立命館大学数理科学科

Date: 5 December (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:40 

Speaker 1: Jorge Ignacio González Cázares (Universidad Nacional Autónoma de México),16:30–17:30 

Title: Markov Chain Monte Carlo: how and why? II

Abstract: We will review classical and widely used Markov Chain Monte Carlo (MCMC) methods. We will begin by reviewing Markov Chain theory and posing the MCMC problem. We motivate the problem from the point of view of stochastic optimisation problems arising in statistics and machine learning. We introduce Metropolis Hastings and Gibbs samplers and show its simplicity using code. We will also hint at its connections with the popular SGD algorithm.

Speaker 2: Andrea Macrina (University College London), 17:40–18:40

Title: Continuous-Time Quantile Processes with Applications in Finance and Insurance

Abstract: We develop a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: the first, which we largely focus on, features a dynamic random quantile level that allows for direct interpretation of the resulting quantile process characteristics such as location, scale, skewness, and kurtosis, in terms of the model parameters. The second type are function--valued quantile diffusions and are driven by stochastic parameter processes, which determine the entire quantile function at each point in time. By the proposed construction method, quantile processes are obtained by transforming the marginals of a diffusion process under a composite map consisting of a distribution and a quantile function. Sub-classes of quantile diffusions are explored, with emphasis placed on the Tukey family of models whereby skewness and kurtosis are directly parameterised and thus the composite map is explicable with respect to such statistical behaviours. As an example of an application of quantile diffusions, we show how probability measure distortions, a form of dynamic tilting, can be induced. Though particularly useful in financial mathematics and actuarial science, examples of which are given in this work, measure distortions feature across multiple research areas. For instance, dynamic distributional approximations (statistics), non-parametric and asymptotic analysis (mathematical statistics), dynamic risk measures (econometrics), behavioural economics, decision making (operations research), signal processing (information theory), and not least in general risk theory including applications thereof.

Date: 31 October (Thu)

Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)

Time: 16:30–18:00

Speaker: Taiho Wang (Baruch College) 

Title: Concentrated Liquidity Provision Market Making

Abstract: 
In this talk, we first review the concentrated liquidity provision amendment to the constant product market making (CPMM), a commonly employed trading protocol populated in automated market making. Unlike the plain CPMM, the feature of concentrated liquidity provision enables liquidity providers (LPs) to specify customized price ranges within which they will provide liquidty for the purpose of maximizing the transaction fee payments collected from the liquidity takers. Should the pool price leave the prespecified price range, transaction fee payments to the LP are suspended until the pool price returns to the price range, if the LP has not withdrawn his liquidity. This new feature introduces a set of new coordinates, the price-liquidity coordinates, under which liquidty is regarded as a function of pool price. This function is referred to as the liqudity profile or the liquidity distribution. We exposit the evolutions of the following quantities: the liquidity profile, the pool reserves, the pool price, as well as payments of transaction fees under continuous liquidity provisions and tradings in this framework. We touch upon the problem of optimal liquidity provision in these circumstances. The talk is based on joint works in progress with Jimmy Risk, Shen-Ning Tung, and Wei-Cheng Wang.