Stochastic Differential Equations and Econometrics

Recently, I’ve been reading up on stochastic differential equations. I’ve had some ideas on some projects at my day job, and quickly realized that I would need to write down some evolving processes that were inherently stochastic, thus why I’ve been reading up. This post will be short, and for more advanced readers may even feel like a “duh, you never realized that?” moment for them.

Something just clicked in my head the other day as it relates to my article on ARIMA. As I was playing with a finite difference monte carlo algorithm to solve one of my equation sets, I started to write up the code to do the analysis. That’s when something popped out of my code that felt so jarring that I had to write up a quick blog post. My code, I realized was identical to code to simulate an AR(1) process.

At that moment, I realized that I was more familiar with stochastic differential equations than I had originally thought. I was just used to dealing with a discretized version of them. (I think someone had told me that once upon a time, it just didn’t click for me until recently). All of the sudden, the topic didn’t seem so daunting to me any more. I even realized that some of the forecasting that I had done in the past even has a flavor of being a version of the solution to a Fokker-Plank equation.

I actually think that’s pretty exciting. This means that I can write down a stochastic differential equation that I feel like describes a phenomenon better than a standard econometric model, discretize it, and then fit it to actual data to come up with more interesting (and somewhat more exotic) time-series models. Somehow, the world of econometrics just feels a little bit bigger today.