Essentials Of Stochastic Processes 3rd Edition Richard Durrett - Solutions

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. Prove that the limiting joint distribution of the residual lifetime Y(t) and spent lifetime Z(t) of a random variable with DF F (x) and finite mean u is given by.00 Pr[Y(t) > y, Z(t) > z) == |[1 - F(u)]du, y> 0, z >0.HJy+z
. Let Y be the stationary forward-recurrence time of a random variable X.Suppose that mn = E(X") exists for n = 1, 2, .... Then show that Mp+1 E{Y"} =(n+1)m1
. Let { N(t), t ≥ 0] be a renewal process induced by a RV X. Show that for large t tE{N()}=and E(X)var(X)var{N(t)}~[E(X)]3
. Assume that the lifetime X of a device is random having DF F(x). Show that the expected remaining life of a device aged y (which has already attained age y) is given by 59[1 - F(1)]dt E(X-y | X>y)=1 - F (y)In particular, for the exponential lifetime, this is equal to E(X), indepen-dent of y. (See
. The three-state process. Suppose that an automatic machine can be in three states: working (state 0), failed in mode 1 (state 1), or failed in mode 2 (state 2). Suppose that a failed machine in either mode cannot go to another failed mode. (That is, transitions from state 1 to 2 and from state 2
. Suppose that a queueing system has m service channels. The demand for service arises in accordance with a Poisson process with ratea, and the service time distribution has exponential distribution with parameter b.Suppose that the service system has no storage facility-that is, a demand that
. Consider a Poisson process with parameter X. Given that n events happen by time t, show that the PDF of the time of occurrence Th of the kth event(k < n) is given by n!f (x) =-KI F (1 -* ) **=(k-1)!(n - k)! tk 0 < x < t,=0, x≥ t.Show that E(Tk) = kt/(n+ 1).
. Consider that two independent series of events A and B occur in accor-dance with Poisson processes with parameters a andb, respectively. Show that the number N of occurrences of the event A between two successive occurrences of the event B has geometric distribution with mass function aPr{ N = n]
. A piece of equipment is subject to random shocks that occur in accor-dance with a Poisson process with rate A. The equipment fails due to the cumulative effect of k shocks. Show that the duration of the lifetime T of the equipment has gamma distribution with PDF fix(x). Note that T is the
. If X is an exponential RV, then show that E{X | X> y) =y+ E{X} for all y>0;that is, E(X- y | X > y) = E(X), independent of y.
. Suppose that X and Y are independent exponential RVs with parameters a andb, respectively. Show that(a) W= X + Y has PDF able-ax - e -**f(x)=x > 0, a /b b - a=0, x≤0.(b) Z = min( X, Y) is exponential with parameter a + b.(c) M = max(X, Y) has PDF f (x) = ae ax + be-bx _ (a + b)e-(a+b)x, x >
. If X ;, i = 1, ..., nare IID exponential RVs with parametera, then show that S" = X1 + ... + X ,, has gamma distribution with PDF a"x"-le ª*fa,n(x) =x> 0Г (х)=0, x≤0.Find E(S,} and var {S,}.
. Suppose that messages arrive at a telephone switch board, the interarrival time of messages being exponential with mean 10 minutes. Find the prob-ability that the number of messages received during the five afternoon hours (1-6 P.M.) is (i) exactly 24, (ii) more than 24, and (iii) nil.
. Suppose that customers arrive at a service counter in accordance with a Poisson process at a rate of three per minute. Find the probability that the interval between two successive arrivals is (i) between one and three minutes and (ii) less than one-third of a minute.
. Suppose that customers arrive at a certain service facility center in ac-cordance with a Poisson process { N(t), t ≥ 0} having parameter À. A customer can make a preliminary inquiry as to whether he or she actu-ally needs the facility. Suppose that the proportion of customers actually needing
. Suppose that customers arrive at a bank in accordance with a Poisson process at the rate of two per minute. Find the probability that the number of customers that arrive during a 10-minute period is (i) ex-actly 20, (ii) greater than 20, and (iii) between 10 and 20.
. Consider the Markov chain of Example 1.3.(a) Use (1.2.21) to find Ż, jvj.(b) For pn = q" p, n = 0, 1, 2, ..., p+ q = 1, examine the nonnull per-sistence of the chain. Find V(s) in this case. Consider the particular case q = p/(1 + p), p < 1.
. Consider the Markov chain with S = {0, 1, ..., m), such that Po,0 = q, Poi = p, Pii-1= 9, Pii+1 = p, i=1,2 ,..., m-1 Pm.m-1 =9, Pm,m = p, 0 < p, q < 1, where p + q = 1. (Each of the transitions between other pairs of states has probability 0.) Show that the chain is irreducible and aperiodic.
. In what is called a gambler's ruin problem, consider a gambler who with capital a agrees to play a series of games with an adversary having a capital b(a + b =c, the total capital). The probability of the gambler winning one game (and with it, one unit of money) is p, and that of losing one unit
. Consider a service facility having a limited waiting space for m customers, including the one being serviced. The server serves one customer, if any, at epochs 0, 1, 2, .. .. Assume that the number of arrivals in the intervals(k, k + 1) is given by an IID random variable A with Pr(A = n) =Pn> >
. Show that a transient state cannot be reached from a persistent state.
. Show that if state j is transient, then 8 W 00 P!") < oo for all i E S.
. Show that for a Markov chain with a finite state space S, the probability of staying forever among the transient states is zero.
. Consider a Markov chain with $ = {1,2, 3, 4] and TPM 12 00 31 0 00 1 0 , 0 0 0 22 Is the chain irreducible? Verify that states 1 and 2 are recurrent. Find u1 and p42. (u1 = 5/3, 42 = 5/2).
. Find the invariant measure of a chain with S = (0, 1,2 ,..., m - 1} and a doubly stochastic transition probability matrix.
. A chain with S = {1, 2, 3, } has TPM 1 0 oo A P=0 1 0 Pi > 0, C Pi = 1.Pı P2 P3 Examine the nature of the states. Find p".
. The transition probability matrix of a Markov chain with three states 0, 1, 2, is given by 0.4 0.5 0.1 0.2 0.6 0.2 0.3 0.3 0.4 and the initial distribution is (0.6, 0.3,0.1). Find (i) Pr(X2=3), and (ii)Pr{ X3 = 1, X2 = 0, X1 = 2, Xo = 0). Find the invariant measure of the chain.
On December 20, 2011, stock in Exxon Mobil was selling at 81.63. (a) Use the Black–Scholes formula to compute the value of an April 12 call (t D 0:3123 years) with strike 70, assuming an interest rate of r D 0:01 and the volatility D 0:26. The volatility here has been chosen to make the price
On December 20, 2011, stock in Kraft Foods was selling at 36.83. (a) Use the Black–Scholes formula to compute the value of a March 12 call (t D 0:227 years)with strike 33, assuming an interest rate of r D 0:01 and the volatility D 0:15.The volatility here has been chosen to make the price
On May 22, 1998, Intel was selling at 74.625. Use the Black–Scholes formula to compute the value of a January 2000 call (t D 1:646 years) with strike 100, assuming the interest rate was r D 0:05 and the volatility D 0:375.
Find a formula for the value (at time 0) of cash-or-nothing option that pays off $1 if St > K and 0 otherwise. What is the value when the strike is the initial value, the option is for 1=4 year, the volatility is D 0:3, and for simplicity we suppose that the interest rate is 0.
Show that for any a andb, V.s; t/ D as C bert satisfies the Black–Scholes differential equation. What investment does this correspond to?
The payoff of the Asian option is based on the average price: An D .S0 C CSn/=.n C 1/. Suppose that the stock follows the binomial model with S0 D 4, u D 2, d D 1=2, and r D 1=4. Find the value of the American version of the three period Asian option, .Sn 4/C, i.e., when you can exercise the
In the American version of the callback option, you can buy the stock at time n at its current price and then sell it at the highest price seen in the past for a profit of Vn D max0mn Sm Sn. Compute the value of the three period version of this option when the stock follows the binomial model
Consider the three-period binomial model with S0 D 16, u D 3, d D 1=2 and r D 1=3. An American limited liability call option pays minf.Sn 10/C; 60g if exercised at time 0 n 3. In words it is a call option but your profit is limited to$60. Find the value and the optimal exercise strategy.
Continuing with the model of previous problem S0 D 27, u D 4=3, d D 2=3, and r D 1=9, we are now interested in finding value VS of the American straddle jS3 30j. Comparing with the values VP and VC of the call and the put computed in the previous problem we see that VS VP C VC. Explain why this
Consider the binomial model of Problem 6.15 S0 D 27, u D 4=3, d D 2=3, and r D 1=9 but now (a) find value and the optimal exercise strategy for the American put option .30 S3/C, and (b) find the value of the American call option.S3 30/C.
Consider the binomial model with S0 D 27, u D 4=3, d D 2=3, and r D 1=9. (a) Find the risk neutral probability p. (b) Find value Vn.a/ of the put option .30 S3/C and the hedging strategy n.a/. (c) Check your answer to (b) by computing V0 D E.V3=.1 C r/3/.
Consider the binomial model with S0 D 64, u D 2, d D 1=2, and r D 1=4.(a) Find the value Vn.a/ of the call option .S3 125/C and the hedging strategyn.a/. (b) Check your answer to (a) by computing V0 D E.V3=.1 C r/3/. (c) Find the value at time 0 of the put option.
Consider now a four period binomial model with S0 D 32, u D 2, d D 1=2, and r D 1=4, and suppose we have a put .50 S4/C with a knockout barrier at 100. Show that the knockout option as the same value as an option that pays off.50 S4/C when S4 D 2, 8, or 32, 0 when S4 D 128, and 18 when S4 D
Assume the binomial model with S0 D 54, u D 3=2, d D 2=3, and r D 1=6.and consider a put .50 S3/C with a knockout barrier at 70. Find the value of the option.
Suppose S0 D 27, u D 4=3, d D 2=3 and interest rate r D 1=9. The European“cash-or-nothing option” pays $1 if S3 > 27 and 0 otherwise. Find the value of the option Vn and for the hedge n.
Consider the three-period binomial model with u D 3, d D 1=2 and r D 1=3 and S0 D 16. The European prime factor option pays off $1 for each factor in the prime factorization of the stock price at time 3 (when the option expires). For example, if the stock price is 24 D 2331, then the payoff is 4 D
In the putback option at time 3 you can buy the stock for the lowest price seen in the past and sell it at its current price for a profit of V3 D S3 min 0m3 Sm Suppose that the stock follows the binomial model with S0 D 4, u D 2, d D 1=2, and r D 1=4. (a) Compute the value function Vn.a/ and
The payoff of the Asian option is based on the average price: An D .S0C C Sn/=.nC1/. Suppose that the stock follows the binomial model with S0 D 4, u D 2, d D 1=2, and r D 1=4. (a) Compute the value function Vn.a/ and the replicating portfolio n.a/ for the three period call option with strike
Consider the two-period binomial model with u D 3=2, d D 2=3, interest rate r D 1=6 and suppose S0 D 45. What is the value of the European call option with strike price 50, i.e., the option with payoff .50 S2/C. Find the stock holdings 0,1.H/, and 1.T/ need to replicate the option exactly.
Consider the two-period binomial model with u D 2, d D 1=2 and interest rate r D 1=4 and suppose S0 D 100. What is the value of the European call option with strike price 80, i.e., the option with payoff .S2 80/C. Find the stock holdings 0,1.H/, and 1.T/ need to replicate the option exactly.
Suppose Microsoft stock sells for 100 while Netscape sells for 50. Three possible outcomes of a court case will have the following impact on the two stocks:Microsoft Netscape 1 (win) 120 30 2 (draw) 110 55 3 (lose) 84 60 What should we be willing to pay for an option to buy Netscape for 50 after
The Cornell hockey team is playing a game against Harvard that it will either win, lose, or draw. A gambler offers you the following three payoffs, each for a $1 bet:win lose draw Bet 1 0 1 1.5 Bet 2 2 2 0 Bet 3 .5 1.5 0(a) Assume you are able to buy any amounts (even negative) of these bets. Is
It was crucial for our no arbitrage computations that there were only two possible values of the stock. Suppose that a stock is now at 100, but in one month may be at 130, 110, or 80 in outcomes that we call 1, 2, and 3. (a) Find all the (nonnegative) probabilities p1, p2, and p3 D 1 p1 p2 that
A stock is now at $60. In a year its price will either be $75 or $45. (a) Assuming that the interest rate is r D 0:05 find the price of a put .60 S1/C. (b) How much stock 0 do we need to sell to replicate the option. (c) Verify that having V0 in cash and 0 in stock replicates the option
A stock is now at $110. In a year its price will either be $121 or $99. (a)Assuming that the interest rate is r D 0:04 find the price of a call .S1 113/C. (b)How much stock 0 do we need to buy to replicate the option. (c) Verify that having V0 in cash and 0 in stock replicates the option
Let Zn be a branching process with offspring distribution pk with p0 > 0 and D Pk kpk > 1. Let ./ D P 1kD0 pkk. (a) Show that E.ZnC1 jZn/ D ./Zn .(b) Let be the solution < 1 of ./ D and conclude that Pk.T0 < 1/ D
Generating function of the time of gambler’s ruin. Continue with the set-up of the previous problem. (a) Use the exponential martingale and our stopping theorem to conclude that if 0, then ex D Ex../V0 /. (b) Let 0 / D 1=s, then use (a) to concludeWhy must the answer in (b) be of the form f
Variance of the time of gambler’s ruin. Let X1; X2; : : : be independent with P.Xi D 1/ D p and P.Xi D 1/ D q D 1 p where p 1 C Cn and let V0 D minfn 0 W Sn D 0g. (5.13) implies ExV0 D x=.1 2p/. The aim of this problem is to compute the variance of V0. If we let Yi D X i .p q/and note that
Wald’s second equation. Let Sn D X1C CXn where the Xi are independent with EXi D 0 and var .Xi/ D 2. Use the martingale from the previous problem to show that if T is a stopping time with ET < 1, then ES2T D 2ET.
Let Sn D X1 C C Xn where the Xi are independent with EXi D 0 and var .Xi/ D 2. By example 5.3 S2 n n 2 is a martingale. Let T D minfn W jSnj > ag.Use Theorem 5.10 to show that ET a2= 2. For simple random walk 2 D 1 and we have equality.
Consider a favorable game in which the payoffs are 1, 1, or 2 with probability 1/3 each. Use the results of Example 5.17 to compute the probability we ever go broke (i.e., our winnings Wn reach $0) when we start with $i.
An unfair fair game. Define random variables recursively by Y0 D 1 and for n 1, Yn is chosen uniformly on .0; Yn1/. If we let U1;U2; : : : be uniform on.0; 1/, then we can write this sequence as Yn D UnUn1 U0. (a) Use Example 5.4 to conclude that Mn D 2nYn is a martingale. (b) Use the fact
Let Xn be the Wright–Fisher model with no mutation defined in Example 1.9.(a) Show that Xn is a martingale and use Theorem 5.11 to conclude that Px.VN (b) Show that Yn D Xn.N Xn/=.1 1=N is a martingale.(c) Use this to conclude that (N-1) x(N-x)(11/N)" Px(0
Lognormal stock prices. Consider the special case of Example 5.4 in which Xi D ei where i D normal. ; 2/. For what values of and is Mn D M0 X1 Xn a martingale?
Brother–sister mating. Consider the six state chain defined in Exercise 1.66.Show that the total number of A’s is a martingale and use this to compute the probability of getting absorbed into the 2,2 (i.e., all A’s state) starting from each initial state.
Four children are playing two video games. The first game, which takes an average of four minutes to play, is not very exciting, so when a child completes a turn on it they always stand in line to play the other one. The second one, which takes an average of eight minutes, is more interesting so
Three vendors have vegetable stands in a row. Customers arrive at the stands 1, 2, and 3 at rates 10, 8, and 6. A customer visiting stand 1 buys something and leaves with probability 1/2 or visits stand 2 with probability 1/2. A customer visiting stand 3 buys something and leaves with probability
At a local grocery store there are queues for service at the fish counter (1), meat counter (2), and café (3). For i D 1; 2; 3 customers arrive from outside the system to station i at rate i, and receive service at rate 4 C i. A customer leaving station i goes to j with probabilities p.i; j/ given
At registration at a very small college, students arrive at the English table at rate 10 and at the Math table at rate 5. A student who completes service at the English table goes to the Math table with probability 1/4 and to the cashier with probability 3/4. A student who completes service at the
Queues in series. Consider a k station queueing network in which arrivals to server i occur at rate i and service at station i occurs at rate i. In this problem we examine the special case of the feed-forward system in which p.i; i C 1/ D pi for 1 i < k. In words the customer goes to the next
Feed-forward queues. Consider a k station queueing network in which arrivals to server i occur at rate i and service at station i occurs at rate i. We say that the queueing network is feed-forward if the probability of going from i to j < i has p.i; j/ D 0. Consider a general three station
Consider a three station queueing network in which arrivals to servers i D 1; 2; 3 occur at rates 3; 2; 1, while service at stations i D 1; 2; 3 occurs at rates 4; 5; 6. Suppose that the probability of going to j when exiting i, p.i; j/ is given by p.1; 2/ D 1=3, p.1; 3/ D 1=3, p.2; 3/ D 2=3, and
Consider a production system consisting of a machine center followed by an inspection station. Arrivals from outside the system occur only at the machine center and follow a Poisson process with rate . The machine center and inspection station are each single server operations with rates 1 and 2.
A professor receives requests for letters of recommendation according to a Poisson process with rate 8 per month. When k are on his desk he feels stressed and writes them at rate 2k (per month). (a) Find the stationary distribution of the number of letters to be written. (b)What is the average time
At present the Economics department and the Sociology department each has one typist who can type 25 letters a day. Economics requires an average of 20 letters per day, while Sociology requires only average of 15. Assuming Poisson arrival and exponentially distributed typing times find (a) the
Customers arrive at a two-server station according to a Poisson process with rate 4. Upon arriving they join a single queue to wait for the next available server.Suppose that the service times of the two servers are exponential with rates 2 and 3 and that a customer who arrives to find the system
Customers arrive at the Shortstop convenience store at a rate of 20 per hour.When two or fewer customers are present in the checkout line, a single clerk works and the service time is three minutes. However, when there are three or more customers are present, an assistant comes over to bag up the
Queue with impatient customers. Customers arrive at a single server at rate and require an exponential amount of service with rate . Customers waiting in line are impatient and if they are not in service they will leave at rate ı independent of their position in the queue. (a) Show that for any
Consider a taxi station at an airport where taxis and (groups of) customers arrive at times of Poisson processes with rates 2 and 3 per minute. Suppose that a taxi will wait no matter how many other taxis are present. However, if an arriving person does not find a taxi waiting he leaves to find
Two teaching assistants work in a homework help room. They can each serve up to two students, but get less efficient when there are helping two at once. Students arrive at rate 3 but will not go in the room if there are already four students there.To cut a long story short the number of students in
Carrboro High School is having a car wash to raise money for a Spring Break trip to Amsterdam. Eight students are working at the car wash. Jill and Kelly are on the sidewalk holding signs advertising the car wash. If there are one, two, or three cars getting washed, then the other six students
There are two tennis courts. Pairs of players arrive at rate 3 per hour and play for an exponentially distributed amount of time with mean one hour. If there are already two pairs of players waiting new arrivals will leave. (a) Find the stationary distribution for the number of courts occupied. (b)
Two people who prepare tax forms are working in a store at a local mall.Each has a chair next to his desk where customers can sit and be served. In addition there is one chair where customers can sit and wait. Customers arrive at rate 4 but will go away if there is already someone sitting in the
Two queues in series. Consider a two-station queueing network in which arrivals only occur at the first server and do so at rate 2. If a customer finds server 1 free he enters the system; otherwise he goes away. When a customer is done at the first server he moves on to the second server if it is
Consider an M=M=4 queue with no waiting room, for example, four prostitutes hanging out on a street corner. Customers arrive at times of a Poisson process with rate 4. Each service takes an exponentially distributed amount of time with mean 1/2. If no server is free, then the customer goes away
Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 minutes to clean the kitchen. Kelly takes an exponentially distributed amount of time with mean 40 minutes to clean the bath room. The
Al, Betty, Charlie, and Diane are working at the math club table during freshmen move-in week. Their attention spans for doing the job are independent and have exponential distributions with means 1/2, 2/3, 1, and 2 hours. (Rates are 2, 3/2, 1, 1/2.) (a) What is the expected amount of time until
A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. Formulate a Markov chain with states 0 D all parts working, 1, 2, 3 D one part failed, and 4 D two failures.
Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confusion, and Depression according to the following transition rates when t is the time in months:A B C D A 4 3 1 0 B 4 6 2 0 C 2 3 6 1 D 0 0 2 2(a) Find the long run fraction of time he spends in these
Consider the salesman from Problem 4.1. She just left Atlanta. (a)What is the expected time until she returns to Atlanta? (b) Find the answer to (a) by computing the stationary distribution.
A small company maintains a fleet of four cars to be driven by its workers on business trips. Requests to use cars are a Poisson process with rate 1.5 per day.A car is used for an exponentially distributed time with mean 2 days. Forgetting about weekends, we arrive at the following Markov chain for
A taxi company has three cabs. Calls come in to the dispatcher at times of a Poisson process with rate 2 per hour. Suppose that each requires an exponential amount of time with mean 20 minutes, and that callers will hang up if they hear there are no cabs available. (a) What is the probability all
A computer lab has three laser printers and five toner cartridges. Each machine requires one toner cartridges which lasts for an exponentially distributed amount of time with mean 6 days. When a toner cartridge is empty it is sent to a repairman who takes an exponential amount of time with mean 1
A computer lab has three laser printers that are hooked to the network.A working printer will function for an exponential amount of time with mean 20 days. Upon failure it is immediately sent to the repair facility. There machines are worked on by two repairman who can each repair one printer in an
A computer lab has three laser printers, two that are hooked to the network and one that is used as a spare. A working printer will function for an exponential amount of time with mean 20 days. Upon failure it is immediately sent to the repair facility and replaced by another machine if there is
Solve the previous problem for a two-pump self-serve station under the assumption that customers will go to another station if there are at least four cars in the station, i.e., two being served and two waiting.
Customers arrive at a full-service one-pump gas station at rate of 20 cars per hour. However, customers will go to another station if there are at least two cars in the station, i.e., one being served and one waiting. Suppose that the service time for customers is exponential with mean six minutes.
Two people are working in a small office selling shares in a mutual fund. Each is either on the phone or not. Suppose that calls come in to the two brokers at rate 1 D 2 D 1 per hour, while the calls are serviced at rate 1 D 2 D 3.(a) Formulate a Markov chain model for this system with state
Consider the set-up of the previous problem but now suppose machine 1 is much more important than 2, so the repairman will always service 1 if it is broken.(a) Formulate aMarkov chain model for the this system with state space f0; 1; 2; 12g where the numbers indicate the machines that are broken at
Consider two machines that are maintained by a single repairman. Machine i functions for an exponentially distributed amount of time with rate i before it fails.The repair times for each unit are exponential with rate i. They are repaired in the order in which they fail. (a) Formulate a Markov
A small computer store has room to display up to three computers for sale.Customers come at times of a Poisson process with rate 2 per week to buy a computer and will buy one if at least 1 is available. When the store has only one computer left it places an order for two more computers. The order
A machine is subject to failures of types i D 1; 2; 3 at rates 1 D 1=24,2 D 1=30, 3 D 1=84. A failure of type i takes an exponential amount of time with rate 1 D 1=3, 2 D 1=5, and 3 D 1=7. Formulate a Markov chain model with state space f0; 1; 2; 3g and find its stationary distribution.
A hemoglobin molecule can carry one oxygen or one carbon monoxide molecule. Suppose that the two types of gases arrive at rates 1 and 2 and attach for an exponential amount of time with rates 3 and 4, respectively. Formulate a Markov chain model with state space fC; 0; g whereCdenotes an attached

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Essentials Of Stochastic Processes 3rd Edition Richard Durrett - Solutions

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