MAT00018M Stochastic Processes 2019-20
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MAT00018M
MMath and MSc Examinations 2019-20
Stochastic Processes
1 (of 4). (a) Give a definition of a Poisson process N = {N(t) : t 2 0} with rate λ . [4]
(b) Write down the formula for the transition probabilities
P (N(t) = jIN(s) = i)
for any t 2 s 2 0 and j 2 i 2 0. [4]
(c) Assuming that
P (N(1) = j) = e_3 , j = 0, 1, 2, 3, . . .
calculate, providing explanation for each step of your calculations, E[N(1)].
[6]
1 (of 4) cont.
(d) The photons emitted by an LED result from the recombination of electron- hole pairs. Assume that the number of photons is modelled using a Poisson process with rate of 10 photons per second.
(i) What is the probability that exactly 8 photons have been emitted during the first 1 second? [2]
(ii) What is the probability that at most 1 photon has been emitted during the
first 1 second? [2]
(iii) What is the probability that at least 1 photon has been emitted during the
first 1 second? [2]
(iv) What is the probability that exactly 2 photons arrive during the first 1
second and exactly 3 photons arrive during the first 3 seconds? [2]
(v) It is known that exactly 4 photons have been emitted during the first 3 seconds. What is the probability that exactly 1 photon has been emitted in the first 1 second? [2]
(vi) How many seconds would you expect to have to wait until the first photon has been emitted? [2]
(vii) Alice was counting the emitted photons. After she has counted the first 1000 photons, she was replaced by her colleague Bob.
What is the probability that exactly 5 photons have been counted by Bob
during the first 2 seconds on his watch? [2]
State explicitly any properties of the Poisson process used in your calcula- tions.
[Total: 28]
2 (of 4). (a) Give a definition of a Brownian Motion W = {W (t) : t 2 0}.
(b) Assuming that W (t), for t > 0, has density
1 x2
pt (x) = ^2πt e_ 2t , x e R,
calculate the probability
P (W (1) 二 0) .
(c) Assuming that W (t), for t > 0, has density
1 x2
pt (x) = ^2πt e_ 2t , x e R, show that for t 2 0 the following equalities hold:
E [e4W (1/2)] = e4 ,
E [(W (1))2] = 1,
E [(W (5))3 e_(W (5))2 ] = 0.
State without proof any result you are using. [13]
[Total: 23]
3 (of 4). (a) Suppose that N = N(t), t 2 0 is a birth-death process with birth rates
λ0 , λ 1 , λ2 , . . . 2 0 and death rates µ 1 , µ2 , . . . 2 0.
Write down the Master (or the forward Kolmogorov) equations for the tran- sition probabilities
pij (t) = P╱N(t) = jIN(0) = i、, t 2 0,
where i, j are natural numbers such that j 2 i 2 0.
[4]
(b) Assume that b, a > 0 are two fixed numbers. Consider a system consisting of K telephone lines. We assume that K 2 3. If n lines are busy, where n 二 K, then with probability nbh, one of them will be freed within small time interval of length h > 0. If n lines are busy, where n 二 K _ 1, then with probability ah, a new call will arrive. The probability that within that time interval two or more conversations will terminate or two or more calls will arrive, is negligible.
Let N(t), for t 2 0, be the number of occupied telephone lines. Assume that N = (N(t)), t 2 0, is a birth-death process.
Find explicit expressions for the birth and death rates of the process N.
[4]
(c) Explain why the stationary distribution for the birth-death process from the previous part exists.
Find the stationary distribution for the birth-death process from the previous part.
Hint: You can use a general formula for the stationary distribution of a birth- death process without deriving it.
[8]
[Total: 16]
4 (of 4). Suppose that W = {W (t)}, t 2 0 is a Brownian Motion.
(a) Define a step process and then define the It integral for such a process. Let ξ = (ξ(t))t>0 be a stochastic processes given by
!(1 if 0 二 t 二 3,
ξ(t) = 二(二) 5(4)
(
Note that the It integral I(ξ) is also denoted by )05 ξ(s) dW (s).
Compute the mean value of this It integral. [6]
(b) State (without a proof) the It isometry. Compute the variance of this It integral I(ξ) from part (a).
[4]
(c) State the It Lemma in the simple form.
Assume that T > 0 is a given number. Using the It Lemma show that
.0 T tW (t) dW (t) = T (W (T))2 _ .0 T (W (t))2 dt _ T2 .
[6]
Using the last result show that
E[.0 T tW (t) dW (t)] = 0.
[3]
(d) Using the It Lemma or otherwise, find the solution to the following stochas- tic differential equation
dX(t) = 3X(t)dt + 4X(t) dW (t), with initial condition X(0) = 1.
(1)
[14]
[Total: 33]
2022-08-13