随机过程考试代考 ST302代写 马尔科夫链代写 统计考试代考

ST302 Stochastic Processes

随机过程考试代考 Each question has several subquestions, whose marks are shown in square brackets. Your answers should be justified by showing your work

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Instructions to candidates

This paper contains FIVE questions (for a total of 100 marks). AnswerALL FIVE.

Question 1: 20 marks

Question 2: 20 marks

Question 3: 28 marks

Question 4: 24 marks

Question 5: 8 marks

Each question has several subquestions, whose marks are shown in square brackets.

Your answers should be justified by showing your work and explaining your steps.

Make sure that your five-digit candidate number and course code are written or included at the top of every page of your answers.

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3 hours

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1.

Consider a discrete time Markov chain (X_n)_n≥0 with infinite state space S = Z. Let the chain be determined by the transition probabilities

p(i, i + 1) = p and p(i, i − 1) = q ∀_i ∈ Z,

where p+q = 1 with q < 1/2. Furthermore, let the starting point X₀ = i₀ < 0 be a negative integer and consider the first time that the Markov chain becomes positive, namely

τ₊ := inf{n ≥ 0 : X_n > 0}.

2. 随机过程考试代考

Consider a Markov chain with a finite state space S = {s₁, s₂, s₃, s₄}, modelling four different states that your insurance company can be in at the end of the fiscal year. We refer to s_i as state i. In state 1, things are not so good and your company has made a loss for the year of 1 million pounds. In state 2, your company breaks even, earning precisely 0 pounds. In state 3, things are amazing, earning you 10 million pounds for the year. In state 4, things are also quite good with earnings of 4 million pounds for the year. The transitions between the states are modelled by the transition matrix

If you find it helpful, you are welcome to use drawings to justify your answers. However, it is also fine to just refer to the entries of the transition matrix.

(a) [1 mark]

Suppose p(1, 4) = 0 and p(4, 1) = p(4, 4) = 0.4. What are then the values of p(1, 3) and p(4, 3)?

(b) [5 marks] In the setting of (a), identify the closed communication classes, the transient states, and the recurrent states. Briefly explain your reasoning in deducing these clasifications. Is the Markov chain irreducible? 随机过程考试代考

(c) [5 marks] Now suppose instead that p(1, 4) = 0.2 while p(4, 1) = p(4, 3) = 0. How do your answers to (b) change? Again, you should briefly explain how you arrive at the different clasifications.

(d) [5 marks] Finally, consider P with p(1, 4) = 0.2 and p(4, 1) = p(4, 4) = 0.4. Recalling that every finite Markov chain has a stationary distribution π, you may use that

π = (0.3, 0.55, 0.05, 0.1)

is a stationary distribution for this transition matrix P. Write down what it means that this π is a stationary distribution. Explain, by appealing to a theorem from lectures, why these probabilities can in fact be interpreted as the long-run probabilities of being in states 1, 2, 3, and 4, respectively (you need to brieflfly justify why the theorem can be applied). Explain whether or not the theorem applies in (b) and (c).

(e) [4 marks] Let P be as in (d). In the long run, how often do you expect to be making 10 million pounds for the year, and what are the average yearly earnings that you expect of your insurance company?

3. 随机过程考试代考

Suppose claims are arriving at your insurance company at a rate of λ > 0, meaning that the total number of claims N_t that have arrived by time t is a Poisson process with intensity λ. These claims are not handled immediately, but are instead collected in a queue. At random times T₁, T₂, . . . all the claims currently in the queue are dealt with and so the queue is reset to zero. The waiting times between resets of the queue, namely S_k := T_k −T_k−1, are assumed to be i.i.d. with S_k ∼ Exp(γ) for some γ > 0. We are interested in understanding the continuous time Markov chain (X_t)_t≥0, where X_t is the number of claims in the queue at time t (currently waiting to be dealt with) starting from X₀ = 0.

(a) [3 marks] Identify the Q matrix of (X_t)_t≥0, by specifying the entries Q(i, j) for all i, j in the state space. Briefly explain why the entries are the way they are.

(g) [3 marks] Derive the limits of p_t(0) and p_t(1) as t → ∞. Explain how this relates to (d), by referring to a theorem from lectures (very briefly explaining why the assumptions are satisfied). In the long run, what proportion of the time do we expect there to be precisely 1 claim waiting to be dealt with?

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