# 1.1 Martingale sequences: examples and further patterns  (Page 2/3)

 Page 2 / 3

Suppose the observed state is j and the action is $a\in A$ . Two results ensue:

1. A return $r\left(j,a\right)$ is realized
2. The system moves to a new state

Let:

${Y}_{n}=$ state in n th period, $0\le n\le N\phantom{\rule{3.33333pt}{0ex}}-1$

${A}_{n}=$ action taken on the basis of ${Y}_{0},\phantom{\rule{0.166667em}{0ex}}{A}_{0},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{Y}_{n-1},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1},\phantom{\rule{0.166667em}{0ex}}{Y}_{n}$

[ A 0 is the initial action based on the initial state Y 0 ]

A policy π is a set of functions $\left({\pi }_{0},\phantom{\rule{0.166667em}{0ex}}{\pi }_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,{\pi }_{N-1}\right)$ , such that

${A}_{n}={\pi }_{n}\left({Y}_{0},\phantom{\rule{0.166667em}{0ex}}{A}_{0},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{Y}_{n-1},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1},\phantom{\rule{0.166667em}{0ex}}{Y}_{n}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le n\le N-1$

The expected return under policy π , when ${Y}_{0}={j}_{0}$ is

$R\left(\pi ,\phantom{\rule{0.166667em}{0ex}}{j}_{0}\right)=E\left[\sum _{k=0}^{N-1}r\left({Y}_{k},\phantom{\rule{0.166667em}{0ex}}{A}_{k}\right)\right]$

The goal is to determine π to maximize $R\left(\pi ,\phantom{\rule{0.166667em}{0ex}}{j}_{0}\right)$ .

Let ${Z}_{k}=\left({Y}_{k},\phantom{\rule{0.166667em}{0ex}}{A}_{k}\right)$ and ${W}_{n}=\left({Z}_{0},\phantom{\rule{0.166667em}{0ex}}{Z}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{Z}_{n}\right)$ . If $\left\{{Y}_{k}:\phantom{\rule{0.277778em}{0ex}}0\le k\le N-1\right\}$ is Markov, then use of (CI9) and (CI11) shows that for any policy the Z -process is Markov. Hence

$E\left[{I}_{M}\left({Y}_{n+1}\right)|\phantom{\rule{3.33333pt}{0ex}}{W}_{n}\right]=E\left[{I}_{M}\left({Y}_{n+1}\right)|\phantom{\rule{3.33333pt}{0ex}}{Z}_{n}\right]\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}n:\phantom{\rule{0.277778em}{0ex}}0\le n\le N-1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}\text{Borel}\phantom{\rule{4.pt}{0ex}}\text{sets}\phantom{\rule{0.277778em}{0ex}}M$

We assume time homogeneity in the sense that

$P\left({Y}_{n+1}=j|{Y}_{n}=i,\phantom{\rule{0.166667em}{0ex}}{A}_{n}=a\right)=p\left(j|i,\phantom{\rule{0.166667em}{0ex}}a\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{invariant}\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}n,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}i,\phantom{\rule{0.166667em}{0ex}}j\in \mathbf{E},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}a\in A$

We make a dynamic programming approach

Define recursively ${f}_{N},{f}_{N-1},\cdots ,{f}_{0}$ as follows:

${f}_{N}\left(j\right)=0,\forall j\in \mathbf{E}$ . For $n=N,N-1,\cdots ,2,1$ , set

${f}_{n-1}\left(j\right)=max\phantom{\rule{0.166667em}{0ex}}\left\{r\left(j,\phantom{\rule{0.166667em}{0ex}}a\right)+\sum _{k\in \mathbf{E}}{f}_{n}\phantom{\rule{3.33333pt}{0ex}}\left(k\right)p\left(k|j,\phantom{\rule{0.166667em}{0ex}}a\right):\phantom{\rule{0.277778em}{0ex}}a\in A\right\}$

Put

${X}_{n}=\sum _{k=1}^{n}\left\{{f}_{k}\left({Y}_{k}\right)-E\left[{f}_{k}\left({Y}_{k}\right)|{W}_{k-1}\right]\right\}$

Then, by A4-2 , $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG, with $E\left[{X}_{n}\right]=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}0\le n\le N$ and

${f}_{n-1}\left({Y}_{n-1}\right)\ge r\left({Y}_{n-1}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1}\right)+\sum _{k\in \mathbf{E}}{f}_{n}\left(k\right)p\left(k|{Z}_{n-1}\right)=r\left({Y}_{n-1}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1}\right)+E\left[{f}_{n}\left({Y}_{n}\right)|{W}_{n-1}\right]$

We may therefore assert

$0=E\left[{X}_{N}\right]=E\left(\sum _{k=1}^{N}\left\{{f}_{k}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{k}\right)-E\left[{f}_{k}\left({Y}_{k}\right)|{W}_{k-1}\right]\right\}\right)\ge E\left(\sum _{k=1}^{N}\left\{{f}_{k}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{k}\right)+r\left({Y}_{k-1},\phantom{\rule{0.166667em}{0ex}}{A}_{k-1}\right)-{f}_{k-1}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{k-1}\right)\right\}\right)$
$=E\left[\sum _{k=0}^{N-1}r\left({Y}_{k},\phantom{\rule{0.166667em}{0ex}}{A}_{k}\phantom{\rule{3.33333pt}{0ex}}\right)+{f}_{N}\left({Y}_{N}\right)-{f}_{0}\left({Y}_{0}\right)\right]=\phantom{\rule{3.33333pt}{0ex}}E\left[\sum _{k=0}^{N-1}r\left({Y}_{k},\phantom{\rule{0.166667em}{0ex}}{A}_{k}\phantom{\rule{3.33333pt}{0ex}}\right)\right]-E\left[{f}_{0}\phantom{\rule{3.33333pt}{0ex}}\left({Y}_{0}\right)\right]$

Hence, $R\left(\pi ,\phantom{\rule{0.166667em}{0ex}}{Y}_{0}\right)\le E\left[{f}_{0}\left({Y}_{0}\phantom{\rule{3.33333pt}{0ex}}\right)\right]$ . For ${Y}_{0}={j}_{0},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}R\left(\pi ,\phantom{\rule{0.166667em}{0ex}}{j}_{0}\right)\le {f}_{0}\left({j}_{0}\right)$ . If a policy π * can be found which yields equality, then π * is an optimal policy.

The following procedure leads to such a policy .

• For each $j\in \mathbf{E}$ , let ${\pi }_{n-1}^{*}\left({Y}_{0},\phantom{\rule{0.166667em}{0ex}}{A}_{0},\phantom{\rule{0.166667em}{0ex}}{Y}_{1},\phantom{\rule{0.166667em}{0ex}}{A}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{A}_{n-2},j\right)={\pi }_{n-1}^{*}\left(j\right)$ be the action which maximizes
$r\left(j,\phantom{\rule{0.166667em}{0ex}}a\right)+\sum _{k\in \mathbf{E}}{f}_{n}\left(k\right)p\left(k|j,\phantom{\rule{0.166667em}{0ex}}a\right)=r\left(j,\phantom{\rule{0.166667em}{0ex}}a\right)+E\left[{f}_{n}\left({Y}_{n}\right)|{Y}_{n-1}=j,\phantom{\rule{0.166667em}{0ex}}{A}_{n-1}=a\right]$
Thus, ${A}_{n}^{*}={\pi }_{n}^{*}\left({Y}_{n}\right)$ .
• Now, ${f}_{n-1}\left({Y}_{n-1}\right)=r\left({Y}_{n-1},\phantom{\rule{0.166667em}{0ex}}{A}_{n-1}^{*}\right)-E\left[{f}_{n}\left({Y}_{n}\right)|{Z}_{n-1}^{*}\right]$ , which yields equality in the argument above. Thus, $R\left({\pi }^{*},\phantom{\rule{0.166667em}{0ex}}j\right)={f}_{0}\left(j\right)$ and π * is optimal.

Note that π * is a Markov policy, ${A}_{n}^{*}={\pi }_{n}^{*}\left({Y}_{n}\right)$ . The functions f n depend on the future stages, but once determined, the policy is Markov.

## A4-9 doob's martingale

Let X be an integrable random variable and Z N an arbitrary sequence of random vectors. For each n , let ${X}_{n}=E\left[X|{W}_{n}\right]$ . Then $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG.

$E\left[|{X}_{n}|\right]=E\left\{|E\left[X|{W}_{n}\right]|\right\}\le E\left\{E\left[|X|\phantom{\rule{0.166667em}{0ex}}|{W}_{n}\right]\right\}=E\left[|X|\right]<\infty$
$E\left[{X}_{n+1}|{W}_{n}\right]=E\left\{E\left[X|{W}_{n+1}\right]|{W}_{n}\right\}=E\left[X|{W}_{n}\right]={X}_{n}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

## A4-9a best mean-square estimators

If $X\in {\mathbf{L}}^{2}$ , then ${X}_{n}=E\left[X|{W}_{n}\right]$ is the best mean-square estimator of X , given ${W}_{n}=\left({Z}_{0},\phantom{\rule{0.166667em}{0ex}}{Z}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{Z}_{n}\right)$ . $\left({X}_{\mathbf{N}},\phantom{\rule{0.166667em}{0ex}}{Z}_{\mathbf{N}}\right)$ is a MG.

## A4-9b futures pricing

Let X N be a sequence of “spot” prices for a commodity. Let t 0 be the present and ${t}_{0}+T$ be a fixed future. The agent can be expected to know the past history ${U}_{{t}_{0}}=\left({X}_{0},\phantom{\rule{0.166667em}{0ex}}{X}_{1},\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}{X}_{{t}_{0}}\right)$ , and will update as t increases beyond t 0 . Put ${Y}_{k}=E\left[{X}_{{t}_{0}+T}|{U}_{{t}_{0}+k}\right]$ , the expected futures price, given the history up to ${t}_{0}+k$ . Then $\left\{{Y}_{k}:\phantom{\rule{0.277778em}{0ex}}0\le k\le T\right\}$ is a Doob's MG, with $Y={X}_{{t}_{0}+T}$ , relative to $\left\{{Z}_{k}:\phantom{\rule{0.277778em}{0ex}}0\le k\le T\right\}$ , where ${Z}_{0}={U}_{{t}_{0}}$ and ${Z}_{k}={X}_{{t}_{0}+k}$ for $1\le k\le T$ .

## A4-9c discounted futures

Assume rate of return is r per unit time. Then $\alpha =1/\left(1+r\right)$ is the discount factor . Let

${V}_{k}=E\left[{\alpha }^{T-k}{X}_{{t}_{0}+T}|{U}_{{t}_{0}+k}\right]={\alpha }^{T-k}{Y}_{k}$

Then

$E\left[{V}_{k+1}|{U}_{{t}_{0}+k}\right]={\alpha }^{T-k}E\left[{Y}_{k+1}|{U}_{{t}_{0}+k}\right]={\alpha }^{T-k-1}{Y}_{k}>{\alpha }^{T-k}{Y}_{k}={V}_{k}\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

Thus $\left\{{V}_{k}:\phantom{\rule{0.277778em}{0ex}}0\le k\le T\right\}$ is a SMG relative to $\left\{{Z}_{k}:\phantom{\rule{0.277778em}{0ex}}0\le k\le T\right\}$ .

Implication from martingale theory is that all methods to determine profitable patterns of prediction from past history are doomed to failure.

## A4-10 present discounted value of capital

If $\alpha =1/\left(1+r\right)$ is the discount factor, X n is the dividend at time n , and V n is the present value, at time n , of all future returns, then

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