### Module Code

math | scheme – 这是一个math的practice, 考察math的理解, 是比较典型的math等代写方向

```
MAT00018M
RESIT
```

### M math and MSc Resit Examinations 2021-

#### Department:

Mathematics

#### Title of Exam:

Stochastic Processes

#### Time Allowed:

You have 2 hours plus 1 hour to upload your solutions.

#### Allocation of Marks:

The marking scheme shown on each question is indicative only.

```
Question: 1 2 3 4 Total
Marks: 25 25 25 25 100
```

#### Instructions for Candidates:

Answer all **four** questions. It is important to show your working and reasoning in order to demonstrate your knowledge and understanding.

#### Queries:

If you believe that there is an error on this exam paper, then please use the Queries link below the exam on Moodle. This will be available for the first hour after the release of this exam. After that, if a question is unclear, then answer it as best you can and note the assumptions youve made to allow you to proceed.

#### Submission:

Please write clearly and submit a single copy of your solution to each question. Any handwritten work in your electronic submission must be legible. Black ink is recommended for written answers. View your submission before uploading. Number each page of your solutions consecutively. Write the exam title, your candidate number, and the page number at the top of each page. Upload your solutions to the Exam submission link below the exam on Moodle (preferably as a single PDF file). If you are unable to do this, then email them to [email protected].

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Page 1 (of 7)
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#### A Note on Academic Integrity

We are treating this online examination as a time-limited open assessment, and you are therefore permitted to refer to written and online materials to a id you in your answers. However, you must ensure that the work you submit is entirely your own , and for 6 hours after the exam is released, you must not:

- communicate with departmental staff on the topic of the assessment (except by means of the query procedure detailed overleaf),
- communicate with other students on the topic of this assessment,
- seek assistance on this assessment from the academic and/or disability support services, such as the Writing and Language Skills Centre, Maths Skills Centre and/or Disability Services (unless you have been recommended an exam support worker in a Student Support Plan),
- seek advice or contribution from any third party, including proofreaders, friends, or family members.

We expect, and trust, that all our students will seek to maintain the integrity of the assessment, and of their awards, through ensuring that these instructions are strictly followed. Failure to adhere to these requirements will be considered a breach of the Academic Misconduct regulations, where the offences of plagiarism, breach/cheating, collusion and commissioning are relevant see Section AM.1.2.1 of the Guide to Assessment (note that this supersedes Section 7.3).

```
Page 2 (of 7)
```

1 (of 4). State without proof any theorem and definition you use in your answer.

```
(a) Suppose that a Markov process N ={N(t) : t 0 }taking values in N=
{ 0 , 1 , 2 , 3 ,} is a birth-death process with birth rates i0, i= 0 , 1 ,...
and death rates i0, i= 1 , 2 ,.... Define a new process M={M(t) : t 0 }
by the following time change
```

```
M(t)=N
```

```
(t
a
```

##### )

```
, t 0 ,
```

```
where a >0 is a certain fixed number. Prove that this new process M =
{M(t) : t 0 }is a birth-death process. Find its birth and death rates in
terms of the birth and death rates of the original process N={N(t) : t 0 }.
[8]
```

```
(b) Consider a population of a certain endangered species of mammals living on
a remote planet. Each individual independently gives birth to new individ-
uals at a constant rate and dies at a rate . In addition there is a constant
rate of individuals joining the population from a neighbouring planet. We
assume that the size of the population is described by a birth-death process
N={N(t) : t 0 }with the birth rates given by
```

```
i=i+, i= 0 , 1 , 2 , 3 , (1.1)
```

```
and the death rates given by
```

```
i=i, i= 1 , 2 , 3 ,. (1.2)
```

```
Let ni(t), for i= 0 , 1 , and t0, be the expected number of individuals at
time t, provided number of individuals at time 0 is equal to i, i.e.
```

```
ni(t)=
```

```
j= 0
```

```
jp i j(t), (1.3)
```

```
where pi j(t) are the transition probabilities defined by, for i,j= 0 , 1 ,,
```

```
pi j(t)=P(N(t)=j|N(0) =i), t 0. (1.4)
```

```
Assume the following values for the parameters , and above,
```

```
= 3 , = 2 , = 3. (1.5)
```

```
Using the Master Equation find n 0 (t), t0.
[9]
```

```
Page 3 (of 7) continued on next page
```

continued from previous page MAT00018M (R)

1 (of 4) cont.

```
(c) Consider the birth-death process N =
```

##### (

```
N(t) : t 0
```

##### )

```
from part (b) of this
Question but with a different choice of the parameters:
```

```
= 1 , = 2 , = 3. (1.6)
Does there exist a stationary distribution of the process N? If yes, find it.
[8]
```

```
[Total: 25]
```

```
Page 4 (of 7)
```

2 (of 4). State without proof any theorem and definition you use in your answer.

```
A car parking of a certain supermarket has unlimited capacity. It opens at 8:00.
Modern electrical cars arrive at the parking with rate A=3 per minute. Outdated
petrol cars arrive with rate B=2 per minute. There are no other types of cars.
You need to assume that the numbers of cars arriving at the car park are modelled
by two independent Poisson processes with rates Aand B.
```

```
(a) What is the average number of electrical cars arriving in the car park during
the first 1 minute, i.e. between 8:00 and 8:01? [3]
```

```
(b) What is the probability that the number of petrol cars arriving in the car park
during the first 2 minutes is smaller than 2 (i.e. less or equal to 1)? [5]
```

```
(c) Assume that during the first 3 minutes of car park being operated, the number
of all cars arriving is 6. What is the probability that the number of all cars
that had arrived in the first 1 minute is equal to 2?
[6]
```

```
(d) Assume that during the first 2 minutes of car park being operated, the number
of electrical cars arriving is 5. What is the probability that the number of
electrical cars arriving in the first 5 minutes is 10?
[5]
```

```
(e) Kim is a manager of the car park. Kims team will work till the time when
the one thousands petrol car arrives. Then their duty will be taken over by a
team lead by Lucy. What is the expected number of petrol cars arriving to the
car park in the first 1 hour of Lucy team job? Justify your answer by stating
a result you use.
[6]
```

```
[Total: 25]
```

```
Page 5 (of 7) Turn over
```

3 (of 4). State without proof any theorem and definition you use in your answer.

```
In the whole question we assume that a stochastic process W ={W(t) : t 0 }
defined on a probability space (,F,P) is a Brownian Motion.
```

```
(a) Show that
```

##### E

##### [

##### W(2)W(4)

##### ]

##### = 2 ,

##### E

##### [

##### (W(2))^2 W(4)

##### ]

##### = 0.

##### [4]

```
Find the expected value below:
```

```
E
```

##### [

##### (W(2))^3 W(4)

##### ]

##### .

##### [2]

```
(b) Show that for every t0, the following equality holds:
```

##### E

##### [

```
e^3 W(t)
```

(^92) t 2 ] = 1. (3.7) [6] Explain why the above equality holds trivially when t=0. [2] (c) Let =

##### (

```
(t) : t 0
```

##### )

```
be a stochastic processes given by
```

```
(t)=
```

```
1 , if 0 t< 2 ,
W(2), if 2 t< 4 ,
W(4) W(3), if 4 t< 7 ,
0 , if t 7.
```

```
(i) Demonstrate that is the step process. [4]
```

```
(ii) Evaluate the Ito integral I() of a step process with respect to the Brow-
nian Motion W ={W(t) : t 0 }and compute the variance of the random
variable I(). [7]
```

```
[Total: 25]
```

```
Page 6 (of 7)
```

4 (of 4). State without proof any theorem and definition you use in your answer.

```
In the whole question we assume that a stochastic process W ={W(t) : t 0 }
defined on a probability space (,F,P) is a Brownian Motion.
```

```
(a) Find real numbers aand bsuch that the following equality is satisfied
```

```
e^2 t+^3 W(t)=a+b
```

```
t
```

```
0
```

```
e^2 s+^3 W(s)ds +c
```

```
t
```

```
0
```

```
e^2 s+^3 W(s)dW (s), t 0. (4.5)
```

##### [10]

```
(b) Use the previous part (a) (or otherwise), calculate, for t0,
```

```
E
```

##### [

```
e^2 t+^3 W(t)
```

##### ]

##### . (4.6)

##### [5]

```
Hint to part (b). Denote by y(t) the expression (4.6) and using part (a) derive
an integral equation y(t). Transform that equation into a differential one and
solve it.
(c) Suppose that a process X=(X(t)) is a solution of following stochastic differ-
ential equation
```

```
dX (t)=
```

##### [

```
eX(t)
```

##### 1

##### 2

```
e^2 X(t)
```

##### ]

```
dt +eX(t)dW (t), (4.7)
```

```
Define a new process Y=(Y(t)) by the formula
```

```
Y(t)=eX(t)
```

```
Show that this process satisfies following stochastic differential equation
```

```
dY (t)=dt +dW (t), (4.8)
```

```
Find the solution to the equation (4.8) such that Y(0) =1 and deduce a closed
form for the solution of the equation (4.7) such that X(0) =0. [10]
```

```
[Total: 25]
```

```
Page 7 (of 7) End of examination.
```