离散状态随机过程代写 Discrete State随机过程代写

Discrete State Stochastic Processes

Final Exam 526

离散状态随机过程代写 1.(2pts+3pts+6pts+3pts+4pts) This exercise is known in physics as the problem of Maxwell’s demon, which is a thought experiment that

1.(2pts+3pts+6pts+3pts+4pts)

This exercise is known in physics as the problem of Maxwell’s demon, which is a thought experiment that considers the possibility of a system that may violate the second law of thermodynamics. We imagine two urns in which there are two types of particles that are in constant motion. The first type of particle moves slowly, and the second moves quicky. In the right urn sits a demon who constructed a small door placed at the openning connecting the two urns. When the demon sees that a slow particle is about to escape the right urn, the demon shuts the door and prevents that particle from entering the left urn.

The illustration of our setup is in the figure. There are seven fast moving particles (in red) and six slow moving particles (in blue). In the image you can see the demon that just closed the door between the two urns blocking a slow particle from moving from the right to the left urn. In general, we will assume n particles total with q ∈ [0, 1] such that qn particles are fast and (1 − q)n are slow. Eventually, all of the slow particles will be trapped in the right urn, a low entropy state (a high degree of order) and thus the system would violate the second law of thermodynamics.

Other than the interventions by the demon the dynamics proceed as follows, at each time step, we pick a particle uniformly at random and move it to the other urn. Of course when the demon intervenes (when we try to move a slow particle from the right urn to the left one), the usual dynamics are violated. We let X_k denote the number of fast particles in the left urn at time step k. The next 5 exercises are in increasing level of difficulty.

(1) 离散状态随机过程代写

Design the sample space Ω, the state space S and explain why each Xk is a random variable. Why is X = {X_k}_k∈N a stochastic process?

(2) Show that why X = {X_k}_k∈N is a Markov chain on the state space S and calculate the (Markov) transition probabilities of X.

(3) Compute E(X_k) for X₀ = z for a fixed 0 ≤ z ≤ qn. Complute lim_k→∞ E(X_k) as k → ∞ and discuss the limiting expression for q ∈ (0, 1) and also the endpoints q = {0, 1}.

(4) Compute the stationary distribution of X = {X_k}_k∈N.

(5) Let L_k be the total number of particles at time k in the left urn. Construct the sample and state space (Ω and S) for (X_k , L_k). Justify that (X, L) = {(X_k , L_k)}_k∈N is a Markov chain and compute its transition probabilities. Compute the stationary distribution of (X, L).

In an election A receives 200 votes while B receivies only 100. Assume that the probability of getting a vote is identical (50%) for A and B. What is the probability that A is always ahead throughout the count?

3.(3pts+3pts) 离散状态随机过程代写

Consider a barbershop which have two barbers, each with his own barber chair. Suppose that there is room for at most 5 customers, with 2 in service and 3 waiting. Assume that potential customers arrive according to a Poisson process at rate 6 per hour. Customers arrive when the shop is full will leave without receiving service. Assume that the duration of each haircut is an independent exponential random variable with a mean of 15 minutes. Customers are served in a first-come first-served manner by the first available barber.

(a) Let X_t be the number of customers at time t. Write the generator matrix Q and find a stationary distribution. You can provide formulas for the values of π_i , it is not necessary to compute the numerical values.

(b) What is the long-run proportion of time there are two customers in service plus two customers waiting? What is the long-run proportion of time that barber A is busy (assume one named A and the other B)?

5.(4pts+4pts)

Let K, Q, and N be independent Poisson processes with rates µ, θ > λ > 0, respectively.

(1) Prove for every k ∈ N and t ≥ 0 we have P(N_t ≥ k) = E(1_{Qt≥k}L_t) where L_t = (λ/θ)^Qt e^{−(λ−θ)t}.

(2) Prove that P(N_t ≥ K_t) ≤ e^{−(√µ−√λ)2t} for all t ≥ 0.

6. (3pts+3pts+3pts+3pts) 离散状态随机过程代写

Consider the Bachalier model S_t = σW_t where W is a Brownian motion. Let X_T = (S_T − K)⁺ = max{S_T − K, 0} be the payoff of a European call option with strike K > 0. The price of the option is

8.(6pts) 离散状态随机过程代写

Consider a process X_t satisfying the following properties:

(1) X_t has independent increments.

(2) For 0 ≤ s ≤ t

X_t − X_s ∼ N(µ(t − s), σ²(t − s)).

(3) Sample paths of Xt are continuous functions of t.

(4) X₀ = 0.

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