T*****w
发帖数: 802 | 1 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 ru... 阅读全帖 |
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w**********y
发帖数: 1691 | 2 sorry..有错..第二个martingale好像不对..跟老板meet..回来再检查..
E(S_1)=p-(1-p)=2p-1
Define X_n=S_n-(2p-1)n
1.X_n is martingale
2.x_n^2-n is martingale
3. exp(aX_n-0.5a^2*n) is martingale
Part a:
From 3, exp[a(S_n-(2p-1)n)-0.5a^2*n] is martingale
let a=-2(2p-1)
then exp(-2(2p-1)S_n) is martingale
exp(-2(2p-1)S_tau) is martingale
E(exp(-2(2p-1)S_tau))=1
so P(s_tau=A)*exp[-2(2p-1)A]+(1-P)*exp[-2(2p-1)B]=1
solution: P(s hit A first)=(exp[-2(2p-1)B]-1)/(exp[-2(2p-1)B]-exp[-2(2p-1)A])
Part b:
From the linear weighted |
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a*********r
发帖数: 139 | 3 A local martingale is a pretty technical concept for novices. A martingale
is a local martingale, but the converse is not true. In financial
engineering, people use positive local martingales most of the time. The
positiveness implies many nice properties. Class D which is a very strong
condition. If a local martingale is of class D, then it must be uniformly
integrable and it is a martingale. |
|
发帖数: 1 | 4 初级概率的martingale定义让人觉得限制很多，martingale的定义最好基于filtration
，可以看作从未来到现在的不断平均化，而且martingale有很多表示方法很有用，非
martingale很多情况下利用概率转换可以变成martingale，加一个R-N derivative
process就行，很方便
俺就知道个皮毛 |
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l******i
发帖数: 1404 | 5 1. For short, there is a theorem stating that:
For any stochastic process Xt,
Xt is a martingale under some probability measure Q
(Or Xt is a Q-martingale)
if and only if dXt=Vt*dWt for some adapted process Vt.
So t*Wt is not martingale.
If you don't buy it, here is the strict proof:
2. Mathematical proof:
Definition of martingale is for time s
Let's prove it can not be hold for t*Wt.
d(t*Wt)=Wt*dt+t*dWt
t*Wt=s*Ws+integrate from s to t of Wu*du+integrate from |
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T*****w
发帖数: 802 | 6 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～ |
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t******m
发帖数: 255 | 7 一般martingale都是local martingale.但是反之不成立，谁能给个例子是local
martingale但不是martingale?谢了 |
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d****d
发帖数: 2919 | 8 俺也问一个，之前看版上有人问的题，
markov chain 不一定是martingale，martingale也不一定是markov chain。
分别举出例子。
markov 不是martingale的很容易，
反过来的例子，我就举不出来了。。
哪位给说个明显的啊？ |
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c**a
发帖数: 316 | 9 【 以下文字转载自 Quant 讨论区 】
发信人: ccca (cc), 信区: Quant
标 题: 一个百思不得其解 的 Martingale stopping time 问题
发信站: BBS 未名空间站 (Mon Sep 1 11:39:04 2014, 美东)
Problem: B(t) a standard Brownian motion, how is the expecting time it first
hits either -1 or 1.
Solution:
X(t)=exp(B(t)-0.5t) is a martingale. By option sampling theorem, X(t)
stopped at B(t)=-1 or 1 is also a martingale.
Since X(0)=1, we have 0.5*X(1)+0.5*X(-1)=1.
Or exp(1-0.5t)+exp(-1-0.5t)=2,
Or exp(-0.5t)=2/(e+1/e)
Or t = sqrt(-2 ln(2/(e+1/e)).
What is ... 阅读全帖 |
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z****i
发帖数: 406 | 10 probability of getting A before B: B/(A+B);
probability of getting B before A: A/(A+B);
(gambler ruin problem)
(S_n)^2 - n is a martingale, stopped at A or B is also a martingale;
if stopped at A, E(A^2-n) = 0, therefore E(n) = A^2.
if stopped at B, E(B^2-n) = 0, therefore E(n) = B^2.
So expected time = B/(A+B)*A^2 + A/(A+B)*B^2 = AB.
Don't know how to construct a martingale if p is not 1/2. Anybody give a
hint please? |
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o****e
发帖数: 80 | 11 判断f(t,w(t))是不是martingale,通常都是用ito lemma求 df=mu *dt+sigma*dw(t),只
要drift term－－－mu 不等于0,
这就不是martingale。
为什么drift term 不等于0，就不是martingale? |
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Q***5
发帖数: 994 | 12 For most of the setups in financial problems, you may just consider local
martingale as a process whose differential form has 0 drifting. i.e. dX =
0dt + something* dW, as the name suggests, it only concerns a `local'
property.
But in order for 'local' martingale to become a 'global' martingale, some extra
'global' requirements (i.e. DL property) have to be met, in order to make
sure that the process is not too 'wild'. |
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n*********3
发帖数: 21 | 13 We all know the common symmetric random walk problem as below:
start from 0, move on axis by brownian motion, then we have probability of
first reaching a to be b/(a+b), and probability of first reaching -b to be a
/(a+b), also we know the expectation of reaching a or b is a*b.
P(a)+P(b) = 1
E[W_t] = a*P(a) + b*P(b) = 0 //martingale
E[W_t^2-t] = a^2*P(a) + b^2*P(b) - E[t] = 0 //martingale
Hence derive the solution (suppose t is stopping time).
The question is ho... 阅读全帖 |
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r*******y
发帖数: 1081 | 14 so it is not a martingale.
it is an interesting problem, W(t) is a martingale but
W(t/2) is not a martingale. |
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m******2
发帖数: 564 | 15 根据Martingale的理论，既然可以把S/e^r（T-t)做成Martingale
完全也可以把S的运动转成driftless，然后把synthetic call portfolio做成
Martingale吧？
这样就不可以么？
还是这样违反pde的解呢？ |
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l******i
发帖数: 1404 | 16 Set W(t) to be standard Wiener process.
If X(t) follows that dX(t) = \mu dt + \sigma dW(t),
then X(t) is Markovian but not martingale.
If X(t) follows that dX(t) = (\int_0^t X(s) ds) dW(t),
then X(t) is martingale but not Markovian. |
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T*******x
发帖数: 8565 | 17 我说一下我的一种理解。
martingale可以看成是，在越来越多的知识下，看同一个随机变量，得到的看法。也就
是说这个随机变量的物理没有变，只是看法在变。而看法的变化也要满足物理没有变这
个限制。
举个例子，比如测量恒星表面温度这个函数吧，它是恒星表面的每个点，的一个函数。
这个物理量假定它在那，不变。但是我们的测量手段有限。比如最开始，恒星在我们看
来就是一个点，我们虽然知道它不是一个点，但是没有手段分辨恒星表面是个面，所以
恒星表面温度这个函数，我们只能给它一个数，也就是表面所有点都是一个数。这个数
实际上也就是恒星表面温度的平均数。反过来看，这个数值对应的定义域，是恒星表面
所有的点，这也是这个集合上的唯一一个可测集，literally。
可以想象我们的测量手段越来越发展，也就是能够分辨或测量越来越细致的恒星表面温
度。比如在下一个阶段，我们有能力把恒星表面分辨出四个区域，并测量标注每个区域
的温度，那么恒星表面温度就是四个数，每个区域一个数，或者说每个区域里的所有点
给同一个数。可测集数量增加了，函数的标注细致了，平均值不能变。再下一个阶段，
分辨力测量手段越来越细致，可测集... 阅读全帖 |
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m**********e
发帖数: 2808 | 18 不是准备写长篇，不过现在明显low watt，尤其灌了大堆关于食物的水后。。。
我指的是，统计意义而言，哪怕“事业的辉煌抵消年龄的劣势且有余”也是illusion。
。。
martingale的一个最明显的特征就是不能arbitrage。相对起金融市场，婚姻市场更符
合martingale的特征。。。 |
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B*********h
发帖数: 800 | 19 ☆─────────────────────────────────────☆
pen (谢谢你) 于 (Mon Aug 28 23:28:01 2006) 提到:
http://www.wilmott.com/messageview.cfm?catid=8&threadid=11322&STARTPAGE=1&FTVAR_MSGDBTABLE=
These are two different notions
Markov processes are random processes that have no memory. Whereas
martingales have their
average value constant in a strong sense.
1. A biased coin (scoring +1 for H, -1 for T). Not a martingale, but still
markov.
2. Now you have a fair coin but more complex rules:
if your previous thro |
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b***k
发帖数: 2673 | 20 ☆─────────────────────────────────────☆
daviduon (David) 于 (Sat Feb 2 18:35:04 2008) 提到:
X为brownian motion, 请证明 exp(X) 和 X^3 是否为martingale?
☆─────────────────────────────────────☆
yww (petite) 于 (Sat Feb 2 19:23:35 2008) 提到:
ITO搞一下就行了
都不是
☆─────────────────────────────────────☆
daviduon (David) 于 (Sat Feb 2 19:33:29 2008) 提到:
谢谢,我也刚刚做了一下.
只是有一个疑问,一般面对证明martingale的问题的时候,是否都用Ito去求drift是否为
0?
☆─────────────────────────────────────☆
fishdaddy (无) 于 (Sat Feb 2 22:11:42 2008) |
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p*****k
发帖数: 318 | 21 ilovekl, some hints:
(this ought to be easy to find the answer by yourself)
what is the definition of "martingale"?
a martingale does not have to have expectation of zero, you can
add in any nonzero constant. so in the definition,
which quantity is zero? |
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w**********y
发帖数: 1691 | 22
不是我说的吧?你再看看几天俺发的贴..
万能判断法则就是,ito lemma的drift是0.
martingale的定义是什么?-conditional expectation of the increment is zero.
E(X_t-X_s|F_s)=0 或者 E(X_t|F_s)=X_s
sin(w_t)的expectation是什么?--0
sin(w_t)是martingale么?--不是.
为什么?--E(sinw_t)=0. 但是 E(sinw_t-sinw_s|F_s)=E[sin(Wt-Ws+Ws)-
sinWs|F_s]=sinWs*{E[cos(Wt-Ws))]-1}!=0.
或者说E(sinW_t|F_s)!=sinW_s |
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t**x
发帖数: 12 | 23 最近在看一些大牛们以前的问题解答， 经常碰到说用martingale, 可是我只知道
martingale的最基本性质， 也就是条件期望不变。
大家给出的解答有时想不明白， 所以想问一下大家，到底用了martingal的那些性质？
或者那位大侠能有推荐的书就更好了， 谢谢 |
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o****e
发帖数: 80 | 24 a martingale stopped at stopping time is still a martingale
这个理论成立的前提条件是什么？我已经第二次被人问到这个了 |
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B****n
发帖数: 11290 | 25 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
... 阅读全帖 |
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w*****e
发帖数: 197 | 26 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。 |
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w*****e
发帖数: 197 | 27 Just treat future process as a stock with a special dividend process. Then
it is not hard to see why it is martingale. But for interest rate future
like EuroDollar, I am not sure what you are saying is true in general.
Remember, anything can be a martingale, as long as you are willing to work
with a particular kind of risk neutral measure.
On the other hand, future RATE is a rather hard thing to model. You always
start with some kind of framework that works with short rate or forward rate
. I am... 阅读全帖 |
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z****i
发帖数: 406 | 28 Thanks a lot.
Just took a look at Shreve's book. It doesn't seem to be quite obvious to
show the martingale property though.
For ED futures, let F(t; T1, T2) be the time-t future rate (simply
compounded), from T1 to T2.
Let L(t; T1, T2) be the forward Libor rate.
Then we have F(t; T1, T2) is a martingale under risk-neutral measure, so is
the ED futures price of course.
Moreover, F(t; T1, T2) is the expectation of L(t; T1, T2) under risk-neutral
measure.
Future rates are quoted directly from mark... 阅读全帖 |
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t******m
发帖数: 255 | 29 Can someone give an accessible explanation of the difference between local
martingale and martingale? Many thanks. |
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w**********y
发帖数: 1691 | 30 说说俺的理解:
Stochastic Calculus的顺序是这样的:
1. 定义了Brownian Motion: W_t
2. 仿照黎曼积分的方式定义了基于BM的Ito 积分 (取每个区间的前端点): \int_0^t f
(w,t) dW_t
这时候问题就来了,到底什么样的f才能用来定义Ito积分..当然f必须是adapted
measurable function.然后我们希望定义这个积分的expectation和variance都存在的
话, 就定义出了一个空间 L2(dp*dt) = {f: E(\int_0^t f^2(w,t) dW_t)<\infty}
这个时候出现一个问题就是: 我们当然希望连续函数f都可以很好的定义ito积分.但是
举个例子,f=exp(W^4)就不在上面的L2空间里面..所以那个条件太强了..所以我们把这
个条件放宽,定义一个新空间,变成LOC2(dp*dt) = {f: P((\int_0^t f^2(w,t) dW_t)<
\infty)}=1.这个就定义出来了一个更general的空间..这个区别就是:新空间里面允许f可以有
很大很大的值... 阅读全帖 |
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A*******u
发帖数: 66 | 32 记得有这么一个结论：The futures price is a martingale under the risk-neutral
measure. 但自己推出来却不对，请高人看看错在哪里：
Suppose dS=rSdt+\sigma*SdW, where W is a BM under RN measure.
Then dF=d(S*exp(rt))=2rFdt+\sigma*FdW
由于有drift项，表明F在RN measure下不是martingale，与之前结论矛盾。
不胜感激！ |
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c**********e
发帖数: 2007 | 33 You can also use martingale. The first 3 martingales are
for t: W_t^2 - t
for t^2: W_t^4 - 6t*W_t^4 + 3t^2
for t^3: W_t^6 - 15t*W_t^4 + 45t^2*W_t^2 - 15t^3
a |
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c**a
发帖数: 316 | 34 I think we can disprove it.
E(Wt+Wt/2 | Ws +Ws/2) = E(Wt|Ws+Ws/2) + E(Wt/2|Ws+Ws/2)
=E(B1 + Ws|Ws+Ws/2) + E(B2+Ws/2|Ws+Ws/2)
=E(B1+ B2|Ws+Ws/2) + E(Ws+Ws/2|Ws+Ws/2)
Hence, Wt+Wt/2 is a martingale iff the first term is zero.
B1 =W(t)-W(s) and its conditional expectation is zero.
B2 = W(t/2)-W(s/2) and its conditional expectation is generally not zero.
If we take the derivative we have
0.5dw(t/2) +dw(t)
since we have dw(t/2), we dont know if it is a martingale. |
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a**n
发帖数: 3801 | 35 martingale都是相对于一个information set而言的
如果你的information来自于W_t， 那X_t=W_{t/2}当然不是martingale |
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k**u
发帖数: 60 | 36 W(t/2)=sqrt(1/2) W(t)
but W(t/2) is a weak soultion, don't know whether it follows the filtration
generated by W(t) exactly.
不过， 感觉W(t/2)+ W(t) 是martingale.
纠结的地方是这个martingale,的filtration是什么样的。 |
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d**t
发帖数: 183 | 37 那样的话你就不知道怎么discount了.
根据Martingale的理论，既然可以把S/e^r（T-t)做成Martingale完全也可以把S的运动
转成driftless，然后把synthetic call portf........
★ Sent from iPhone App: iReader Mitbbs Lite 7.28 |
|
k*****y
发帖数: 744 | 38 我觉得是不是可以这么想：martingale只要每一步是fair play就行了，但是赢输多少
可以由整条historic path决定不是只依赖于当前的值。 |
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A**u
发帖数: 2458 | 39 硬币
XXXX
如果 前N次连续出现H的最大值是M的话，
第N+1次， 投H 得M刀， 投T 得-M刀.
是Martingale, 单不是markov |
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k*****y
发帖数: 744 | 40 我觉得书上可能是说：Chapter 5那里证明了Z是martingale，这里S刚好也是Z这样的形
式，所以是martingale。然后就像你证明的那样，再考虑Dumand说的"0"那个boundary。 |
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m******2
发帖数: 564 | 41 费牛劲证明Stopped Martingale也是Martingale
这怎么用在Barrier Option的证明上的？
有联系吗？？？ |
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y**********0
发帖数: 425 | 42 从Qd转换到Qf
likelihood ratio is d(Ld)=Ld*ud(t)*dWp(dWp is Brownian Motion under
P(real)
martingale)
d(Lf)=Lf*uf(t)*dWp
从Qd到Qf的
L=Lf/Ld
那么 dL=L*(uf(t)-ud(t))*dWp
但是这里就有问题了，从Qd转换到Qf时上面的dWp应该是dWd吧(under Qd martingale)
从Qd到Qf的likelihood ratio的kernel 应该是uf(t)-ud(t)对吧，书上是这么说的，但
是得到的公式却出现了dWp而不是dWd。问题在哪里呢？谢谢。 |
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y**********0
发帖数: 425 | 43
equations
hope you read more books about martingale.
Here we are talking about likelihood ratio, not the martingale dynamics of
asset. |
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Q***5
发帖数: 994 | 44
not a very good attitude -- but I will ignore it this time :-)
asset.>
These two are closely related, think about numeraries. |
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s******g
发帖数: 129 | 45 Cant type pinyin now. my apology.
Q:
On average, how many times must a 6-sided die be rolled until a 6 turns up?
clearly, the answer is 6 times. However, is there a way to solve the problem
using martingale?
I assume X_i = 1, if you have 6, or 0 otherwise. S_n = sum of (X_1,... X_n)
It seems that (S_n)^2- n^2/36 - 5*n/36 is a martingale. But how come we end
up with n = 6?
Many thanks! |
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c**a
发帖数: 316 | 46 Problem: B(t) a standard Brownian motion, how is the expecting time it first
hits either -1 or 1.
Solution:
X(t)=exp(B(t)-0.5t) is a martingale. By option sampling theorem, X(t)
stopped at B(t)=-1 or 1 is also a martingale.
Since X(0)=1, we have 0.5*X(1)+0.5*X(-1)=1.
Or exp(1-0.5t)+exp(-1-0.5t)=2,
Or exp(-0.5t)=2/(e+1/e)
Or t = sqrt(-2 ln(2/(e+1/e)).
What is wrong? |
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b****l
发帖数: 23606 | 47 你们这是什么jb系啊？martingale也就算了，基本上经济学上用的多，理工科
不怎么用，妈的马尔可夫，到处都在用，连ee，气象都用这个。居然系里没人
知道？这搞笑吧 |
|
b****l
发帖数: 23606 | 48 martingale的理论意义强，构建随机积分的时候有用，
但实际意义并不大。除了在经济学和金融里面用，别的我真不知道还有哪里用？ |
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T*******x
发帖数: 8565 | 49 马尔可夫比较好理解。而martingale不是很容易理解，尤其是从它的定义式转换到物理
意义。但是它们在概率随机过程中是到处出现的，所以是最基本的。最基本的不一定是
最好理解的。
谁来讲讲吧。 |
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T*******x
发帖数: 8565 | 50 是从过去到现在不断平均化吧？我觉得这个确实抓住了重点。但是还有其他表述。我也
是皮毛。不过其实martingale的意义本来就在皮毛上，我是说本就是一个有物理意义的
东西，常识就可理解。但是数学表述上不容易把这个常识表达出来。
filtration |
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