Brownian motion到底是什么东西?

hoollly

http://en.wikipedia.org/wiki/Brownian_motion

Wiener process是什么来的?
我读书时候有读到brownian motion,它们的解释就是随机运动,也就是无数个小分子撞击一个大分子,而那个大分子的运动就叫做brownian motion
但是疑问来了，既然brownian motion是随机的，那么它的数学公式是算什么的？既然是随机的，算出来的答案可能是每次一样的么？

mikata

The keyword here is "stochastic" (ie. This means that even if the initial condition (or starting point) is known, there are more possibilities the process might go to, but some paths are more probable and others less.). In short, probability of an event not only changes with initial condition, it also changes with time (ie. Markov chain). Keep in mind that probability is just a measure of likeliness, not exactness.

hoollly

多谢回复我如此冷门的贴.

In short, probability of an event not only changes with initial condition, it also changes with time (ie. Markov chain).

想问你,那么计算stochastic的probability,standard deviation和mean是如何得到的.以什么为参考.

因为我有学金融数学,里面有提及到stochastic process,模拟出一段走势,里面有说monte carlo simulation is not depending on the past path,but the current path.
如果stochastic的过程是和过去irrelevant,那么设置其stochastic的parameter是哪里拿来的~

等着回复...

mikata

The answer for your question lies in the studies of correlation of random variables (ie. how strong are random variables related, by studying statistical pattern or distribution) and these paragraphs have partly answered your question. "One approach to stochastic processes treats them as functions of one or several deterministic arguments ('inputs', in most cases regarded as 'time') whose values ('outputs') are random variables: non-deterministic (single) quantities which have certain probability distributions...Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical correlations..

hoollly

我理解了,就是大概是:
用stochastic process模拟出来的走势,是根据过去的variable,而不是historical path.
比如一个股票过去从1元涨到10,再到回2,今天重复看回,发觉又是1元到10元,但是如果判断会跌回2,就是根据historical path方式判断.
如果是根据当初涨到10元,跌回2元的variable,multivariate计算.这就是stochastic process方式.(由于variable的不同,未必跌回2元,如果和它correlate的variable竟然升到100的话,那么它也会涨到100去,而不是跌到2元.)
那么,这是否意味stochastic process中间一些variable,probability的判断其实和expected value一样,是多数属于主观判断.(比如根据empirical,揣测,inference)
stochastic process的意思,是不是就好象expected value那样,给人一个想象/决策空间?并不在于提供什么精确数字和答案?

mikata

Your partly right. In short "stochastic" loosely speakingis "What you expect now for the future (though it's unknown, random) has to rely on past events and experience (ie. statistics)". The modelling example like monte carlo correctly reflects such characteristics by using probability and statistical theory.

hoollly

嗯嗯,多谢回复.我想我需要很长的时间继续消化目前的知识和揣测.真复杂,我学这个时候,弄来弄去,都不明白stochastic的sample是哪里採出来的,真复杂~