The true jump method

In htis post, I’ll discuss a simulation technique for generating statistically exact jump times when the rate is state-dependent, $\lambda(X_t)$.

Diffusing diffusivities, stochastic subordination

In this post, I’ll discuss some recent explanations for anomalous, yet Gaussian diffusion, including diffusing diffusivities and stochastic subordination.

Stochastic limits, part 2: tails, memory, and the Joseph and Noah effects

In the previous post about limit theorems of stochastic processes, we considered when everything goes right, leading to Gaussian-like behavior. Here we’ll discuss when things go wrong, particularly when memory effects and infinite moments are introduced.

Stochastic limits, part 1: CLT, Donsker’s FCLT

One way to understand the structure of randomness is to experience a lot of it. We’ll use $\lim_{n\to\infty} X_1 + \cdots + X_n$ as a case study, and along the way bump into classical ideas like the Central Limit Theorem (CLT) and Brownian motion.