A Short Note About Markov Process, Transition Semi-group etc.

Markov process 的機率特性 (意指 pdf, fdd 等) 由 transition probability 完全決定。所以先介紹 transition probability 的前身 -- measure kernal ：

Def. Measure Kernal
Let (X,A) and (Y,B) be measurable spaces. A measure kernal k is a function form X,B to R

that is a measure on B given a point in X, and an A-measurable given a event in B.
When the measure is a probability, k is called a probability kernal.

當 (Y,B) = (X,A) 的時候， probability kenals 可以`相乘’：
Def. Product of Probability Kernals
\mu and \nu are probability kernals, their product is defined as

where x is a point in X and B is a event in \mathcal{B}, sorry for the bad notations.
Remark. 我們可以把 k(x,B) 想像成一個 partical 從 x 出發，最後落到 B  的機率。上面的乘法 (\mu \nu)(x,B) 可以解釋成一個 partical 從 x 出發，先經過 y ，再落到 B 上的機率。
Remark2. 注意一下乘法的順序， \mu\nu 就好像是 \mu 作用在 \nu 上面。的確，按照定義 \mu 是當作 measure。

Example. When \mu is the delta distribution concentrated on x

then it is an identity

當然， Probability kernals 還可以跟 X 上的 measure 相乘：

可以解釋為 \pi 作用在 \mu 上面 (因為 \pi 作為 measure)。另外一個對偶的解釋是 \mu 作用在 \pi 上變成 X 上一個新的 measure 。

Def. Transition Semi-group
A transition semi-group is a collection of probability kernals
indexed by two time variable s and t (s<=t) such that
1) for all s equals to t

2) for all s<=t<=u
 
A transition semi-group is called time-homogeneous if

for all s and t.
Remark. Semi-group 就是有乘法、有結合律的集合。
Remark2. 實際上這是一個 Monoid ，因為 delta 扮演 identity 的角色。

我們知道給定 Markov process 有對應的 transition semi-group ，那給定 transition semi-group 是否有對應的 Markov process 呢？答案是肯定的。以下定理如是說：

Thm. Existence of Markov Process Give a Transition Semi-group
Given a transition semi-group and a collection of distribution \pi_t on (X,A) such that

for all s and t, then there exists a Markov process X(t) having the same probability kernals and distributions. The converse is also true that a Markov process defines a transition semi-group via

Remark. 證明存在就用 Kolmogorov ExtensionTheorem 。因為給定 transition semi-group 和一個初始 distribution 就足以決定所有的 fdd (finite dimensional distribution) ，且由於 transition semi-group 的定義，這些 fdd 一定 consistent 。

Thm. Markov property is preserved under coraser filter
If a Markov process is adapted to {g_t}, then it is still Markovian for all coraser filters.

A Short Note about Predictable and Optional Process

In the context of stochastic calculus, integrands have to be predictable. However, the precise definition of predictability is not very direct, therefore, usually in the introductory text we turn to require the integrand to be caglad.

The concept of predictability is not at all obscured; predictable processes are generated by adapted, caglad processes in the same way as continuous functions generate measurable functions.

Simply put, all the adapted, caglad processes generate the smallest sigma-algebra on [0,T]*\Omega; predictable processes are the processes that are measurable function on this sigma-algebra.

Similarly, optional processes are the processes generated by adapted, cadlag processes.

One question arises: how close is a general predictable to caglad processes? Or, how do we approximate a predictable process by caglad processes?

Recall a theorem in measure theory: measurable function is almost as good as a continuous function in the sense that for all \epsilon there exist a compact set K (which has less than \epsilon area difference between the domain) such that the measurable function is uniformly continuous on K. (I forgot the detail about this theorem; I might drop some conditions or even the conclusion is wrong.)

Have no idea how to connect these two things.

Log 23

今天完成了 M-PSK 的架構。不過有幾個問題值得注意：

1. Observable equation 加入的 noise power 極低， variation 幾乎完全由 state equation 掌控。這代表加 Levy noise 或是 Gaussian noise 相差不大。

2. 現在用的演算法在多 symbol 的情況下非常耗時。瓶頸在於 matrix exponential 運算的速度太慢。如果沒有大量的 realization 數目，則無法充分 reuse 運算的結果。embedded AR(1) 應該會比較快。

3. 無論用什麼 implimentation ，都要考慮到怎麼推廣到 ARMA(p,q) 的情況。

幾個要探討的模擬議題：

1. 改變 Linewidth 測 SER 。
2.

我找出 Var[Z(1)] = <F,F> 的證明了，不過今天太晚了，明天再打出來。

Log 22 090421

今日進度：

完成 Merton model 的 code 。不過後來加上去的 Levy noise 出奇的小，讓 phase flucturation 小的可憐 XD 。

 

來日計畫：

把 modulation 加進來。
把 code 改寫的漂亮一點。現在太 messy 。

A Short Note about Spectral Representation

One of the most prominent feature of weakly stationary process is the spectral representation. This representation resembles the familiar Fourier transform of deterministic functions. It doesn’t sounds very exciting for engineers, because they’re already used to perform their favorite transformation on whatever the object is. However, intuition could go wrong when dealing with the infinities. Convergence is a subtle but crucial issue, and must be treated carefully.

This note tries to fill the blank of the textbook `Probability and Random Processes’ as the author only gives the proof for discrete WSS process.

There are two parts of the Spectral representation theorem, one regards of the autorrelation function and the other takes care of the process itself. First we make the notations and definition clear:

A random process X(t) is called wide-sense stationary (WSS) if

Thm. Spectral representation for autocovariance function
H1. The process X(t) has finite variance
H2. Its autocovariance function is continuous at 0
C. The autocovariance function can be expressed as

where Sxx(f) is called its power spectral density function, because it has the unit power per Hertz ([X]^2/Hz).

Thm. Spectral representation for X(t)
H. X(t) satisfies the above assumptions and has zero mean.
C. X(t) can be expressed as

where ^X(f) is defined in a distributional sense.

Remark. the original theorem states the result as

where S is a complex random process with independent increments. I preffer to use the oridinary frequency to stick to the engineer’s convention

Sketch of the proof:
We’ll prove the original formula, and defined the spectral process by

Define the inner product space generated by the random process X(t):

the inner product is defined by

It’s easy to see that it is a subspace of the L2 space

Now define the other inner product space

where the measure is defined by

this is well-defined because of the spectral representation for autocovariance function.
The inner product is defined by

There is a natural way to identify each element in H_mu with H_X. We can define a linear mapping T from H_mu to H_X by

It is an isometry(at least formally), because

This enables us to extend the domain of T from H_mu to its closure, because we’re sure T is continuous by this isometry.

Define the spectral process S by

It’s easy to see S(f) is zero mean, it has orthogonal increments and has variance equals to Sxx(f). i.e

But we need to make sure the indicator function belongs to the closure of H_mu, otherwise the process S is not well-defined. This is true because we can approximate the indicator function by the Fourier series of the truncated indicator function. By making the window of truncation longer and longer, the series converges both pointwisely and in L^2 because it is a finite measure space. Therefore, the closure of H_mu must equal to L^2, because H_mu contains all the step functions, and step functions are L^2 dense in L^2.

Define the stochastic integral of a simple function wrt S by

This integration is a linear mapping from the step functions to H_X. In fact, this integration agrees with the mapping T on simple functions, that is

therefore, we can happily extend this integral to all the function on L^2(R,\mu), and get the result

As we said before, the `Fourier transform’of X(t) is defined as

^X(f) has the following property that’s used a lot in engineering literatures:

Because

which motivates us to write formally that

and

which leads to the concludsion that

Take a geometric way of interpreting the spectral representation: The process X(t) can be seen as a L^2 uniformlly continuous path on the sphere of radious R_XX(0). On the other hand, the spectral process S starts form the origin of the L^2 space at frequency -\infty, and `growth’by orthogonal increments to the surface of the sphere at frequency \infty.

The weak derivative of S seems to be at the platform of white noise anasis. If we define the probability space by ^X, then we get the white noise space? That’s something to be explore in the furture.

Log 21 090406

時光飛逝，轉眼已經到了四月。預計四月底以前要把該研究的東西都差不多搞定。留兩個月的時間寫論文和修改。

最近總覺得自己的知識非常有限。光是 white noise 就把我搞暈了。有太多東西想學，而我的能力卻又這麼有限，我不禁懷疑自己有沒有能力念博士。以數學系的標準來說，我弱爆了。以電機系的標準來說，我沒什麼專長。真不知道自己擅長什麼…

在瑞典快兩年了。這是一段悠閒愜意的時光。生活步調閒散悠哉，唸書也是一樣慵慵懶懶的。有時候覺得這樣實在沒什麼競爭力。

今日進度

1. 試了 BW-driven OU + Levy noise 。搭配 spectrum analysis ，果然簡單許多。不過 Fourier transform 2pi 的問題還有待解決。

2. 我想 white noise 跟 Levy process 的關係應該就跟我猜的一樣。不過我還找不到嚴謹的證明。

 

來日計畫：

1. 把 Fourier transform 係數的問題搞定。完成 Lineshape 的討論。

2. 寫模擬。首先要能產生出 CPP 。再來要產生 OU 。後面都是系統配置的問題了。 OU 的部分可以參考 Levy-driven CARMA 那篇 paper 。

Log 20 090401

一大早就被太陽曬醒。這種感覺真好！北邊的冬天真是陰暗寒冷，不過春分以後日照時間迅速增長，北半球的逆襲啊！

今日進度：

今天再把 Levy process 的教材看過一次。這的確不是個簡單的東西。尤其是 Ito-Levy decomposition 的證明，還要用到 Martingale 的定理。我決定先研究沒有 convergence issue 的 Levy process 。簡單的說，就是 Brownian motion + drift + Compound Possion process 。

 

來日計畫：

確認

是否正確。其中

該開始著手模擬了。最大的挑戰應該就是如何產生 Levy process 吧！其他應該都不難。不過我動作得快一點。