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这是一篇英国作业是金融随机微积分相关一个Assignment
Instructions:
Please submit your answers by 4pm on Friday, December 3rd, 2021. No late submissions will be accepted.
Please submit one solution set for each group.
Please include the surnames, first names, and group number (in this order) of each member in the group, in your submission.
Please justify all your answers. However, your answers need to be succinct and to the point; you might get penalized for long-winded answers.
There are 8 questions on this coursework, worth a total of 100 marks. The number of marks that each question is worth is written in brackets.
Question 1 [10 marks]
Answer the following questions:
a. Suggest a possible sample space to describe each of the following scenarios: i. A roulette wheel spin.
ii. The number of claims arising in an insurance portfolio each year. iii. A soccer match.
b. Prove that the following sets are Borel sets by directly showing that they can be obtained from intervals such as (∞,a],a ∈ R through the usual operations (unions, intersections, complements):
i. (2,+∞)
ii. (0,1] ∪ (2,3]
iii. (∞,1) iv. {1}
Question 2 [10 marks]
A fixed-income portfolio contains n zero-coupon bonds subject to default risk, each of them with maturity 1 year. The i-th bond has face value Ci. Define the following random variables:
N = number of defaulted bonds after 1 year.
Li = loss for bond i.
L = L1 + … + Ln, i.e., the resulting total loss.
Note that 0 ≤ Li ≤ Ci, and assume that each default will result in a loss. Finally, let, for each 1 ≤ i ≤ n, Di be the event “Bond i has defaulted after 1 year”.
Answer the following questions:
a. Express N using D1,…Dn and indicator functions.
b. For each of the following relations, state, justifying your claim, whether they are true or false:
i. Di {Li > 0}
ii. σ (N) = σ (1D1,…1Dn)
iii. σ (1Dn) σ1D1,…1Dn1,L iv. σ (1Dn) σ1D1,…1Dn1,N
v. {L = 0} = {N = 0}
vi. σ (L) = σ (N)
c. Assume the so-called zero recovery rule: in case of default, the whole face value is lost, so that Li = Ci in case Di is true. Express L using Di and Ci for i = 1,…,n and indicator functions. Reconsider the relations in part (b) and state whether they are true or false.微积分作业