随机过程英语讲义-15

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Chapter 5
Martingales
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5.0 Martingales origins
Originally, martingale referred to a class of betting strategies that was popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, his probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users.
Figure. 5.3.1
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5.3 Azuma’s Inequality for Martingales
As the formula for this line segment is
y − f ( β ) y − f (−α ) = x−β x − (−α ) ⇔ y=
β α +β
f (−α ) +
or, expanding in a Taylor series,
β 2i +1 /(2i )! =∑ β 2i +1 /(2i + 1)! ∑
i =0 i =0
∞
∞
xi ex = ∑ i =0 i !
∞
which is clearly not possible when β ≠ 0 . Hence, if the lemma is not true, we can conclude that the strictly positive maximal value of f (α , β ) occurs when β = 0 . However, f (α , 0) = 0 and thus the lemma is proven.
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5.3 Azuma’s Inequality for Martingales
Theorem 5.3.3 Azuma's Inequality Let Zn, n ≥ 1 be a martingale with mean μ = E[ Z n ]. Let Z 0 = μ and suppose that for nonnegative constants α i , βi , i ≥ 1,
STOCHASTIC PROCESSES AND ITS APPLICATIONS
Prof. Xia Yuanqing School of Automation Beijing Institute of Technology E-mail:yuanqing.xia@gmail.com Assistant Dai Li School of Automation Beijing Institute of Technology E-mail:daili1887@gmail.com
E[ f ( X )] ≤
β α +β
f (−α ) +
α α +β
f (β )
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5.3 Azuma’s Inequality for Martingales
Proof: Since f is convex it follows that, in the region −α ≤ x ≤ β , it is never above the line segment connecting the points (−α , f (−α )) and ( β , f ( β )) . (See Figure 5.3.1.)
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5.0 Martingales origins
The concept of martingale in probability theory was introduced by Paul Pierre Levy, and much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies. This is the essence of the Martingale.
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5.3 Azuma’s Inequality for Martingales
Now the preceding inequality is true when α = −1 or +1 and when β is large (say when | β |≥ 100 ). Thus, if Lemma 5.3.2 were false, then the function
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5.1 Martingale ,
Why?
= E[ Z n | Z1 , L , Z n ] = Zn .
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5.1 Fra Baidu bibliotekartingale
Why?
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5.1 Martingale
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5.1 Martingale
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5.2 Stopping Times
Definiton:
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5.2 Stopping Times
Corollary
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5.2 Stopping Times
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5.2 Stopping Times
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5.2 Stopping Times
Example :
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5.2 Stopping Times
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5.3 Azuma’s Inequality for Martingales
Let Zi , i≥0 be a martingale sequence. In situations where these random variables do not change too rapidly, Azuma's inequality enables us to obtain useful bounds on their probabilities. Before stating it, we need some lemmas. Lemma 5.3.1 Let X be such that E [X] = 0 and P{−α ≤ X ≤ β } = 1 .Then for any convex function f
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5.3 Azuma’s Inequality for Martingales
We just consider that α ≠ 0 when β ≠ 0 . We can see that
e β + e− β α 1+ α β = 1 + . ⇒ β (e β + e − β ) = e β − e − β e − e− β β
Assuming a solution in which β ≠ 0 implies, upon division, that
e β + e− β α 1+ α β = 1+ . −β e −e β
As it can be shown that there is no solution for which α = 0 and β ≠ 0 (see Problem 5.14) When α = 0 and β ≠ 0, [1] ⇔ [2] ⇔ e β − e − β = 2 β exp{αβ + β 2 / 2} 5.14 shows that the equation Remark e β − e − β = 2 β exp{αβ + β 2 / 2} has no solution when β ≠ 0 .
E[ f ( X )] ≤
E [X] = 0
f ( β ).
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β α +β
f (−α ) +
α α +β
5.3 Azuma’s Inequality for Martingales
Lemma 5.3.2 For 0 ≤ θ ≤ 1
θe
(1−θ ) x
+ (1 − θ )e
−θ x
≤e
x2 / 8
Proposition:
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5.2 Stopping Times
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5.2 Stopping Times
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5.2 Stopping Times
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5.2 Stopping Times
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5.2 Stopping Times
Theorem:
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5.2 Stopping Times
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5.2 Stopping Times
4
Paul Lévy
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Joseph Doob
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5.0 Martingales origins
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5.1 Martingale
Definiton:
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5.1 Martingale
Remark:
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5.1 Martingale ,
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5.1 Martingale
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5.1 Martingale ,
α α +β
f (β ) +
1 [ f ( β ) − f (−α )]x α +β
it follows, since −α ≤ X ≤ β , that
f (X ) ≤
β α +β
f (−α ) +
α α +β
f (β ) +
1 [ f ( β ) − f (−α )] X . α +β
Taking expectations gives the result, that is,
−α i ≤ Z i − Z i −1 ≤ βi .
Then for any n ≥ 0, a > 0 :
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5.3 Azuma’s Inequality for Martingales
Proof: Suppose first that μ = 0 . Now, for any c > 0 Markov's inequality Remark (| x |) E
∂f (α , β ) = 0 ⇒ e β − e − β + α (e β + e − β ) = 2(α + β ) exp{αβ + β 2 / 2} [1] ∂β ∂f (α , β ) = 0 ⇒ e β − e − β = 2 β exp{αβ + β 2 / 2} [2] ∂α
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5.3 Azuma’s Inequality for Martingales
Proof: Letting θ = (1 + α ) / 2 and x = 2 β , we must show that for −1 ≤ α ≤ 1 ,
(1 + α )e
β (1−α )
+ (1 − α )e
− β (1+α )
≤ 2e
β2 /2
or, equivalently,
e β + e − β + α (e β − e− β ) ≤ 2 exp{αβ + β 2 / 2}.
f (α , β ) = e β + e − β + α (e β − e − β ) − 2 exp{αβ + β 2 / 2}
would assume a strictly positive maximum in the interior of the Region R = {(α , β ) :| α |≤ 1,| β |≤ 100}. Setting the partial derivatives of f equal to 0, gives that