10%、20%、30%。保费折扣级别之间的转换规则如下：
(i) 如果保单持有人有一个免赔付年，她将提高一个折扣级别或保持在

(ii) 如果保单持有人在一年内有一次索赔，她保持在同一水平。
(iii) 如果她在上一年有两次或更多的索赔，她将向下移动一个折扣

(a) 求状态空间为 {0%, 10%, 20%, 30%} 的马尔可夫链的转移矩阵

[6 分]
(b) 说明该过程是否 (i) 不可简化和/或 (ii) 非周期性，并说明原因

[4 分]
(c) 确定马尔可夫链的平稳分布。
[8 分]

Pr(N = 1) = 0.3
(d) 评论该方案的平稳分布与
(c) 部分中发现的平稳分布。不计算分布。
[3 分]
[总分：21 分]

【总分：5分】

⌦ = {0, 1, 2,…}。
(a) 求这个泊松过程 N(t) 的生成矩阵 Q。
[3 分]
(b) 令 pij (t) 为在时间长度 t 内从状态 i 到状态 j 的转移概率。
p0 的 Kolmogorov 前向方程
00(t) 和 p0
0j (t) 对于所有 j = 0。
[9 分]
[总分：12 分]

Question 3:
Consider a No Claim Discount (NCD) scheme which o↵ers the following premium discounts: 0%,
10%, 20%, 30%. The transition rules between the premium discount levels are the following:
(i) if the policyholder had a claim-free year, she moves up by one discount level or stays at
(ii) if the policyholder has one claim in a year, she stays at the same level.
(iii) if she had two or more claims during the previous year, she moves down by one discount
level or stays at the minimum premium discount level.
The number of claims made in the previous year is denoted as N. Based on the calculations of
the insurance company, Pr(N = 0) = 0.6 and Pr(N = 1) = 0.2.
(a) Find the transition matrix of the Markov chain with state space {0%, 10%, 20%, 30%}
and draw the transition graph.
[6 marks]
(b) State if the process is (i) irreducible and/or (ii) aperiodic and give reasons for your
[4 marks]
(c) Determine the stationary distribution of the Markov chain.
[8 marks]
A new insurance company applies the same scheme with Pr(N = 0) = 0.6 but calculates
Pr(N = 1) = 0.3
(d) Comment on how the stationary distribution of this scheme would compare with the
stationary distribution found in part (c). Do not calculate the distribution.
[3 marks]
[Total: 21 marks]

Question 4:
An investor has bought shares worth £1, 000 and wants to estimate how much they will be
worth in a year’s time. Describe both a deterministic and stochastic model based on an ex-
pected growth rate of 3% over the year including one example for each model.
[Total: 5 marks]
Question 5:
Let N(t) denote the accumulated number of breakdowns occurred by a machine by time t.It
is also assumed that N(t) can be modelled as a Poisson process with rate 0.2 and state space
⌦ = {0, 1, 2,…}.
(a) Find the generator matrix Q for this Poisson process N(t).
[3 marks]
(b) Let pij (t) be the transition probability from state i to state j in time-length t.Findthe
Kolmogorov forward equations for p0
00(t) and p0
0j (t) for all j = 0.
[9 marks]
[Total: 12 marks] EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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