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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 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

  1. Y N is absolutely fair relative to Z N iff
    Y N Z N and E [ Y n + 1 | W n ] = 0 a . s . n N
  2. Y N is favorable relative to Z N iff
    Y N Z N and E [ Y n + 1 | W n ] 0 a . s . n N
  3. Y N is unfavorable relative to Z N iff
    Y N Z N and E [ Y n + 1 | W n ] 0 a . s . n 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 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

  1. ( X N , Z N ) is a martingale iff ( Y N , Z N ) is absolutely fair.
  2. ( X N , Z N ) is a submartingale iff ( Y N , Z N ) is favorable.
  3. ( X N , Z N ) is a supermartingale iff ( Y N , Z N ) is unfavorable.

Let * be any one of the symbols = , , or .  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

  1. ( X N , Z N ) is a SMG iff ( - X N , Z N ) is a SRMG
  2. We write (S)MG to indicate the same statement can be made for a MG or a SMG with the appropriate choice of = or
  3. We write ( ) to indicate simultaneously two cases:
    • ( ) read as = in all places (for a MG)
    • ( ) read as in all places (for a SMG)

Some basic patterns

Ixa3-1

If ( X N , Z N ) is a (S)MG and X N H N , with H N Z N ,  then ( X N , H N ) is a (S)MG.

Let K n = ( H 0 , H 1 , , 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 } ( ) E [ X n | K n ] = X n a . s .

Ixa3-2

For integrable X N Z N , the following are equivalent

a ( X N , Z N ) is a (S)MG b E [ X n + k | W n ] ( ) X n a . s . n , k N c E [ I C X n + 1 ] ( ) E [ I C X n ] C σ ( W n ) n N d E [ I C X n + k ] ( ) E [ I C X n ] C σ ( W n ) n , k N

  • as a  special case
  • By (CE9) , (a), and monotonicity
    E [ X n + k | W n ] = E { E [ X n + k | W n + k - 1 ] | W n } ( ) E [ X n + k - 1 | W n ] a . s .
    k - 1 iterations yield   E [ X n + k | W n ] ( ) X n a . s .
  • as a  special case
  • By (CE1) and (c),   E [ I C X n + 1 ] = E { I C E [ X n + 1 | W n ] } ( ) E [ I C X n ] a . s . .   Since X n W n a . s . and E [ X n + 1 | W n ] W n a . s . , the result follows from the uniqueness property (E5)
  • By (CE1) , (b), and monotonicity E [ I C X n + k ] = E { I C E [ X n + k | W n ] } ( ) E [ I C X n ]

We thus have d c a b d

Ixa3-3

If ( X N , Z N ) is a (S)MG, then E [ X n + k ] ( ) E [ X n ] ( ) E [ X 0 ]

Ixa3-4

( X N , Z N ) is a (S)MG iff E [ X q - X p | W n ] ( ) 0 a . s . n p < q N

EXERCISE.  Note X q - X p = Y p + 1 + + Y q

IXA3-2

Ixa3-5

If ( X N , Z N ) is an L 2 MG, then

E [ X q - X p ] = 0 p < q N E [ X n ( X q - X p ) ] = 0 n p < q N E [ ( X n - X m ) ( X q - X p ) ] = 0 m < n p < q N E [ X p X q ] = E [ X p q 2 ] p , q N E [ ( X q - X p ) 2 ] = E [ X q 2 ] - E [ X p 2 ] 0 p < q N E [ X p 2 ] = k = 0 p E [ Y k 2 ] p N

  1. E [ X q - X p ] = E { E [ X q - X p | W n ] } = 0 by (CE1b) and Thm IXA3-4
  2. 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
  3. As in b, since X n - X m W n
  4. Suppose p < q .  Then, since X p 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.
  5. 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
  6. By c, E [ Y j Y k ] = 0 for j k .  Hence, E [ X p 2 ] = E [ ( k = 0 p Y k ) 2 ] = j k E [ Y j Y k ] = 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 W n - 1 , n 1 , be bounded (nonnegative).  Set

X n * = k = 0 n H k Y k = k = 0 n Y k * n 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 ] 0 a . s . for H n 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 ] 0 and 0 H n 1 a . s . n N , then 0 E [ X n * ] E [ X n ] n N

E [ Y n + 1 | W n ] H n + 1 E [ Y n + 1 | W n ] ( ) 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 ] ( ) E [ Y n + 1 * ] ( ) 0 n N and E [ Y 0 ] = E [ X 0 ] H 0 E [ Y 0 ] = E [ Y 0 * ]

Hence

E [ X n ] = k = 0 n E [ Y k ] k = 0 n E [ Y k * ] = E [ X n * ] 0

Some important special cases

Ixa3-8

Suppose integrable X N Z N . If X n + 1 - X n ( ) 0 a . s . n N , then ( X N , Z N ) is a (S)MG.

Apply monotonicity and Theorem IXA3-4

Ixa3-9

Suppose X N has independent increments.

  1. If E [ X n ] = c , invariant with n , then X N is a MG.
  2. If E [ X n + 1 - X n ] ( ) 0 , n N ,   then ( X N is a (S)MG.
  1. For any n , consider any C 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 ) ] ( ) 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 ) | ] < n N ,  Let H n = g ( X n ) n N ,

  1. If ( X N , Z N ) is a MG, then ( H N , Z N ) is a SMG.
  2. If g is nondecreasing and ( X N , Z N ) is a SMG, then so is ( H N , Z N )
  • By Jensen's inequality and the definition of a MG
    E [ g ( X n + 1 ) | W n ] g E [ X n + 1 | W n ] = g ( X n ) a . s .
  • By Jensen's inequality
    E [ g ( X n + 1 ) | W n ] g E [ X n + 1 | W n ] a . s .
    Since E [ X n + 1 | W n ] X n a . s . and g is nondecreasing, we have
    g E [ X n + 1 | W n ] g ( X n ) a . s .

Some commonly utilized convex functions

  1. g ( t ) = | t |
  2. g ( t ) = t 2 g is increasing for t 0
  3. g ( t ) = u ( t ) t g ( X n ) = X n + g nondecreasing for all t
  4. g ( t ) = - u ( - t ) t g ( X n ) = X n - g nonincreasing for all t
  5. g ( t ) = e a t , a > 0 g is increasing for all t

Ixa3-11

Consider integrable X N Z N .

  1. If   E [ X n + 1 | W n ] = a X n a . s . n and X n * = 1 a n X n n , then ( X N * , Z N ) is a MG
  2. If   E [ X n + 1 | W n ] a X n a . s . , a > 0 , n and X n * = 1 a n X n n , then ( X N * , Z N ) is a SMG
E [ X n + 1 * | W n ] = 1 a n + 1 E [ X n + 1 | W n ] ( ) 1 a n + 1 a X n = X n * a . s .

The restrictionl a > 0 is needed in the 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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