Stochastic Processes

Stochastic Processes Practice 2

Advanced topics: continuous-time MC, compound Poisson, SDEs, branching processes, and limit theorems

8 Problems

Suggested: 2 hours

Instructions

• Try to solve each problem before viewing the solution
• Click "Show Solution" to reveal the answer and detailed explanation
• Focus on understanding the problem-solving methodology

1Continuous-Time Markov Chain: Transition Probabilities

Problem

A continuous-time Markov chain on {0, 1} has generator matrix:

Q = \begin{pmatrix} -2 & 2 \\ 3 & -3 \end{pmatrix}

(1) Find the transition probability matrix P(t) = e^(Qt).

(2) Find the stationary distribution.

(3) If X(0) = 0, find P(X(1) = 1).

2Compound Poisson Process

Problem

Claims arrive at an insurance company according to a Poisson process with rate λ = 10 per day. Each claim amount is uniformly distributed on [0, 1000] dollars, independent of arrival times and other claims.

(1) What is the expected total claim amount in one day?

(2) Find the variance of the total claim amount in one day.

(3) What is the probability that total claims exceed 6000 dollars in one day?

3Ito Calculus and Stochastic Differential Equations

Problem

Consider the SDE: dX(t) = μX(t)dt + σX(t)dB(t) with X(0) = x₀.

(1) Use Ito's lemma to find the SDE for Y(t) = ln(X(t)).

(2) Solve for X(t) (geometric Brownian motion).

(3) Find E[X(t)] and Var(X(t)).

4Gambler's Ruin

Problem

A gambler starts with $5. Each round, they win $1 with probability p = 0.48 or lose $1 with probability q = 0.52.

They play until reaching $0 (ruin) or $10 (target).

(1) Find the probability of ruin starting from $5.

(2) Find the expected number of rounds until the game ends.

5Branching Process

Problem

In a Galton-Watson branching process, each individual produces k offspring with probability pₖ:

p₀ = 0.2
p₁ = 0.3
p₂ = 0.5

(1) Find the mean offspring m.

(2) Will the population eventually go extinct?

(3) If extinction occurs, find the extinction probability starting with Z₀ = 1.

6Stationary Process and Ergodicity

Problem

Let {Xₙ, n ≥ 0} be a stationary stochastic process with mean μ and autocovariance function γ(h) = Cov(Xₙ, Xₙ₊ₕ).

(1) Show that γ(h) = γ(-h).

(2) If γ(h) = σ²ρ^|h| for |ρ| < 1, find the variance of the sample mean $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ .

(3) Under what conditions does $\bar{X}_n \to \mu$ as n → ∞?

7Stopped Brownian Motion

Problem

Let B(t) be standard Brownian motion. Define the stopping time T = inf{t: B(t) = 2}.

(1) Show that E[B(T)] = 2.

(2) Find E[T].

(3) Is E[B(T)²] = E[T]?

8Limit Theorems for Markov Chains

Problem

Consider an irreducible, aperiodic Markov chain with stationary distribution π.

(1) State the ergodic theorem for Markov chains.

(2) If f is a function on the state space, what does $\frac{1}{n}\sum_{k=0}^{n-1} f(X_k)$ converge to?

(3) Describe the Central Limit Theorem for Markov chains.