Nondecreasing sequence of numbers

Reference no: EM13966729

1. Let X, Y , and Z be independent real random variables with EeX ∞, E |X | ∞, EY = 0, and EZ 2 ∞. Show that {X, eX + Y + Z 2} is a submartingale where B1 is the smallest σ-algebra making X measurable, and find its Doob decomposition. Hint: See Problem 10.1.5.

2. Let fn := an 1[0,1/n] on [0,1] with Lebesgue measure. For what sequences {an } is { fn } uniformly integrable?

3. Let {Xn } be a submartingale. Show that EXn is a nondecreasing sequence of numbers, and give an example to show that Xn - EXn is not necessarily a martingale. Hint: Consider the uniqueness in the Doob decomposition.

Reference no: EM13966729

Questions Cloud

Explaining the different types of abs : Two specific, real-world examples of asset-backed securities with appropriate maturities and how they compare to a Treasury security of the same maturity in terms of the spread over the Treasury yield

Rather than be in a perfectly competitive market : Rather than be in a perfectly competitive market, in an effort to maximize profits, select the type of market would you rather be in and analyze what you would have to do to make it happen. [Include monopoly, monopolistic competition, and oligopoly.]

Player probability of winning : A player throws a fair coin and wins $1 each time it's heads, but loses $1 each time it's tails. The player will stop when his or her net winnings reach $1, or after n throws, whichever comes sooner. What is the player's probability of winning, an..

Find the equation of a tangent plane to surface : Suppose that curves r1(t)= and r2(s)= both lie on a surface S and intersect at point P(0,4,0). Find the equation of a tangent plane to surface S at point P.

Nondecreasing sequence of numbers : Let {Xn } be a submartingale. Show that EXn is a nondecreasing sequence of numbers, and give an example to show that Xn - EXn is not necessarily a martingale. Hint: Consider the uniqueness in the Doob decomposition.

Identify issues that might affect the success : Identify issues that might affect the success of your proposed intervention and explain how you would address these issues.

Continuous real-valued function : Show that for any metric space S, if A ⊂ S and x ∈ A\ A, then there is a bounded, continuous real-valued function on A which cannot be extended to a function continuous on A∪{x }. Hint: f (t ) := sin(1/t ) for t > 0 cannot be extended continuously..

How does the tone of story serve to reinforce storys theme : What is the tone of the story, in setting, narration, action, and dialogue? How does the tone of the story serve to reinforce the story's theme?

Find the value of k that will meet your objective : Find the value of K that will meet your objective as stated in A). How much of the final $50,000 consists deposites and how much consists of interest?

User Account

All Pages