Fatou's lemma and monotone convergence theorem

In this post, we deduce Fatou's lemma and monotone convergence theorem (MCT) from each other.


Young's, Hölder's and Minkowski's Inequalities

In this post, we prove Young's, Holder's and Minkowski's inequalities with full details. We prove Hölder's inequality using Young's inequality. Then we prove Minkowski's inequality by using Hölder.


Sequential importance sampling-resampling


In this post, we review the sequential importance sampling-resampling for state space models. These algorithms are also known as particle filters. We give a derivation of these filters and their application to the general state space models.


Importance sampling


This simple note reviews the importance sampling. This discussion is adapted from here and here.


Static Parameter Estimation for the GARCH model


In this post, we review the online maximum-likelihood parameter estimation for GARCH model which is a dynamic variance model. GARCH can be seen as a toy volatility model and used as a textbook example for financial time series modelling.


Nonnegative Matrix Factorization


In this post, I derive the nonnegative matrix factorization (NMF) algorithm as proposed by Lee and Seung (1999). I derive the multiplicative updates from a gradient descent point of view by using the treatment of Lee and Seung in their later NIPS paper Algorithms for Nonnegative Matrix Factorization. The code for this blogpost can be accessed from here.


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.