Martingale sequences: the concept, examples, and basic patterns (Page 2/2)

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Notation . When we write $(X_{N}, Z_{N})$ is a martingale (submartingale, supermartingale), we are asserting X _N is integrable, Z _N is a decision sequence, $X_{N} \sim Z_{N}$ , and X _N is a MG (SMG, SRMG) relative to Z _N .

Definition . If Y _N is integrable and Z _N is a decision sequence, then

Y _N is absolutely fair relative to Z _N iff
$Y_{N} \sim Z_{N} and E [Y_{n + 1} | W_{n}] = 0 a . s . \forall n \in N$
Y _N is favorable relative to Z _N iff
$Y_{N} \sim Z_{N} and E [Y_{n + 1} | W_{n}] \geq 0 a . s . \forall n \in N$
Y _N is unfavorable relative to Z _N iff
$Y_{N} \sim Z_{N} and E [Y_{n + 1} | W_{n}] \leq 0 a . s . \forall n \in N$

Notation . When we write $(Y_{N}, Z_{N})$ is absolutely fair (favorable, unfavorable), we are asserting Y _N is integrable, Z _N is a decision sequence, $Y_{N} \sim Z_{N}$ , and Y _N is absolutely fair (favorable, unfavorable) relative to Z _N . IXA2-2

Ixa2-1

If X _N is a basic sequence and Y _N is the corresponding incremental sequence, then

$(X_{N}, Z_{N})$ is a martingale iff $(Y_{N}, Z_{N})$ is absolutely fair.
$(X_{N}, Z_{N})$ is a submartingale iff $(Y_{N}, Z_{N})$ is favorable.
$(X_{N}, Z_{N})$ is a supermartingale iff $(Y_{N}, Z_{N})$ is unfavorable.

Let * be any one of the symbols $=, \geq$ , or $\leq$ . Then by linearity and (CE7)

E [X_{n + 1} | W_{n}] = E [Y_{n + 1} | W_{n}] + E [X_{n} | W_{n}] = E [Y_{n + 1} | W_{n}] + X_{n} * X_{n} a . s . iff E [Y_{n + 1} | W_{n}] * 0 a . s .

Remarks

$(X_{N}, Z_{N})$ is a SMG iff $(- X_{N}, Z_{N})$ is a SRMG
We write (S)MG to indicate the same statement can be made for a MG or a SMG with the appropriate choice of = or $\geq$
We write ( ≥ ) to indicate simultaneously two cases:
- $(\geq)$ read as = in all places (for a MG)
- $(\geq)$ read as $\geq$ in all places (for a SMG)

Some basic patterns

Ixa3-1

If $(X_{N}, Z_{N})$ is a (S)MG and $X_{N} \sim H_{N}$ , with $H_{N} \sim Z_{N}$ , then $(X_{N}, H_{N})$ is a (S)MG.

Let $K_{n} = (H_{0}, H_{1}, \dots, H_{n})$ . By (CE9) , the (S)MG definition, monotonicity, and (CE7)

E [X_{n + 1} | K_{n}] = E {E [X_{n + 1} | W_{n}] | K_{n}} (\geq) E [X_{n} | K_{n}] = X_{n} a . s .

Ixa3-2

For integrable $X_{N} \sim Z_{N}$ , the following are equivalent

$\begin{matrix} a & (X_{N}, Z_{N}) & is a (S)MG \\ b & E [X_{n + k} | W_{n}] (\geq) X_{n} a . s . & \forall & n, k \in N \\ c & E [I_{C} X_{n + 1}] (\geq) E [I_{C} X_{n}] & \forall & C \in σ (W_{n}) & \forall & n \in N \\ d & E [I_{C} X_{n + k}] (\geq) E [I_{C} X_{n}] & \forall & C \in σ (W_{n}) & \forall & n, k \in N \end{matrix}$

b $\Rightarrow$ a as a special case
a $\Rightarrow$ b By (CE9) , (a), and monotonicity
$E [X_{n + k} | W_{n}] = E {E [X_{n + k} | W_{n + k - 1}] | W_{n}} (\geq) E [X_{n + k - 1} | W_{n}] a . s .$
$k - 1$ iterations yield $E [X_{n + k} | W_{n}] (\geq) X_{n} a . s .$
d $\Rightarrow$ c as a special case
c $\Rightarrow$ a By (CE1) and (c), $E [I_{C} X_{n + 1}] = E {I_{C} E [X_{n + 1} | W_{n}]} (\geq) E [I_{C} X_{n}] a . s .$ . Since $X_{n} \sim W_{n} a . s .$ and $E [X_{n + 1} | W_{n}] \sim W_{n} a . s .$ , the result follows from the uniqueness property (E5)
b $\Rightarrow$ d By (CE1) , (b), and monotonicity $E [I_{C} X_{n + k}] = E {I_{C} E [X_{n + k} | W_{n}]} (\geq) E [I_{C} X_{n}]$

We thus have $d \Rightarrow c \Rightarrow a \Leftrightarrow b \Rightarrow d$

Ixa3-3

If $(X_{N}, Z_{N})$ is a (S)MG, then $E [X_{n + k}] (\geq) E [X_{n}] (\geq) E [X_{0}]$

Ixa3-4

$(X_{N}, Z_{N})$ is a (S)MG iff $E [X_{q} - X_{p} | W_{n}] (\geq) 0 a . s . \forall n \leq p < q \in N$

EXERCISE. Note $X_{q} - X_{p} = Y_{p + 1} + \dots + Y_{q}$

IXA3-2

Ixa3-5

If $(X_{N}, Z_{N})$ is an $L^{2}$ MG, then

$\begin{matrix} E [X_{q} - X_{p}] = 0 & \forall & p < q \in N \\ E [X_{n} (X_{q} - X_{p})] = 0 & \forall & n \leq p < q \in N \\ E [(X_{n} - X_{m}) (X_{q} - X_{p})] = 0 & \forall & m < n \leq p < q \in N \\ E [X_{p} X_{q}] = E [X_{p \land q}^{2}] & \forall & p, q \in N \\ E [{(X_{q} - X_{p})}^{2}] = E [X_{q}^{2}] - E [X_{p}^{2}] \geq 0 & \forall & p < q \in N \\ E [X_{p}^{2}] = \sum_{k = 0}^{p} E [Y_{k}^{2}] & \forall & p \in N \end{matrix}$

$E [X_{q} - X_{p}] = E {E [X_{q} - X_{p} | W_{n}]} = 0$ by (CE1b) and Thm IXA3-4
$E [X_{n} (X_{q} - X_{p})] = E {X_{n} E [X_{q} - X_{p} | W_{n}]} = 0$ by (CE1b) , (CE8) , and Thm IXA3-4
As in b, since $X_{n} - X_{m} \sim W_{n}$
Suppose $p < q$ . Then, since $X_{p} \sim W_{p}$ , $E [X_{p} X_{q}] = E {X_{p} E [X_{q} | W_{p}]} = E [X_{p}^{2}]$ by definition of MG. For $q < p$ , interchange $p, q$ in the argument above.
$E [{(X_{q} - X_{p})}^{2}] = E [X_{q}^{2}] - 2 E [X_{p} X_{q}] + E [X_{p}^{2}] = E [X_{q}^{2}] - 2 E [X_{p}^{2}] + E [X_{p}^{2}]$ by d, above
By c, $E [Y_{j} Y_{k}] = 0$ for $j \neq k$ . Hence, $E [X_{p}^{2}] = E [{(\sum_{k = 0}^{p} Y_{k})}^{2}] = \sum_{j} \sum_{k} E [Y_{j} Y_{k}] = \sum_{k = 0}^{p} E [Y_{k}^{2}]$

A variety of weighted sums of increments are useful.

Ixa3-6

Suppose $(X_{N}, Z_{N})$ is a (S)MG and Y _N is the incremental sequence. Let H ₀ be a (nonnegative) constant and let $H_{n} \sim W_{n - 1}, n \geq 1$ , be bounded (nonnegative). Set

X_{n}^{*} = \sum_{k = 0}^{n} H_{k} Y_{k} = \sum_{k = 0}^{n} Y_{k}^{*} \forall n \in N

Then $(X_{N}^{*}, Z_{N})$ is a (S)MG.

$E [Y_{n + 1}^{*} | W_{n}] = E [H_{n + 1} Y_{n + 1} | W_{n}] = H_{n + 1} E [Y_{n + 1} | W_{n}] a . s .$ by (CE8)

For MG case: $E [Y_{n + 1}^{*} | W_{n}] = 0 a . s .$ for arbitrary bounded H _n

For SMG case: $E [Y_{n + 1}^{*} | W_{n}] \geq 0 a . s .$ for $H_{n} \geq 0$ , bounded

The conclusion follows from [link]

Remark . This result extends the pattern in the introductory gambling example. [link] IXA3-3

Ixa3-7

In Theorem IXA3-6 , if $E [X_{0}] \geq 0$ and $0 \leq H_{n} \leq 1 a . s . \forall n \in N$ , then $0 \leq E [X_{n}^{*}] \leq E [X_{n}] \forall n \in N$

$E [Y_{n + 1} | W_{n}] \geq H_{n + 1} E [Y_{n + 1} | W_{n}] (\geq) 0 a . s .$ , by hypothesis, and $H_{n + 1} E [Y_{n + 1} | W_{n}] = E [Y_{n + 1}^{*} | W_{n}] a . s .$ , by (CE8) . Thus, by monotonicity and (CE1b)

E [Y_{n + 1}] (\geq) E [Y_{n + 1}^{*}] (\geq) 0 \forall n \in N and E [Y_{0}] = E [X_{0}] \geq H_{0} E [Y_{0}] = E [Y_{0}^{*}]

Hence

E [X_{n}] = \sum_{k = 0}^{n} E [Y_{k}] \geq \sum_{k = 0}^{n} E [Y_{k}^{*}] = E [X_{n}^{*}] \geq 0

Some important special cases

Ixa3-8

Suppose integrable $X_{N} \sim Z_{N}$ . If $X_{n + 1} - X_{n} (\geq) 0 a . s . \forall n \in N$ , then $(X_{N}, Z_{N})$ is a (S)MG.

Apply monotonicity and Theorem IXA3-4

Ixa3-9

Suppose X _N has independent increments.

If $E [X_{n}] = c$ , invariant with n , then X _N is a MG.
If $E [X_{n + 1} - X_{n}] (\geq) 0, \forall n \in N$ , then $(X_{N}$ is a (S)MG.

For any n , consider any $C \in σ (U_{n})$ . By independent increments, ${I_{C}, (X_{n + 1} - X_{n})}$ is independent. Hence, $E [I_{C} X_{n + 1}] - E [I_{C} X_{n}] = E [I_{C} (X_{n + 1} - X_{n})] = E [I_{C}] E [(X_{n + 1} - X_{n})] (\geq) 0$ . The desired result follows from Theorem IXA3-2(c) .

Ixa3-10

Suppose g is a convex Borel function on an interval I which contains the range of all X _n and

$E [| g (X_{n}) |] < \infty \forall n \in N$ , Let $H_{n} = g (X_{n}) \forall n \in N$ ,

If $(X_{N}, Z_{N})$ is a MG, then $(H_{N}, Z_{N})$ is a SMG.
If g is nondecreasing and $(X_{N}, Z_{N})$ is a SMG, then so is $(H_{N}, Z_{N})$

a. By Jensen's inequality and the definition of a MG
$E [g (X_{n + 1}) | W_{n}] \geq g (E [X_{n + 1} | W_{n}]) = g (X_{n}) a . s .$
b. By Jensen's inequality
$E [g (X_{n + 1}) | W_{n}] \geq g (E [X_{n + 1} | W_{n}]) a . s .$
Since $E [X_{n + 1} | W_{n}] \geq X_{n} a . s .$ and g is nondecreasing, we have
$g (E [X_{n + 1} | W_{n}]) \geq g (X_{n}) a . s .$

Some commonly utilized convex functions

$g (t) = | t |$
$g (t) = t^{2}$ g is increasing for $t \geq 0$
$g (t) = u (t) t$ $g (X_{n}) = X_{n}^{+}$ g nondecreasing for all t
$g (t) = - u (- t) t$ $g (X_{n}) = X_{n}^{-}$ g nonincreasing for all t
$g (t) = e^{a t}, a > 0$ g is increasing for all t

Ixa3-11

Consider integrable $X_{N} \sim Z_{N}$ .

If $E [X_{n + 1} | W_{n}] = a X_{n} a . s . \forall n$ and $X_{n}^{*} = \frac{1}{a^{n}} X_{n} \forall n$ , then $(X_{N}^{*}, Z_{N})$ is a MG
If $E [X_{n + 1} | W_{n}] \geq a X_{n} a . s ., a > 0, \forall n$ and $X_{n}^{*} = \frac{1}{a^{n}} X_{n} \forall n$ , then $(X_{N}^{*}, Z_{N})$ is a SMG

E [X_{n + 1}^{*} | W_{n}] = \frac{1}{a^{n + 1}} E [X_{n + 1} | W_{n}] (\geq) \frac{1}{a^{n + 1}} a X_{n} = X_{n}^{*} a . s .

The restrictionl $a > 0$ is needed in the $\geq$ case.

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Source: OpenStax, Topics in applied probability. OpenStax CNX. Sep 04, 2009 Download for free at http://cnx.org/content/col10964/1.2

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