HW4

Consider a simple model of the hunting behaviour of a fox. In a single day, the fox catches either 0 or 1 rabbits, with probabilities 0.3 and 0.7, respectively. The outcome of hunting on day m is independent of all other days.We describe this system through a discrete-time, discrete-state stochastic process R(m), where R is the total number of rabbits caught between the end of day 0 (defining a starting point of our observation) and the end of day m. m = 0,1,2,3... is thus a discrete time parameter, and R(0) = 0 by definition (a) It can be shown that pr(m) – the probability of catching R = r rabbits in m days – is given by a binomial distribution (see Appendix A.2.1). Here, m is the number of “trials”, r the number of successes and q = 0.7 the probability of success. Hence give expressions for:i The mean number of rabbits caught after m days. ii The standard deviation of the total number rabbits caught after m days. iii The standard deviation of the mean number of rabbits caught per day…

This is a Stochastic Process question. Each customer makes claims following a Poisson process with rate 0.44 per year.For Scheme A, there are 3 discount levels: 0%, 10% and 20%.For a customer in Scheme A, he or she will move between discount levels throughout the year at continuous times.The annual premium to be paid at the start of each year is determined by the discount level he or she is in at that point of time, irrespective of the movements between the discount levels throughout the year.If a customer in Scheme A makes a claim, he or she will immediately move to a worse discount level, if possible, or remain in the 0% level.Independently of the customer making claims, for each customer in Scheme A independently the insurance company will perform a 'bump' where the customer will move to a better discount level if possible (or remain iin 20% otherwise) every so often as a customer loyalty programme with time between bumps exponentially distributed with mean 2 years. Find the…

Subject: Stochastic process Question 1 is attached to the image section below

Explain autoregressive conditional heteroskedasticity (ARCH) & generalized ARCH (GARCH) model?

Consider the following stochastic system. Let Xn be the price of a certain stock (rounded to the nearest cent) at the time that the stock market closes on the n-th day starting today.   Would it be appropriate to model this system as a Discrete-time Markov Chain?

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Define the following terms: Stationarity Heteroskedasticity Cointegration

why is the covariance of a deterministic and a stochastic process 0? This relats to Arithmetic Bronian Motion

Let {N_1(t)} and {N_2(t)} be two independent  Poisson processes with rates λ1=1 and λ2=2, respectively.  Find the probability that the second arrival in N_1(t) occurs before the third arrival in N_2(t). Round answer to 4 decimals.

Consider an economy with two states of nature and one time period. An option pays $3 in state 1 and $0 in state 2, and is being traded for the price of $1. A forward contract pays $3 in state 1 and -$2 in state 2. Using this information, compute the risk-free rate and the risk-neutral probabilities for each state of nature.

The manager of a market can hire either Mary or Alice. Mary, who gives you service at an exponential rate 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can hired at a rate of $C per hour. The manager estimates that, on the average, each customer’s time is worth $1 per hour and should be accounted for in the model. Assume customers arrive at a Poisson rate of 10 per hour. a)  What is the average cost per hour if Mary is hired? If Alice is hired? b)  Find C if the average cost per hour is the same for Mary and Alice.

Peter takes the course Basic Stochastic Processes this quarter on Tuesday, Thursday, and Friday. The classes start at 10:00 am. Peter is used to work until late in the night and consequently, he sometimes misses the class. His attendance behaviour is such that he attends class depending only on whether or not he went to the latest class. If he attended class one day, then he will go to class next time it meets with probability 1/2. If he did not go to one class, then he will go to the next class with probability 3/4. Describe the Markov chain that models Peter’s attendance. What is the probability that he will attend class on Thursday if he went to class on Friday?

Peter takes the course Basic Stochastic Processes this quarter on Tuesday, Thursday, and Friday. The classes start at 10:00 am. Peter is used to work until late in the night and consequently, he sometimes misses the class. His attendance behaviour is such that he attends class depending only on whether or not he went to the latest class. If he attended class one day, then he will go to class next time it meets with probability 1/2. If he did not go to one class, then he will go to the next class with probability 3/4. Suppose the course has 30 classes altogether. Give an estimate of the number of classes attended by Peter and explain it.

In a binomial model, give an example of a stochastic process that is both a martingale and Markov.

1. Define a Stochastic process and briefly discuss the meaning of measurability of a stochastic process. 2. Consider the ARMA(1,1) model yt = 0.8yt-1 + et + 0.5et-1 with et ~ WN(0, σe2). Derive the Wold representation of yt. 3. Consider the ARMA(2,1) process Φ(L)Xt = Θ(L)et with Φ(L) = 1 − 1.3L + 0.4L2 , Θ(L) = 1 + 0.4L and et ∼ WN(0, σe2). Obtain its Wold representation. 4.Consider the ARMA(2,2) process given by Xt =0.4Xt−1+0.45Xt−2+et+et−1+0.25et−2 with et ∼WN(0,σe2). 5. Consider the MA(1) process yt = et+1.5et−1 with et ∼ WN(0,σe2). Is the above MA(1) a Wold representation? Why or Why not? If not, obtain a suitable Wold representation.

Suppose Xn is an IID Gaussian process, withµX[n]=1, and σ2 X[n]=1Now, another stochastic process Yn = Xn − Xn−1. Please find:(a) The mean µY (n).(b) The variance σ2Y (n).(c) The auto-correlation RY (n, k)

Which of the following are reasons why a process would be considered stochastic? 1. we lack of data on the variable of interest/generating process 2. we lack quantitative understanding of the generating process 3. the generating process itself is random  4. the process can be modelled with certainty   5. the generating process changes over time