c*******h
发帖数: 1096 | 1 【 以下文字转载自 Mathematics 讨论区 】
发信人: cockroach (冬冬), 信区: Mathematics
标 题: a question on brownian motion
发信站: BBS 未名空间站 (Thu Feb 10 20:22:29 2011, 美东)
Let W(t) be the standard Brownian motion. It is known that the covariance
matrix K has entries K(i,j)=min{i,j}. Now, if t is a vector instead of a
scalar (I even don't know the name of this random process), what does the
covariance matrix look like? |
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k****2
发帖数: 2 | 2 小弟在学随即过程，可是学的不太明白，有几个问题想请高手指点一下。
#1.
W(t) is standard brownian motion starting at 0.
let
X=integral W(t)dt from 0 to 1.
show that X is a normal distribution and find Cov(X,W(t))
#2.Let Y(t)=exp{W(t)} be geometric brownian motion .evaluate the diffusion
coefficients
Lim E[Y(t+d)-Y(t)|Y(t)=y]/d=b(y) d -> 0+
Lim E[{Y(t+d)-Y(t)}^2|Y(t)=y]/d=a(y) d -> 0+
希望高手讲解一下，谢谢 |
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j****j
发帖数: 270 | 3 Hi all,
I am looking for the Radon-Nikodym derivative when both Brownian Motion and
Poisson Process present? For example, when the Bt and Nt are independent, we
have a Radon-Nikodym derivative which is the product of the RN derivative
of the Brownian Motion part and the RN derivative of the Poisson Process
Part. But what if they are not independent? |
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B****n
发帖数: 11290 | 4 謝謝 想再請問一下
那如果把1/2換成小於1/2的任何正數 是不是就成立了
Brownian Motion的trajectory究竟滿足什麼holder condition的條件
我手頭只有Brownian Motion and Stochastic Calculus這本書 可是沒有找到
如果不容易答 告訴我哪本書有也可以
謝謝 |
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b***k
发帖数: 2673 | 5 ☆─────────────────────────────────────☆
finalguy (答案) 于 (Wed Nov 28 15:04:44 2007) 提到:
two brownian motions,B1 and B2.
B1(0)=0, B2(0)=1.
t denotes the time when B1 and B2 meet each other for the first time. What
is the distribution of t?
☆─────────────────────────────────────☆
eventhorizon (洒水车) 于 (Wed Nov 28 15:14:54 2007) 提到:
Just to show my new suit,
P(t
where Phi is the CDF of standard normal distribution.
Use reflection symmetry of Brownian motion.
☆── |
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r**a
发帖数: 536 | 6
I did not understand why it is not straight forward.
Suppose W_t is a standard brownian motion and t_0 is the time at which W_{t_
0}=m. Then you may construct a new standard brownian motion B_t=W_{t+t_0}-m.
Then ask what the property of the zeros of B_t is. This will be exactly the
same as the example I mentioned.
BTW, the page number might be wrong, since i used the 3rd edition. If u
check the 4th edition, the page number would be 370 and the section number
would be "8.4.1". |
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c**********e
发帖数: 2007 | 7 Let Ws be a Brownian bridge, that is a Brownian motion constrained such that
W0=0 an Wt=x. What is the SDE satisfied by the constrained process? |
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p*********g
发帖数: 453 | 8 请问怎么用SAS估计Brownian Motion的parameters？
For example,
dSt = adt + bdWt <-- Partial Differential Equation
St是已知数据，Wt是Brownian Motion，请问这么估计a和b， 用SAS？
谢谢！ |
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c********y
发帖数: 30813 | 11 【 以下文字转载自 Quant 讨论区 】
发信人: skykive (skykive), 信区: Quant
标 题: 【Brownian Motion】一道题求解
发信站: BBS 未名空间站 (Thu Dec 22 06:41:02 2011, 美东)
Let S_0=0 and for n \in N define
T_n = inf { t >= S_{n-1} | B_t >= 1} and S_n = inf{ t>= T_n | B_t <= -1}
Show that P(lim n -> infinity T_n = infinity) = 1 |
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n*******e
发帖数: 2057 | 12 最近看介电谱图，讲alpha movement是micro-brownian motion of chain segment, 与
glass transition有关，讲beta movement是 local motion of chain segment. 这个所
谓的微布朗运动是不是segment整体可以移动？ |
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B****n
发帖数: 11290 | 13 Assume X is a brownian motion in R^d
Does a radnom variable M exist such that for almost every w
|X(w,t)-X(w,s)|<=M(w)||s-t||^{1/2} for every s,t in R^d and
EM^2<\infty?
Here, ||.|| is the Euclidean distance in R^d
簡單的說就是Brownion Motion是不是滿足Holder condition with constant 1/2 我印
象中好像是對的 請大俠指點一下 謝謝 |
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c*******h
发帖数: 1096 | 14 Let W(t) be the standard Brownian motion. It is known that the covariance
matrix K has entries K(i,j)=min{i,j}. Now, if t is a vector instead of a
scalar (I even don't know the name of this random process), what does the
covariance matrix look like? |
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c*******h
发帖数: 1096 | 15 I guess it is called Brownian random field. Can anyone offer some books that
I
can consult? Thanks. |
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l******r
发帖数: 18699 | 16 It is called a Brownian sheet when the index is multi-dimensional
while the process is uni-dimensional
that |
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m*******r
发帖数: 4468 | 17 非物理专业，
弱问一下，
怎么从brownian motion 中算出水分子的大小？ |
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B*********h
发帖数: 800 | 18 ☆─────────────────────────────────────☆
ilikexck (问天向) 于 (Sat Jul 15 14:36:21 2006) 提到:
根据Martingale convergence theorem, Brownian Motion converges to a random
vairiable B with probability one.
What is B's distribution?
☆─────────────────────────────────────☆
erain (红花会大老板) 于 (Sat Jul 15 18:09:27 2006) 提到:
There are 3 types convergence. w/p 1 convergence is the strongest one.
Here you should fix a specific time t then you can talk about two random variables' convergence result.
so B |
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m****d
发帖数: 331 | 19 Thanks a lot.
Brownian Bridge is:
dZt=[(b-Zt)/(1-t)]dt+dWt, t is [0,1), Z0=a; |
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m****d
发帖数: 331 | 20 如果Z(t)是Brownian motion, 如何证明E[(Z(t+h)-Z(t))^4]=3h^2 ? |
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b***k
发帖数: 2673 | 21 ☆─────────────────────────────────────☆
nevertrue (Blank) 于 (Sun Dec 9 20:16:45 2007) 提到:
x is a brownian motion with drift dx=mdt+dz. If x starts from 0, what is the
probability that x hits 2 before hitting -4?
Can anyone give a solution in detail? Thanks.
☆─────────────────────────────────────☆
findle (It is not a big deal) 于 (Sun Dec 9 20:33:25 2007) 提到:
then exp{-2mX(t)}is a martingale.
the
☆─────────────────────────────────────☆
chopinor (lonelycat) 于 (Sun Dec 9 20:46:33 2 |
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m**********s
发帖数: 87 | 22 Suppose that x is a Brownian motion with drift m and unit variance, i.e. dx
=m dt + dz. If x starts at 0, what is the probability that x hits 3 before
hitting -5? |
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w**********4
发帖数: 4 | 23 The condition for y_n to be martingale is 1/2(t^2 + 1/t) = 1
How did you get this?
Can we transfer the random work to Brownian motion, then choose
Yt=exp(-4/9*Xt) for martingale, where Xt=0.5t+1.5Wt.
E[Yt]=1=p*exp(-4/9*(-1))+(1-p)*0
We can get p=exp(-4/9), close to (sqrt(5)-1)/2
What make the difference?
Thanks |
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S******y
发帖数: 1123 | 24 Looking for Java code for Brownian
Motion Animation -
(i.e. I can generate random
numbers for direction, co-
ordinates, etc, but I do not know
how to write Java swing source
code to do a live animation demo).
Thanks. |
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m*****n
发帖数: 3575 | 25 简单Brownian Motion，请问怎么在起点和终点的条件下，推导中点的的分布？
是不是通过Bayesian啊，具体怎么算？谢谢 |
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a**m
发帖数: 102 | 26 who have any thoughts to show the following statement:
The expected number of times that a brownian motion W hits a particular value in a
given interval of time is infinity. |
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f*******r
发帖数: 198 | 28 CS MS，统计或是微积分什么的不太强。之前面了个quant,一大堆Morkov chain的问题
，都不会，听说quant还挺喜欢Brownian movement的问题。
下星期可能又要面一个hedge fund的quant,临死抱佛脚一把，希望从IT民工转矿工。不
知道大家有什么建议没有。
谢谢。 |
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c***z
发帖数: 6348 | 29 google了半天还是不知道。
如果是导数，物理意义是啥？速度？
倒是有篇说BM处处不可导。
Theorem 1.30 (Paley, Wiener and Zygmund 1933). Almost surely, Brownian
motion is nowhere differentiable.
谢谢大侠！ |
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j*****4
发帖数: 292 | 30 dW represents the rate that the Brownian motion accumlates at. |
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s*****g
发帖数: 77 | 31 stock current price is 50. A contract will pay 1 dollars if the stock hits
100. THe stock price follows geometry brownian motion. What is the price of
the contract? |
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e********5
发帖数: 422 | 32 We have to assume that there is no expiration date for this contract.
Then the contract worth $.50
if it is sold for more, you can write 100 such contract, and sell it
to 100 people. You will get 50+X dollars. Use that $50 to buy 1 share of
stock and put X dollars in your pocket (or riskless bank). By assuming stock
price follows geometric brownian motion, the stock price will eventually
goes to $100 (with probability 1), sell it and give those 100 people $1 each
. You will earn X dollars + ris... 阅读全帖 |
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r********e
发帖数: 169 | 33 geometric brownian motion 不一定到100吧
it
stock
eventually
each
100 |
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s*****g
发帖数: 77 | 34 谢谢，我个人觉得你这个解法挺对的，但是我不理解Geometric brownian motion的这
个条件起到什么作用了吗？
stock
each |
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e********5
发帖数: 422 | 35 shreve的书上有 好像是 brownian motion reach level m的概率是1 但是reach m的期
望是无穷大
我也是猜的 不知道这个GBM有啥用 在这里 |
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r********e
发帖数: 169 | 36 你说的这个结论对Brownian motion是对的。GBM不一定。绿皮书里有详解 |
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d******r
发帖数: 193 | 38 2D brownian motion在时间0～T中离原点距离的最大值的数学期望怎么算啊？ |
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r**a
发帖数: 536 | 39 来看1d brownian motion。根据Doob's不等式，以及W(s)是martingale，我们有下面的不等式 E[sup_{0\leq s\leq T}|W(s)|^2]=4E[W(T)^2]=4T |
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r**a
发帖数: 536 | 40 我给的那个应该是个不等式。对于1d brownian motion, 有下面的结论
E(sup_{0\leq s\leq T}|W(t)|)=\sqrt{\pi T/2}
似乎可以用reflection principle来证明。我记得shreve书里面有讲到这个东西在条件概率。 |
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x*****i
发帖数: 287 | 41 I remembered a property (maybe not precisely), but don't know to prove it.
It is like
If Brownian motion crosses a level m, then it crosses it infinitely many
times in any short period of time.
Thanks a lot. |
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u*****n
发帖数: 28 | 42 这个应该是求一个integration over brownian bridge。把BB 的微分转化成 BM的微分
，就可以了吧，不过似乎写出很繁琐。 |
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h*****u
发帖数: 204 | 43 B_t=\int_0^t sign(W(s))dW（s）
sign(x)=1，if x>=0
sign(x)=-1，if x<0,
We want to show B_t is a Brownian Motion.
I try to use the levy's theorem. but I don't know how to show B_t has
continuous paths. Thanks. |
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k*****y
发帖数: 744 | 44 只考虑finite sum的话，类比tree的情况就是把相应的有些地方往上往下交换一下，直
观上应该还是brownian的。
但是取limit的话，貌似对于任何一条path w, B_t(w)会等于abs(W_t(w))。哪里错了？ |
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l******i
发帖数: 1404 | 45 俺在2楼说了：http://www.mitbbs.com/article0/Quant/31308681_0.html
俺个人脚得可“微”就是连续路径，也就是dB_t存在就行了。
对任意固定的t, dB_t = sign(W(t))dW（t），
在Ito integral定义下也就是B_(t+dt)-B_t=sign(W(t)) (W_(t+dt)-W_t)，
你让dt趋于0，
由于Brownian motion是连续的，sign(W_t)*(W_(t+dt)-W_t)就趋于0，
那么B_(t+dt)-B_t也就趋于0。
注意对任意固定的t，sign(W_t)已经固定，所以我觉得连续性和integrad没有什么关系。
不过这只是我个人的想法，楼主就不大赞同我的观点。 |
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j*******a
发帖数: 101 | 46 Brownian motion， W(t+s) = W(t) + W(s) 感觉是对的，想确认确认。多谢 |
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e******o
发帖数: 757 | 47 一个two dimensional brownian motion starting from (1,1). 当这个它撞上X轴时撞
上负轴的几率是多大？
谢谢 |
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t**********e
发帖数: 68 | 48 打算在给定expected return和Volatility的情况下通过geometric brownian motion生
成几千个one-year price time series，用测试结果的平均值来考验某个trading
strategy。相比historical backtesting,我想用这种方法是不是更有效的测试
strategy的有效性，避免over fitting？
问题是得出的结论似乎与historical backtesting相差比较大，看了我生成的2000个
samples,发现虽然其平均return是我给定的30%，但其分布很广，从-0.6到3.6(见附图
），只有1/4的sample returns落在10%-50%的区间内。
我怀疑如果想测试这个strategy在30% return情况下的表现，这些偏离过远的samples
得出的结果是不是应该剔除？如果是的话该如何剔除？
我是菜鸟，问题也许比较初级，请见谅。多谢！ |
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p****s
发帖数: 3184 | 49 按军板规矩，要不是我幻想出来的，你全家死光？
第58楼，虎肉加了还特地加了？！以示特立独行
quote below
“谁告诉你 Stopped Brownian motion 一定是 martingale 的？！”
你是否想像虎肉那样扣着字眼继续论证一下standard Brownian motion和stopped
Brownian motion的关系？
我的理解是stopped Brownian motion是standard Brownian motion的一个构造子集，
类似进入absorbing state的Markov chain和Markov chain的关系。虎肉傻逼之处是以
为教材上写错了，装逼原创了虎肉鞅论装成了傻逼 |
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i**w
发帖数: 71 | 50 背景:fresh physics PhD
还没offer,但年前基本上不会再折腾了。
有重复。基本上都是很标准的题。
简单的题如果人家想问倒你也是很容易的。
面试书recruiter推荐
1) Mark Joshi: "Quant Job Interviews: Questions and Answers". I have
heard very good things about this book.
2) Xinfeng Zhou, "A Practical Guide to Quantitative Finance Interviews"
个人觉得非常有用， 大部分问题都在这两本上。
算法，C++, stochastic calculus 就看比较标准的几本。
- sqrt(i)=?
- You and me roll a dice,first one gets a six wins. You roll first. what
is the probability of you winning?
- A stair of n steps. Each time you st... 阅读全帖 |
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