白色兒子槍 2018-10-18 00:32:01

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Exotic 2018-10-18 00:34:06 講埋markov martingale 嘛

利申: 行家

MacGhosFy 2018-10-18 00:37:10 完全睇唔明partII, 證明大學唔take係正確

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吾不會數學 2018-10-18 00:37:42 連probability都無點學stochastic calculus

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非典型廢柴 2018-10-18 00:50:08 留名
下年minor要學

會考都無C 2018-10-18 01:15:09 有啲用ms word有啲又用latex Beamer class嘅

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Btw數撚留名

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ihavetoldyou 2018-10-18 01:32:51 算撚數

連狗淨係識數浪畫圖

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葉山美空 2018-10-18 01:40:45 留名學野

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，其實香港有冇data analysis 呢行

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ICON 2018-10-18 01:50:23 認真 樓主如果用呢種文筆出書 應該會幾好賺

宇智波月巴 2018-10-18 01:54:20 2b.) Ito's lemma and isometry

Ito's lemma同isometry可以話係令Stochastic calculus真正workable嘅神兵利器
之後好多result都需要用到呢條lemma同埋isometry
所以呢一part係非常非常重要嘅foundation!!!

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之不過係真正打大佬之前
我地其實仲有兩個concept要build up --- 佢地分別就係Information (F) 同 Martingale
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(a). Information/Filtration (or in rigorous term "σ-field")
下圖係"比較formal"嘅definition

大家都係唔洗比啲鬼畫符嚇親

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我地首先簡化咗個情況先 假設我地依家有n個time point
然後我地假設埋每個timepoint嘅stock price係 S(t_0) = 100, S(t_1) = 99.5 ,...., S(t_(n-1)) = 97
如果time now = t_(n-1) 咁information generated by S at time = t_(n-1)就包含上面所有野
個concept其實就類似係咁
(其實唔止包含咁少野 有興趣知多啲可以睇埋下面optional個段)



跟住我地又黎abuse notation

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正常黎講我地只可以話一啲event/set係喺information/filtration入面
但因為stochastic variable個domain其實係sample space(一堆event)
所以我地就直接abuse notation(好似上圖咁)
如果given information up to time t, Z可以完全determine到嘅話
我地就話 "Z is adapted to the filtration"

[Content below is optional to readers]
點解我會話呢個只係"比較formal"嘅definition?
因為information/filtration其實即係一個sample space嘅σ-field
咩係σ-field? 大家有讀過probability嘅話應該都會見過呢三兄弟 (Ω,F,P) 中間嘅F就係σ-field
而σ-field本身就已經有一個非常formal嘅definition (contains ∅ and Ω, close under complement, close under countable union)

我假設大家睇得呢段野都有學過咩叫σ-field啦

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(想學多啲就睇下measure theory)
依家你可以幻想下我地model緊stock price嘅movement
一個簡單啲嘅情況係想像有n個discrete嘅time point
咁係t = 1嘅時候 我地嘅information 其實就係類似係 F1 = {∅,{S_0},{S_1},{S_1嘅complement},Ω}
where {S_n} = the event that stock price at time n equals to S_n
當然呢個非常非常informal嘅寫法 不過大家可以見到其實呢個information set即係σ-field
而F1係會包含係F2入面 如此類推
(In other words, information/filtration is essentially a collection of subsets of σ-field)

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(b). Martingale
下圖係formal definition


其實最重要嘅property係第三點
如果given information up to time s (s<=t)
咁呢個stochastic variable X(t)嘅conditional expectation 只會係X(s)
In other words, the expectation is only up to current knowledge
另一種講法就係當呢個X係martingale 咁樣先會係fair game

點解係fair game?
大家試想下 如果個conditional expectation係細過X(s)
咁即係代表in long run, 你係會不停輸錢 (The technical term is supermartingale)
而相反X(s) > E[X(t)|F_s] => in long run不停贏錢 (The technical term is submartingale)
只有兩者相等先會係fair game

下圖係兩個好trivial嘅related result

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(c). Ito's isometry


上圖就係大家期待而久嘅ito's isometry

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點解ito's isometry咁有用? 就係因為佢話左比我聽expectation of stochastic integral可以點計

[Content below is optional to readers]
第一句嘅assumption其實即係話

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大家首先諗下普通integral係點樣嘅先 最後面個"ds"嘅"s"應該係deterministic嘅
但係依家stochastic integral顧名思義 "dW(s)"嘅"W(s)"係stochastic(random)嘅
所以依靠平時計riemann integral (or lebesgue integral)嘅方法係冇任何幫助

不過既然係random嘅野 咁用返我地學random variable嘅logic
我地照樣可以搵呢樣野嘅mean (E) 同 variance (VAR)係咩
而ito's isometry就已經直接話左比我地聽呢兩樣野分別點計

比個好簡單嘅例子大家:

(希望大家follow到example嘅steps 唔明可以隨時問

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)

相信醒目嘅朋友已經即刻知道呢個isometry有幾勁
如果我地知道E(Y)同Var(Y(t)) 加埋dW(t) follows Normal 呢個property
我地可以rewrite Y as the following:

咁樣就可以方便我地去用電腦simulate呢個Y

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(因為我地知道點樣sim standard normal嘅variable)
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(d). Application of Ito's isometry
Ito's isometry 對martingale有兩個重要嘅result 請睇下圖


1.) Wiener process係martingale (其實佢仲係markov process添 因為無論你知道幾多history都好 都只係最latest嘅information對你有用)

2.) 呢個result話咗比我聽 如果一個stochastic process X要係martingale 咁佢一定要冇drift term
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打住咁多先 ito's lemma下個cm講 我唔夠位了

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宇智波月巴 2018-10-18 01:55:19 gg我唔識呢堆野

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RMSC嘅course冇教到咁深

宇智波月巴 2018-10-18 01:56:09 因為有啲野我係直接由notes (pdf) crop過黎

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懶得再係word打一次

bzbz 2018-10-18 01:57:11 難到仆街

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宇智波月巴 2018-10-18 01:59:32 This is the magic of abusing notation

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你要記得我由頭到尾都冇話過佢d到
我只係將∆t->0
然後用dW(t) 代替 W(t+∆t) - W(t), same for dX(t)

宇智波月巴 2018-10-18 02:00:50 我講ito's lemma嘅時候就會一次過處理曬呢堆野

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Captain_America 2018-10-18 02:47:46 屌，HKU 啲 stochastic 講咁深入既

垃圾 CU 教到 markov chain , continuous markov chain, birth and death process 就算

就算 ODE 都係輕輕帶過，連 PDE 都無講過

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吾不會數學 2018-10-18 02:49:03 stochastic process同stochastic calculus唔同架喎

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Captain_America 2018-10-18 02:52:40 屌，唔小心將R4005 同 Stat3007 撈亂左

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血小板張六常 2018-10-18 03:16:44 此回覆已被刪除