Martingale sequences: the concept, examples, and basic patterns

The concept, examples, and basic patterns

A classical example

The notion of martingales and related concepts seem to have originated in studies of games of chance similar to the following.  Suppose

•       $Y_n=$ a gambler's “gain” on the n th play of a game
•       $Y_0=$ the original capital or “bankroll”

Set $X_n=0$ for ${\displaystyle n<0,X_n=\sum _{k=0}^n{Y}_k\forall n\ge 0}$ .  Thus, X _n is the capital after n plays, and

Put $U_n=(X_0,X_1,\cdots ,X_n)$ and $V_n=(Y_0,Y_1,\cdots ,Y_n)$ .   For any $n\in \mathbf{N}$ , $U_n=g_n\left(V_n\right)$ and $V_n=h_n\left(U_n\right)$ or, equivalently, $\sigma \left(U_n\right)=\sigma \left(V_n\right)$ .  Hence $E\left[H\right|U_n]=E\left[H\right|V_n]\mathrm{a}.\mathrm{s}.$

If Y _N is an independent class with $E\left[Y_n\right]=0\forall n\ge 1$ , the game is considered fair .  In this case, we have by (CE5) , (CE7) , and hypothesis

Also   $E[X_{n+1}-X_n|U_n]=E\left[Y_{n+1}\right|V_n]=0\mathrm{a}.\mathrm{s}.$

Gamblers seek to develop a “system” to improve expected earnings.  We examine some typical approaches and show their futility.  To keep the analysis simple, consider asimple coin-flipping game.  Let

•       $H_k=$ event of a “head” on the k th component trial
•       $T_k=H_k^c=$ event of a “tail” on the k th component trial

The player has a system .  He decides how much to bet on each play from the pattern of previous events.  Let $B_n[I_{H_n}-I_{T_n}]=B_n{Z}_n$ be the result of the n th play, where   $|B_n|$ is the amount of the bet;   $B_n>0$ indicates a bet on a head;   $B_n<0$ indicates a bet on a tail;   $B=0$ indicates a decision not to bet.

Systems take various forms;  here we consider two possibilities.

1. The amount of the bet is determend by the pattern of outcomes of previous tosses
   $$B_n=g_{n-1}(I_{H_1},I_{H_2},\cdots ,I_{H_{n-1}})Y_n=B_n{Z}_n{Z}_n=I_{H_n}-I_{T_n}=2I_{H_n}-1$$
2. The amount bet is determined by the pattern of previous payoffs
   $$B_n=g_{n-1}(Y_1,Y_2,\cdots ,Y_{n-1})=h_{n-1}(B_1,I_{H_1},\cdots ,B_{n-1}{I}_{H_{n-1}})Y_n=B_n{Z}_n$$

Let $Y_0=X_0=C$ , a constant.  Since C is independent of any random variable, $E\left[H\right|C]=E[H]$ . In either scheme, by (CE8) , (CI5) , and the fact $E\left[Z_k\right]=0$

It follows that

The “fairness” of the game is not altered by the betting scheme, since decisions must be based on past performance.  In spite of simple beginnings, the extension and analysis of these patterns form a major thrust of modern probability theory.

Formulation of the concept

In the following treatment,

• $X_{\mathbf{N}}=\{X_n:n\in \mathbf{N}\}$ is the basic sequence   $\mathbf{N}=\{0,1,\cdots \}$
• $Y_{\mathbf{N}}=\{Y_n:n\in \mathbf{N}\}$ is the incremental sequence

We suppose Z _N is a decision sequence and $X_{\mathbf{N}}\sim Z_{\mathbf{N}}$ ; that is, $X_n=g_n\left(W_n\right)=g_n(Z_0,Z_1,\cdots ,Z_n)$ .

• $X_{\mathbf{N}}\sim Z_{\mathbf{N}}$ iff $Y_{\mathbf{N}}\sim Z_{\mathbf{N}}$
• If $X_{\mathbf{N}}\sim H_{\mathbf{N}}$ and $H_{\mathbf{N}}\sim Z_{\mathbf{N}}$ , then $X_{\mathbf{N}}\sim Z_{\mathbf{N}}$ .  In particular, if $H_n=k_n\left(U_n\right)=k_n(X_0,X_1,\cdots ,X_n)$ , then $X_{\mathbf{N}}\sim H_{\mathbf{N}}$ .

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

1. X _N is a martingale (MG) relative to Z _N iff
   $$X_{\mathbf{N}}\sim Z_{\mathbf{N}}\text{and}E\left[X_{n+1}\right|W_n]=X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$$
2. X _N is a submartingale (SMG) relative to Z _N iff
   $$X_{\mathbf{N}}\sim Z_{\mathbf{N}}\text{and}E\left[X_{n+1}\right|W_n]\ge X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$$
3. X _N is a supermartingale (SRMG) relative to Z _N iff
   $$X_{\mathbf{N}}\sim Z_{\mathbf{N}}\text{and}E\left[X_{n+1}\right|W_n]\le X_n\mathrm{a}.\mathrm{s}.\forall n\in \mathbf{N}$$