In this talk, I'll discuss various notions of "mixing" for sequences of random variables. Loosely, mixing means some form of asymptotic indpendence, and is a useful concept in situations where random variables "feel i.i.d." in some sense, but are not actually independent. The most important of these notions is "ergodic." I will discuss two equivalent definitions of "ergodic" and compare notions of mixing in probability theory with those in ergodic theory. Unfortunately there is some disagreement in terminology between the two fields, and most Internet sources seem to assume the ergodic theorists's perspective. I'm here to help you navigate this treacherous landscape. In addition, I'll state a couple of theorems that will help explain why mixing conditions are useful.

In this talk, I'll discuss various notions of "mixing" for sequences of random variables. Loosely, mixing means some form of asymptotic indpendence, and is a useful concept in situations where random variables "feel i.i.d." in some sense, but are not actually independent. The most important of these notions is "ergodic." I will discuss two equivalent definitions of "ergodic" and compare notions of mixing in probability theory with those in ergodic theory. Unfortunately there is some disagreement in terminology between the two fields, and most Internet sources seem to assume the ergodic theorists's perspective. I'm here to help you navigate this treacherous landscape. In addition, I'll state a couple of theorems that will help explain why mixing conditions are useful.

The Black-Scholes equation is a foundational result of mathematical finance. In this talk I will build the tools needed to derive it, including Brownian Motion, Itô integration, and Itô's formula. In part 2 I will derive the Black-Scholes equation and give an overview of what it can tell us.

The Black-Scholes equation is a foundational result of mathematical finance. In this talk I will derive the Black-Scholes equation and give an overview of what it can tell us.

Abstract here