Math 106 Stochastic processes

Week 3

Probability theory revisited (1.2)

This is where things get a bit more mathy. To give a proper presentation of stochastic processes on uncountable state spaces, we need to revisit some basic ideas in probability theory and give them a measure theoretic foundation. While is possible to understand a lot about stochastic processes without measure theory, I feel it’s important to be familiar with the ideas because it makes a lot of literature that would otherwise be impenetrable accessible. At the level of this course you can mostly think of it as a new language to talk about things we understand intuitively, we are not going to be doing technical measure theory proofs.

  • D 1.2 Note that if we are in a finite state space this is just new terminology to talk about outcomes and events.
  • Also take a look at Appendix C, but you won’t need to know this for exams

General stochastic processes and filtrations (5.1,5.2)

Notes: Filtration and stopping times

Gaussian processes (5.4)

Notes: Gaussian processes

Exercises

  • 1.11
  • 1.12
  • 1.13
  • 5.1
  • 5.2 (prove for discrete time)
  • 5.4
  • Write code to generate samples of a mean zero Gaussian process with given covariance function at times $t_1,\dots,t_n$. Experiment with different paramaters in the Matern kernel family Matérn covariance function.
  • Consider the stationary Ornstein–Uhlenbeck (OU) process restricted to a finite interval $[0,T]$. Its covariance kernel is \(K(s,t) = \frac{\sigma^2}{2\theta}\, e^{-\theta \lvert t-s\rvert}, \qquad s,t\in[0,T].\) Show that any eigenfunction $\phi$ satisfying (E) also satisfies the second–order ODE \(\lambda\,\phi''(t) = \theta^2\,\lambda\,\phi(t) - \sigma^2\,\phi(t), \qquad t\in(0,T),\) together with boundary conditions at $t=0$ and $t=T$ that can be derived from the integral representation of $(\mathcal{K}\phi)(t)$ at the endpoints. Hint: Write the operator action explicitly by splitting the integral at $s$: \((\mathcal{K}\phi)(s) = \int_0^s e^{-\theta(s-t)}\,\phi(t)\,dt \;+\; \int_s^T e^{-\theta(t-s)}\,\phi(t)\,dt.\) Differentiate with respect to $s$ to obtain first–order relations, differentiate once more, and use the eigenvalue equation.

(I recommend focusing on the exam practice problems – see midterm practice problems)