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V n = k = 1 α k X n + k so that V n + 1 = k = 1 α k X n + k + 1 = k = 2 α k - 1 X n + k
= 1 α k = 1 α k X n + k - X n + 1 = ( 1 + r ) V n - X n + 1

Note that V n + 1 ( ) V n iff r ( ) X n + 1 / V n . Set Y n = E [ V n | U n ] . Then Y n + 1 = ( 1 + r ) E [ V n | U n + 1 ] - X n + 1 a . s . so that

E [ Y n + 1 | U n ] = ( 1 + r ) Y n - E [ X n + 1 | U n ]

Thus, ( Y N , X N ) is a (S)MG iff

r ( ) E [ X n + 1 | U n ] E [ V n | U n ] = Expected return next period, given U n Expected present value, given U n

Summary: convergence of submartingales

The submartingale convergence theorem

If ( X N , Z N ) is a SMG with lim n E [ X n + ] < , then there exists X W such that X n X a . s .

Uniform integrability and some convergence conditions

Definition . The class { X t : t T } is uniformly integrable iff

sup { E [ I { | X t | > a } | X t | ] : t T } 0 as a

Any of the following conditions ensures uniform integrability:

  1. The class is dominated by an integrable random variable Y .
  2. The class is finite and integrable.
  3. There is a u.i. class { Y t : t T } such that | X t | | Y t | a . s . for all t T .
  4. X integrable implies Doob's MG { X n = E [ X | W n ] : n N } is u.i.

Definition . The class { X t : t T } is uniformly absolutely continuous iff for each ϵ > 0 there is a δ > 0 such that P ( A ) < δ implies sup T { E [ I A | X t | ] : t T } < ϵ .

X T = { X t : t T } is u.i. iff both (i) X T is u.a.c., and (ii) sup T { E [ | X t | ] } < .

Definition . X n P X iff P ( | X n - X | > ϵ ) as n for all ϵ > 0 .

X n L p X iff E [ | X n - X | p ] 0 as n

X n X a . s . implies X n P X

If (i) X n L p X , (ii) X n X a . s . and (iii) lim n E [ X n | Z ] exists a.s., then

lim n E [ X n | Z ] = E [ X | Z ] a . s .

Suppose ( X N , Z N ) is a (S)MG. Consider the following

  • lim n E [ X n + ] < or, equivalently, sup n E [ | X n | ] < - - - - - - - - - - - -
  • X N is uniformly integrable.
  • X N + is uniformly integrable.
  • X n L 1 X .
  • X n + L 1 X .
  • There is an integrable X W such that
    X n X a . s . and E [ X | W n ] ( ) X n a . s . n N
  • Condition (c) with even for a MG.
  • There is an integrable X with E [ X | W n ] ( ) X n a . s . n N
  • Condition (d) with even for a MG.

Then

  1. Each of the propositions (a) through (d ' ) implies (A), hence SMG convergence.
  2. (a) (a + )
  3. (a) (b) (c) (d)
  4. (a + ) (b + ) (c ' ) (d ' )
  5. For a MG, (d) (a), so that (a) (b) (c) (d)

The notion of regularity is characterized in terms of the conditions in the theorem.

Definition . A martingale ( X N , Z N ) is said to be martingale regular iff the equivalent conditions (a), (b), (c), (d) in the theorem hold.

A submartingale ( X N , Z N ) is said to be submartingale regular iff the equivalent conditions (a + ), (b + ), (c ' ), (d ' ) in the theorem hold.

Remarks

  1. Since a MG is a SMG, a martingale regular MG is also submartingale regular.
  2. It is not true, in general that a submartingale regular SMG is martingale regular. We do have for SMG (a) (b) (c) (d).
  3. Regularity may be viewed in terms of membership of X in the (S)MG. The condition E [ X | W n ] ( ) X n a . s . is indicated by saying X belongs to the (S)MG or by saying the (S)MG is closed (on the right) by X .

Summary

For a martingale ( X N , Z N )

  1. If martingale regular, then X n X W a . s . and X n = E [ X | W n ] a . s . n N E [ X n + k | W n ] = E { E [ X | W n + k ] | W n } = E [ X | W n ] = X n a . s . and E [ X 0 ] = E [ X n ] = E [ X ] n N
  2. If submartingale regular, but not martingale regular, then X n X W a . s . but E [ X | W n ] X n a . s . n N and E [ X 0 ] = E [ X n ] E [ X ] < n N

For a submartingale ( X N , Z N )

Either martingale regularity or submartingale regularity implies

X n X W a . s . and X n E [ X n + 1 | W n ] E [ X | W n ] a . s . n N

and E [ X 0 ] E [ X n ] E [ X ] < n N

If X N is uniformly integrable , then E [ X n ] E [ X ] .

If ( X N , Z N ) is a MG with E [ X n 2 ] < K n ı N , then the proceess is MG regular, with

X n X a . s . E [ ( X - X n ) 2 ] 0 and E [ X n ] = E [ X ] n N

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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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