代写37007 Probability Theory and Stochastic Analysis Sample Exam Autumn 2024代做回归

37007 Probability Theory and Stochastic Analysis

Sample Exam Autumn 2024

Assume all RVsand SPs are adapted to suitable (filtered) probability spaces.

Question 1.

Consider the joint-Gaussian process

(a) Find the distribution of

(b) Find

(c) Let Bt, t  ≥ 0, be a standard Brownian motion and set

Find Pearson’scorrelation corr(xt, xs ) where 0 ≤ s < t.

(d) Let Bt, t ≥ 0, be a standard Brownian motion, x  ∈ ℝ and set

Determine if xt  is a Markov process.

Question 2.

Consider the Poisson process Nt, t  ≥ 0 , with intensity λ  =  1.

(a) Calculate

P(N1  = 4, N3  = 5|N5  = 7).

Now define the compound Poisson

with Nt  as above and where Yk, k  ∈ {1,2, … , Nt }, has the distribution

Assume that Nt  and the Yk  are all independent from each other.

(b) Find E[eiuxt].

(c) Let Bt, t  ≥ 0, be a standard Brownian, m, r  ∈ ℝ and σ  > 0 and consider the process

adapted to the filtered probability space (Ω, ℱ, Q, (ℱt )t≥0). Find m such that

Question 3.

Consider the homogenous, discrete-time Markov chain xt, t  = 0, 1,2 …, taking states xt   ∈ {x1  = 8, x2  = −4, x3  = −6} with one-step transition matrix and initial distribution

(a) Calculate E[x3 |x1  = −4].

(b) Using the criteria of Theorem 5 on page 28 of Chapter 5 notes, determine if xt is ergodic.

Now consider the homogenous, continuous-time Markov chain Yt, t  ≥ 0, taking states Yt  ∈ {1,2,3} with transition matrix (when jump occurs)

Assume that the expected values of weighting times for jumps from states 1, 2 and 3 are 1/2, 1/6 and 1/7 respectively.

(c) Find a stationary distribution of yt.

(d)Find the moment generating function of T2, where T2  is the waiting time for a jump from yt  = 2. Make sure to list any conditions(s) necessary for this function to be properly defined.

Question 4.

Consider the process

where Bt, t   ≥ 0, is astandard BM.

(a) Using Definition 1 page 6 Chapter 11 Notes, derive the drift function for xt. (b) Find the Ito representation of xt.

Now let

yt  = ext,   t ≥ 0.

(c) Find the Ito representation of yt.

(d) Write down the Kolmogorov forward equation for

where

with

Question 5.

(a) Let Bt, t ≥ 0, be a standard Brownian motion and consider the process

Determine if xt  is a martingale, submartingale or supermartingale.

(b) Let ξt, t ∈ {0, 1, … }, be a sequence of independent RVs and set

Find the Doob-Meyer decomposition of exp(st ).

(c) Let Bt, t ≥ 0, be a standard Brownian motion on (Ω, ℱ, P) and define

where σ(t) > 0 is anon-random function of time such that

Determine the distribution of

on the probability space (Ω, ℱ, Q) defined by

with IA  the indicator of event A.