也是以前MS的面試題了
: Given an example of a martingale that is not a Markov process and an example
: of a Markov process that is not a martingale
: 我在mathematically 能看出兩個的區別，但是班上大牛能不能給個清楚的解釋, 為什
: 麼兩個其實沒有關系了。 我怎麼覺得從定義上理解，Martingale 是Markov Process的一個
: 特例呢？ 去構造example的思路是什麼？
: Markov Process：最重要的特點就是future is independent with history before time s!
: Martingale 最重要的特點就是 driftless, ， 但是 it may depend on the history Ft?
: 我比較在和漿糊。。。
: 多謝～
|
T*****w
发帖数: 802 | 8 多谢楼上的讨论，清楚多了
1) Markov process is about entire probability distribution, which is
independent with history before time s.
2) Martingale is about expectation ONLY (not the distribution), which
depends only on time s, not the earlier time.
再补充一些网上看到的例子:
【Markov but Not a martingale】
1. A biased coin (scoring +1 for H, -1 for T). Not a martingale, but
still markov
2.dX_t = a dt + \sigma dW_t is Markov but is not a martingale.
【Martingale but not a Markov process】
1. A fair coin, but more complex rules:
if previous throw was a head then you get +1 for H, -1 for T on the next
throw. If your previous throw was a tail then you get +2 for H and -2
for T on the next throw. Your score is still a martingale, but no longer
markov because the actual scores that are possible depend on what you
have thrown already.
2. Ito integral:
dY = f(Y). dz
This is a martingale, this is a basic property if Ito integrals. But if
f is a function of the path that Y takes then it is not markov.
3. dX_t = (\int_0^t X_s ds) dW_t is a martingale but is not Markov. |
w*****e
发帖数: 197 | 9 为什么不用离散时间的例子
非常简单，一个过程X(n)每一个时刻以等概率
翻倍或者不变，这个过程肯定是markov的，也
肯定不是martingale。
反方向的例子类似，X(n)以等概率要么增加
S(n-1) = X(1) + X(2) + ... + X(n-1)
要么减少S(n-1)，这个肯定是martingale，
但是很明显不是markov的。 |
