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Jin Ma. A colleague, a thinker, a scholar, a husband and father, but most of all: a friend. In a previous lifetime, or so it seems, I worked at Purdue University, in West Lafayette, Indiana. During my time at Purdue, I had the good fortune to hire Jin Ma as a postdoc, fresh from his PhD at the University of Minnesota. He was so outstanding we were able to change his status after two years, from a postdoc to tenure track. The rest was inevitable: Associate Professor, and then Professor. 

This article presents a Hawkes process model with Markovian baseline intensi- ties for high-frequency order book data modeling. We classied intraday order book trading events into a range of categories based on their order types and the price change after their arrivals. In order to capture the stimulating eects between mul- tiple types of order book events, we use multivariate Hawkes process to model the self- and mutually-exciting event arrivals. We also integrate a Markovian baseline intensities into the event arrival dynamic, by including the impacts of order book liquidity state and time factor on the baseline intensity. A regression-based non- parametric estimation procedure is adopted to estimate the model parameters in our Hawkes+Markovian model. To eliminate redundant model parameters, LASSO reg- ularization is incorporated into the estimation procedure. Besides, model selection method based on Akaike Information Criteria is applied to evaluate the eect of each part of the proposed model. An implementation example based on real LOB data is provided. Through the example we studied the empirical shapes of Hawkes excitement functions, the eects of liquidity as well as time factors, the LASSO vari- able selection, and the explanation power of Hawkes and Markovian elements to the dynamics of order book. 

Microlending, where a bank lends to a small group of people without credit histories, began with the Grameen Bank in Bangladesh, and is widely seen as the creation of Muhammad Yunus, who received the Nobel Peace Prize in recognition of his largely successful efforts. Since that time the modeling of microlending has received a fair amount of academic attention. One of the issues not yet addressed in full detail, however, is the issue of the size of the group. Some attention has nevertheless been paid using an experimental and game theory approach. We, instead, take a mathematical approach to the issue of an optimal group size, where the goal is to minimize the probability of default of the group. To do this, one has to create amodel with interacting forces, and to make precise the hypotheses of the model. We show that the original choice of Muhammad Yunus, of a group size of five people, is, under the right, and, we believe, reasonable hypotheses, either close to optimal, or even at times exactly optimal, i.e., the optimal group size is indeed five people. 

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change. 

Abstract This paper provides a new explanation for closed‐end fund (CEF) discounts and premiums using the local martingale theory of asset price bubbles. This is a rational asset pricing model that is shown to be consistent with the existing empirical evidence on CEF discounts/premiums. Additional testable implications of the model are derived, which await subsequent research for their resolution. This bubble theory also applies equally well to understanding discounts and premiums on exchange traded funds.