The EM Algorithm


In this post, we review the Expectation-Maximization (EM) algorithm and its use for maximum-likelihood problems.


Stochastic gradient descent

In this post, I introduce the widely used stochastic optimization technique, namely the stochastic gradient descent. I also implement the algorithm for the linear-regression problem and provide the Matlab code.


Gaussianity, Least squares, Pseudoinverse


In this post, we show the relationship between Gaussian observation model, Least-squares and pseudoinverse. We start with a Gaussian observation model and then move to the least-squares estimation. Then we show that the solution of the least-squares corresponds to the pseudoinverse operation.


The use of Ito-Doeblin formula to solve SDEs


These notes are mostly based on the book Stochastic Calculus for Finance vol. II, Chapter 4. I give a few propositions and focus on exercises of Shreve by make use of the Ito-Doeblin formula. The use of Ito-Doeblin formula is almost purely practical to solve continuous-time stochastic models. My treatment is slightly different from the Shreve since I emphasize on the differential forms of the formulas.