(a) Let 0 , 1 , 2 , be independent r v s and E i 0 for all i Let X 0 0 and X t Z 1 Z t for t 1 where Z k k1 k Are Zs independent Show that X t is a martingale (b) Let E i m 0 Write Doobs decomposition

Question: (a) Let 0 , 1 , 2 , ... be independent r.v.s and E{ i } = 0 for all i.

(a) Let ξ₀, ξ₁, ξ₂, ... be independent r.v.’s and E{ξ_i} = 0 for all i. Let X₀ = 0 and X_t = Z₁+...+Z_t for t ≥1 where Z_k = ξ_k−1ξ_k. Are Z’s independent? Show that X_t is a martingale.

(b)∗ Let E{ξ_i} = m = 0. Write Doob’s decomposition.

Step by Step Solution

★★★★★

3.32 Rating (149 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a The neighbor terms Zk k15k and Zk1 Skk1 may be dependent since they contain the common te... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Probability And Stochastic Modeling Questions!

a. Example 13-1: Batch Reactor with an Exothermic Reaction Wolfram 1. Adiabatic Case: Use Wolfram to see whether you can find a trajectory that is ready to ignite and whose trajectory looks like a...

Suppose we have the following information for an economy: GDP deflator 90 100 110 120 130 Aggregate Expenditure 550 500 450 400 350 Output 150 300 450 600 750 a. Plot the AD and AS curves in a...

The following additional information is available for the Dr. Ivan and Irene Incisor family from Chapters 1-6. On December 12, Irene purchased the building where her store is located. She paid...

1.8. Let {Xn}, where n = 0, 1, 2, ... , be independent and identically distributed according to the normal distribution

Let 1 , 2 , ... be independent r.v.s uniformly distributed on [0,a]. Let X 0 = 1 and X t =C t 1 ... t . For which constant C t is X t a martingale?

Problem 1 (20+20 pts). i) Assume that three events A, B and C are independent. Show that A, B and C' are ndependent. 'ii) Let 0 = {1, 2, ...,60} be a sample space with the probability rule P(A) =...

5. A group is a set G together with a binary operation o such that: the operation o has closurefor all x, y in G, x o y is in G associativity (x o y) o z = x o (y o z)for all x, y, z in G ...

FINANCIAL DECISION MAKING QUESTION 1 Nenia Limited (Nenia) is an insurance business with operations in Scotland, England and Wales. At the recent monthly Board of Directors meeting, the CFO raised...

6. Let = {0, 1, 2} be an alphabet. (a) Write out some strings contained in the set * (b) Let L be the language defined by the set of all strings from 2. that do not contain a 1 i.e. only strings with...

Home mortgage lenders usually charge points on a loan to avoid exceeding a legal limit on interest rates or to be competitive with other lenders. As an example, for a two-point loan, the lender would...

In each of the following "transport networks" two companies, ci and C2, produce a certain product that is used by two manufacturers, mi and m2. For the network shown in part (a) of Fig. 13.22,...

Find out the average weight of a group of five items from the given table:

Refer to the Johnson Matthey section on corporate governance in their annual report and accounts 2010, shown on pages 192206, and illustrate their areas of compliance (or not) under the UK Corporate...

Describe a recent purchase you made. Indicate why that particular product was available at a store and, in particular, at the store where you bought it.

Define the functions of marketing in your own words. Using an example, explain how they can be shifted and shared.

Online computer shopping at websites makes it possible for individual consumers to get direct information from hundreds of companies they would not otherwise know about. Co n sumers can place an...

A company retires its bonds at 105. The face value is $100,000 and the carrying value of the bonds at the retirement date is $103,745. The issuer's journal entry to record the retirement will include...

All of the following are considered marketing-related intangible assets except copyright. noncompete agreement. trademark. internet domain name.

Mr. toast purchases a bond that pays out $44,000 in eight years. He then holds the bond for three years and then sells it. If the interest rate is 8%, then what price does he sell it for? $29,742.10,...