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.