**Stochastic Processes **

**Brownian bridge [11 marks]**

Consider the process

where is a standard Brownian Motion.

- Show that is a standard Brownian bridge
**[2 marks]**.

- Determine
**[3 marks]**.

- Show that
**[3 marks]**.

- Using the result from (c), determine
**[3 marks]**.

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**Question 2. Stationarity [9 marks]**

Consider the process

where the and independent from each other. Determine if the following processes are strictly stationary and for those that are write down the covariance function.

**[3 marks]**.

**[3 marks]**.

**[3 marks]**.

**Question 3. MC option pricing [10 marks]**

Consider the geometric Brownian Motion (GBM)

where , , , and is a standard Brownian motion.

An example of a discretely-monitored Asian call option with European payoff has price at given by

where .

- Taking ,
**use Mathematica**to calculate the crude Monte Carlo estimate

where , are random samples of and respectively **[4 marks]**. Also calculate the sample estimate of **[1 mark]**.

- Taking ,
**use R**to calculate the control variate Monte Carlo estimate

where the expected value of the control variate is given by the Black-Sholes European vanilla call price formula

with

and the crude MC estimate of the control variate

**Question 4. Markov chains [10 marks]**

**You may use computational software for calculations, but express your answers using proper mathematical notation.**

** **

Let , , be a homogenous Markov chain taking states with one-step transitional probability matrix and initial distribution

respectively.

- Calculate
**[2 marks]**.

- Calculate a stationary distribution
**[2 marks]**.

Let , , be a homogenous Markov chain taking states with generator matrix

- What is the probability of the state change when the Markov chain jumps
**[2 marks]**?

- Let and let the waiting time

What is **[2 marks]**?

- A Markov chain is ergodic if the limit of the state probability vector, , exists and does not depend on . Using this criterion, determine if ergodic
**[2 marks]**?

**Question 5. ARMA processes [10 marks]**

Consider the process

where , , is a zero-mean white-noise process with variance .

- The process above is not stationary. Explain why
**[2 marks]**and identify an appropriate ARIMA() model**[2 marks]**.

** **

** **

- Describe how the ARIMA() model from (a) could be converted to an ARMA() model
**[2 marks]**.

** **

** **

- Determine if the ARMA() from (b) is invertible
**[2 marks]**.

- Plot the ARIMA() and ARMA() models identified in (a) and (b) respectively with
**[2 marks]**?

** **

**Question 6. Diffusion processes [10 marks]**

Consider the diffusion process

where is a standard Brownian motion.

- Using the definition

find the drift coefficient of **[3 marks]**.

- Find the diffusion coefficient of using the Ito formula
**[2 marks]**.

- Write down the Kolmogorov backward equation for the transition density function of the process
**[2 marks]**.

Define the process

with as above.

- Using the Ito formula, write down the stochastic differential equation of the process
**[3 marks]**.