Questions and Answers of Applications Of The Time Value Of Money

. Using a computer, solve the forward Kolmogorov equation for the Wiener process with drift using your own choice of parameters and a numerical method of solution such as that of Crank-Nicolson. Use
. Show that infinite boundary points are natural and finite boundary points are regular for the OUP.
. Let X be an OUP satisfying dX =( p - ocX) d t a dW. Find Cov[X(s), X(/)].
. Let an OUP satisfy the SDE dX =(p — X)dt adW, with X(0) = x.Show that the transformed process Y = e‘X satisfies dY = e\pdt ad W).Hence 284 Diffusion processes Y(t)= F(0) + /i dt' -f a Jo
. Determine the trajectories followed by an OUP when a = 0. Do these coincide with ElX{t)\X(0) = xj!
. Show that the spatial eigenfunctions of the Kolmogorov equation on finite spatial intervals are orthogonal with weight factor p*.
. Compare the properties of the birth and death process of sections 9.7 and 9.8 with those of the random environment growth process of section 12.7.
. Perform a simulation for the OUP with your own choice of parameter values and determine the same properties as in Exercise 22. In addition, compare the density approached by your simulated process
. Using the method outlined in section 12.7, perform a simulation of the process defined by the stochastic differential equation dX = X[/id/-h adW^ using both I to and Stratonovich interpretations.
. Show that in the stock price model of section 12.7, the mean price is Xof"'.
. Show that in the population model. Equation (12.35), the probability of extinction is as given in the text.
. Show that the Stratonovich SDE dX = f{X)\_pdi + a dW~\ may be transformed to that of a Wiener process by setting Y =1 f{X')dX\so that the transition density of X may be obtained as P(, x,
. Verify the expressions in section 12.7 for the mean and variance of AV.
. Use characteristic functions to prove the convergence of the random walk model of section 12.7 to a Wiener process.
. Show that in the limit the infinitesimal moments of order greater than two vanish in the sequence of processes used to model the activity of a nerve cell, dX^ = c[dN ^ — dN2}.
. The deterministic logistic growth equation dx/dt = x{K — x), where K > 0 is the carrying capacity, can be converted to a model for population growth in a random environment by replacing K with
. Convert the following Stratonovich SDEs to Ito equations: (a) dX =Xál + ^ á W \ (b) dX = X^dt +e~'dW; (c) dX = X^dt+e -' XdW.
. Let X be a standard Wiener process. Use Ito's formula to obtain Ito SDEs satisfied by T = (a) X + (b) X^ + and (c) In X.
. Show that the expected value of the Ito integral of a simple random process is zero.
. Write down a stochastic differential equation (SDE) for an OUP with infinitesimal drift and diffusion given by a(x, t) = — ax + b and t) =respectively. What would be the difference between an I
. Use the formulae of section 12.2 to verify that the stationary density for an unrestricted OUP is as given in section 12.4.
. Repeat Exercise 4 for the OUP with reflecting barriers at 0 and a.
. Use the theory of section 12.2 to verify that the invariant distribution in the case of a Wiener process confined between reflecting barriers is as given in section 12.3. Sketch the density for the
. Verify Equation (12.19) and use it to investigate the behaviour of p^(t;a, x)as / ^ 0 0 for various values of p.
. Show that the spatial eigenfunctions for the Wiener process between reflecting barriers satisfy the orthogonality relation (12.17) and determine the constants A„. (Hint: A handbook of integrals
. Show by integration with the transition density (12.15) of a Wiener process with drift, that the mean and variance are x -h jut and a^t, respectively.
. Outline how you would apply the method of separation of variables to find the transition density for a Wiener process with drift on (0,a) with absorbing barriers at 0 and a.
. For the unrestricted Wiener process, sketch what happens to the transition density as / oo for the cases g0.
. Verify that for a time-homogeneous diffusion process the Kolmogorov equations become Equations (12.6) and (12.7).
. Show that the second infinitesimal moment, defined in Equation ( 12.2), is also given by Var[AA|A(i) = x]Pix,[)= lim A t -> 0 At
. For 5 > 0, / > 0 find the correlation coefficient p(sj) (see section 1.3) of VF(s)and W(t). Assume s < t and s is fixed. What happens to p as / ^ oo?
. Let M = dN/dt, where N is as in Exercise 14. What would a sample path of M look like? Use the results of Exercise 7 to ascertain the mean and covariance functions of M.
. Let N = {N(t)] be a Poisson process with parameter 2. Find the mean and covariance functions of N.
. Use Theorem 6.5 to find the characteristic function of X(5) = Xq -\-jLLs + gW(s).
. Prove that the transition probability density (11.12) of a Wiener process with drift satisfies the heal equation (11.13).
. What is the probability that - sjt < W(t) < J t l
. Find functions f^it) and fjU) = - fi(t) such that a standard Wiener process is between / j and f 2 with probability (a) 0.5, (b) 0.95.
. Compute the probability that the Wiener process with drift X = 2W-\-t satisfies the inequality 0 ^ A(/)^ 1.
. Establish the following results for a Wiener process with drift /it and variance parameter cr:£[A(/)] = Xo~y jit, Var[A(/)] = (j^/, Cov[A(s), X(t)'] = min(5, t).
. The derivative {X'(t)} of a random process {X(t) \ can be defined in the usual way by Exercises 235 X'(t)= lim A t ^ 0 X(t + At)~ Xjt)At Use the results£[A'(/)] = -E [X (/)], dt Cov[X'(s), X'(/)]
. Show that for a covariance (weakly) stationary process, the mean value function is constant and the covariance function satisfies R(z) = R( - t).
. Prove that if a and b are constants, and X and L are random variables, then(a) Cov[A+a, y+6] =Cov[A, L];(b) thus Cov[A, Y~\ = Cov[A — E[X], Y — E[L]], so means can always be subtracted when
. Show that if 4, B, and C are three random variables, CowlA,(B + Q ] = Cov[^, B] + Cov[/l, C].
. Let AW= W{i + At) - W(t) be an increment in a standard Wiener process.Examine the limits as A/ ^ 0 of (i) E[A fL], (ii) Var[AfL], (iii) E[AfL/A/], and (iv) Var[AH//A/] to provide an indication that
. Show that a continuous time process with independent increments is a Markov process. (Hint: It will suffice to proceed as in Exercise 7.2;examine Pr(X{i^) = z\X(i2) = y, X(i^) = x), where
. Prove that the solution of the logistic differential equation (11.3) is in fact given by (11.4). [Hint: Separate the variables by putting the equation in the form f(y)dy = g(i)dt, and integrate
. Assume, very roughly speaking, that a human population is a branching process. What is the probability of extinction if the proportion of families having 0, 1 or 2 children are 0.2, 0.4 and 0.4
. A branching process has initially one individual. Use the law of total probability in the form Pr (extinction) = ^ Pr (extinction |/c descendants) Pr{k descendants)k to deduce that the extinction
. Use Fig. 10.4b to show graphicalJy that when p> \ .
. For the binary fission branching process, solve the equation P{x) = x to show that the probability of extinction is 218 Branching processes 10.P e x t =- y i -4/7(1 -p)
. Viewing a branching process as a Markov chain, show that the transition probabilities for the binary fission case are To, k odd Pjk= I = n=J} = \ i j \ k/2^^ _ y-k/2^\k/2) ’ k even, where is
. A branching process is referred to as binary fission if an individual leaves either zero or two descendants. That is Pz = P^ Po ^ ^ - p, 0 < p < 1. If Xq = 1 with probability one, find the
. Consider a branching process in which Xq = 1 with probability one. Each individual leaves behind either zero descendants or one descendant with probabilities Pq and pi respectively. Show that the
. Let {Xj^, k = 1,2,...} be i.i.d. with E(X = jj. and Var (X J and lei N be a Poisson random variable with parameter /t, independent of the Xj^.Prove, using generating functions, that= X 1 + X 2 +
. Let X be a non-negative integer-valued random variable with probability generating function f(s) = Xo Prove that E(X) = f'(\)Var(X) = /" ( !)+ / '( ! ) - /'^ ( l) .
. Deduce from the recursion relation ( 10.1 ) that the variance of the population in the Gal ton-Watson process at generation n is Var(X„)^ 1/7(7^ f i ^ \ .
. Let the moment generating function of N(t) in the birth and death process be {¡/(O, t) = = (p(e^, i). From the given expression for
. If M(i) is the second moment, £[A/^(/)| A/(0) = Hq] in the simple birth and death process, prove using the differential-difference equations (9.17), that M satisfies (9.25).
. Show that when A = fi in the simple birth and death process, the expectation of the extinction time is infinite.
. Verify, by direct substitution, that the function (p(sj) given in (9.21)satisfies the partial differential equation of Exercise 16 with initial data(j)(s, 0) =
. Show that the probability generating function (p(sj) for Pr{N(t) =n\N(0) = Hq} in the simple birth and death process satisfies= (Xs-jii)(s- 1 ) ^ .di ds
. Let T be the extinction time in a pure death process with Hq individuals initially. What is the density of 7?
. A herd of 50 ungulates is released on to a large island. The birth and death probabilities are /I = 0.15 and ¡j. = .05 per animal per year. A hunter wishes to visit the island when he can be 95%
. Prove that a negative binomial random variable X^, with probability law given by (9.15), has mean and variance E{X,) = -, Var(X,) = '^.P P (Hint: is the sum of r i.i.d. random variables.)
. In a simple birth process in which À = nQ=\, show that py has a maximum at / = In ( 1 + k).
. For the Yule process, prove that + ^ = 0,1,2,... is given by (9.12).
. In the Yule process, 1 (/) is the probability that the population has increased by one at time t. This quantity satisfies (9.10); i.e., P«o + 1 + + 1 )Pno + I = ^noP„„, P„„ M (0) = 0-Show
. Let X be a continuous time Markov chain with stationary transition probabilities Pr\X(i) = Si,\X{0) = Sj}=p{k,t\j).Give an interpretation of the Chapman-Kolmogorov relation p(/c,i| + i 2 |f) =
. For a continuous time Markov chain the transition probabilities are(equation (9.3)), Show that a Poisson process is a continuous time Markov chain but that the only transition probabilities needed
. Let N, and N2 be two independent simple Poisson processes with rate parameters 2} and ^2 respectively. Define a new process X = {X(t), i ^0}by X{ t )=NM+N2 (l \(i) Find E(X(t)) and Var(2f(/)).(ii)
. If {N{t\ / ^ 0} is a simple Poisson process, find the characteristic function of N{t).
. Name as many as you can of the deficiencies of the simple Poisson process as a realistic model for the growth of a population of, say, humans.
. What, if any, are the differences between a simple Poisson process and a Poisson point process?
. Show how the defining properties of a simple Poisson process enable the joint distribution of , NitJ} to be found for arbitrary 0 ^ /l < /2 < ••• < < 00.
. In a simple Poisson process, let p„(t)= Pr {N{t) = n\N{0) = 0}. Use the relationsto derive differential-difference equations for n = 0.1,2,..., in the same manner in which (9.8) was derived for
. Using the birth and death rates for 1966 given in Table 9.2 and the 1966 population of Australia given in Table 9.1, estimate the 1971 population.Compare with the actual population in 1971. Is the
. What will happen in the Markov chain model of random mating with mutation if 7 ^ 0 but a2 = 0 ?
. In the Markov chain model of random mating with mutation in a population of size A/, find P ifa, == « 2 = a 7 ^ 0. Given an arbitrary initial probability distribution p(0), find p(l) and deduce
. For a simple random walk assume there are reflecting barriers at 0 and 3.That is, when the particle gets to 0 or 3 it goes on the next step to states 1 or 2 (respectively) with probability one.
. Let be the response on trial /i, n = 0,1,2,...(a) Find the stationary probability vector p.(b) Will the probability distribution of approach p as n 0 0?(c) Find the matrix P.(d) Prove, using
. The following learning model, due to Bush and Mosteller, is discussed in Bailey (1964). In a learning experiment let Sj be a correct response and 5 3 an incorrect response. The response at any
. Prove that > 0, /c = 1,2,...P =\ - p p 'q 1 -q_ 0
.The following example is based on an application discussed in Isaacson and Madsen (1976). Farms are divided into four categories: very small (sj, very large (5 2 ), large (5 3 ) and small (S4 ).
. For the Markov chain of Exercise 11, find the average number of years a first-year, second-year and third-year student will remain in university.
. For the Markov chain of Exercise 11, solve the equations (1 - Q) = I to obtain the fundamental matrix = (1 — Q) “ E For the Markov chain of Exercise 11, find a student’s chances of
. The following problem is based upon one in Kemeny and Snell (1960). In each year of a three-year degree course, a university student has probability p of not returning the following year,
. For the simple random walk with absorbing barriers at 0 and 3, verify that the formulas 1 T p 1 “h q 1 - pq - pq for the expected times to absorption from Xq = \, Xq = 2, respectively, agree with
. Prove that the Markov chain {X„} for random genetic drift defined in Section 8.3 is a martingale, (cf. Exercise 14 of Chapter 7.) Use the optional stopping theorem to deduce immediately that the
. For the genetic Markov chain (Section 8.3) with a population of N diploid individuals, find the matrices Q and R. Verify that the matrix FI of absorption probabilities O:'1 - \/2N \/2N 1 - 2/2N
. The matrix PI of absorption probabilities for the simple random walk with absorbing barriers at 0 and 3 was found to be 180 Population genetics and Markov chains n --p q P Verify that(a) the row
. For the Markov chain for random mating with no mutation, the transition matrix when /V = 1 is 1 0 0 1/4 1/2 1/4 0 0 1 If Xq has the distribution p(0) = [0 y, find the probability distributions of X
. Any stochastic matrix defines a temporally homogeneous Markov chain.Which of the following matrices are stochastic?(a) \\/4 C'(c)3/4 (b) 1 / 2 1 / 2 0“0 _ 1 / 2 1/4 0 J 0 0 ."■o 1 0 0 “1/3
. Show that the matrix with elements given by (8.13) is stochastic.
. Establish the Chapman- Kolmogorov equations P„(jmk + /7) _— VZ. Pji Pik • i = 1(Hint: Use matrix multiplication.)
. Complete the proof of Theorem 8.2; that Pr(A:o = Sj^, X , = Sj,,..., = SjJ = 1).for r? ^ 1 . (Hint: Use mathematical induction.)
. A gene is present in human populations which has two alleles A j and A2 . If a group initially has 40 AiAj, 30 A1A2 or A2 A1 and 30 A2 A2 individuals, what will the equilibrium (HW) genotype
. In tennis, if the score in a game reaches deuce, a player must win two points in a row to win a game. Construe the score as a random walk on{0, 1,2,3,4}. If the players have probabilities p and q
. A random process {Xq,X^ is called martingale if E(A'„^ 1 ,..., A'o)= for n = 0, 1,2,.... The theory of such processes is elegant and often used in probability (see, for example, Kannan, 1979, or
. Prove that the expressions given in (7.24) apply for the expected time to absorption of the simple random walk when go.
. Solve the difference equation for the expected time to absorption of a simple random walk at 0 , pD^ , , -D^ + qD,_ 1 = - 1, with boundary conditions Dq = D^ = 0. To do this proceed as with the

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