## Math-Finance seminar

### ath-Fi seminar on 19 Jan.

2023.01.18 Wed up
• Date: 19 Jan. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 18:00-19:30
• Speaker: Tai-Ho Wang (Baruch College)
• Title: Entropy regularized robust optimal order execution
• Abstract:
Order execution, a mission that algorithmic trading departments and execution brokerage agencies embark on regularly, is cast as an entropy-regularized robust optimal control problem. During the course of executing a large order of significant amount, the agent faces with not only the risk of price impact that his own execution would incur towards the transaction price but also the liquidity and uncertainty of the market. The agent’s goal is to maximize an objective functional associated with his profit-and-loss of trading and simultaneously minimize the exeuction risk. It is documented that “a liquid market is one which is almost infinitely tight, which is not infinitely deep, and which is resilient enough so that prices eventually tend to their underlying value”. As such, we model the market’s liquidity and uncertainty by the principle of least relative entropy associated with the market volume. The problem of order execution is thus turned into a relative entropy-regularized (Bayesian) stochastic differential game. Standard argument of dynamic programming applies in this setting which yields that the value function of the differential game satisfies a “Bayesian” Hamilton-Jacobi-Isaacs (HJI) equation. Under the assumptions of linear-quadratic model with Gaussian prior, the Bayesian HJI equation reduces to a system of Riccati and linear differential equations. Further imposing constancy of the corresponding coefficients, the system of differential equations can be solved in closed form, resulting in analytical expressions for optimal strategy and trajectory as well as the posterior distribution of market volume.
In conclusion, numerical examples, comparisons and discussions of the optimal strategy to conventional trading strategies are demonstrated.

### Math-Fi seminar on 5 Dec.

2022.12.04 Sun up
• Date: 5 Dec. (Mon.)
• Place: On the Web (Zoom)
• Time: 10:30-12:00
• Speaker: Pei-Chun Su (Duke University)
• Title: Optimal shrinkage of singular values under noise with separable covariance & Its application to fetal ECG analysis
• Abstract:
​High dimensional noisy dataset is commonly encountered in many scientific fields, and a critical step in data analysis is denoising. Under the white noise assumption, optimal shrinkage has been well developed and widely applied to many problems. However, in practice, noise is usually colored and dependent, and the algorithm needs a modification. We introduce a novel fully data-driven optimal shrinkage algorithm when the noise satisfies the separable covariance structure. The novelty involves a precise rank estimation and an accurate imputation strategy. In addition to showing theoretical supports under the random matrix framework, we show the performance of our algorithm in simulated datasets and apply the algorithm to extract fetal electrocardiogram from the benchmark trans-abdominal maternal electrocardiogram, which is a special single channel blind source separation challenge.

### Math-Fi seminar on 1 Dec.

2022.12.01 Thu up
• Date: 1 Dec. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Ju Yi Yen (University of Cincinnatti)
• Title: Mathematical analysis of automated market makers
• Abstract:
Automated market makers (AMMs) are examples of Decentralized Finance systems. Nowa- days, AMMs are dominated by the Constant Function Market Makers (CFMMs). CFMMs pool liquidity from its takers and providers, and set the relative prices of the two assets within the pool by a mathemat- ical formula. The relative price is determined by the reserves of the two assets in the pool. Notice that the assets in the liquidity pool are risky assets, their performances are impacted by the market risk. In this talk, we describe the stochastic process used for modeling the relation between the pool price and the corresponding market price for assets traded via CFMMs, and present limit theorems of this stochastic process. Our results are deduced from properties of the Brownian motion and its local time process.

### Math-Fi seminar on 24 Nov.

2022.11.22 Tue up
• Date: 24 Nov. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Michael Zierhut (KIER, Kyoto University)
• Title: The Arbitrage Pricing Theory in Incomplete Markets
• Abstract:
The arbitrage pricing theory (APT) is traditionally viewed as a descriptive theory: If asset prices are decomposed into systematic and idiosyncratic components, the latter are negligible for almost all assets in large markets. This paper analyzes its role as a predictive theory: When prices of systematic risk factors are estimated by means of linear regression, these estimates are a lower-dimensional representation of a pricing kernel. Such estimates can be used to predict arbitrage-free prices for new assets. Market structure matters: When markets are complete, there is a unique pricing kernel and factor pricing is always arbitrage-free. When markets are incomplete, this method may select a nonpositive pricing kernel. This leads to a problem that is robust in a topological sense: For an open set of arbitrage-free markets, estimated factor models do not assign arbitrage-free prices out of sample. The critical assumption is therefore not that markets grow large, but that markets grow complete.

### Math-Fi seminar on 17 Nov.

2022.11.11 Fri up
• Date: 17 Nov. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Etsuo Segawa (Yokohama National University)
• Title: 量子ウォークの快適度とグラフの組合わせ構造
• Abstract:

### Math-Fi seminar on 20 Oct.

2022.10.18 Tue up
• Date: 20 Oct. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Luis Iván Hernández Ruíz (Kyoto University)
• Title: Results on Limit Theorems for the Renewal Hawkes Process
• Abstract:
Point processes are often used to model occurrences of events in time. One of such models that has seen applications in Finance is the self-exciting process proposed by Hawkes in 1971, in which previous occurrences of events increase the chance for new events to occur.  In this process, immigrants arrive to the system following a Poisson process, then, each immigrant has the possibility to have offspring. At the same time, each new offspring individual has the possibility to give birth to further offspring. In this work, we present an extension to the original Hawkes process, but we consider that the arrival of immigrants is given by a Renewal process; the interarrival times are still independent, but they follow an arbitrary distribution. Existence is proved by exploiting the cluster structure of the process and we use martingale theory to prove a Law of Large Numbers. We give a conjecture for a functional Central Limit Theorem.

### Math-Fi seminar on 6 Spe.

2022.09.05 Mon up
• Date: 6 Sep. (Tue.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Dan Crisan (Imperial College London)
• Title: Classical and modern results in the theory and applications of stochastic filtering
• Abstract:
Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this talk is to give a historical account of the subject concentrating on the continuous-time framework. I will also present a recent application of filtering to the estimation of partially observed high dimensional fluid dynamics models. In particular, I will introduce a so-called particle filter that incorporates a nudging mechanism. The nudging procedure is used in the prediction step. In the absence of nudging, the particles have trajectories that are independent solutions of the model equations. The nudging presented here consists in adding a drift to the trajectories of the particles with the aim of maximising the likelihood of their positions given the observation data. This introduces a bias in the system that is corrected during the resampling step.  The methodology is tested on a two-layer quasi-geostrophic model for a beta-plane channel flow with O(10^6) degrees of freedom out of  which only a minute fraction are noisily observed.

The talk is based on the papers:

[1] D Crisan, The stochastic filtering problem: a brief historical account, Journal of Applied Probability 51 (A), 13-22
[2] C Cotter, D Crisan, D Holm, W Pan, I Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise,  Journal of Statistical Physics, 1-36, 2020.
[3] D Crisan, I Shevchenko, Particle filters with nudging, work in progress.

### Math-Fi seminar on 26 Jul.

2022.07.25 Mon up
• Date: 26 Jul. (Tue.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Kei Noba (The Institute of Statistical Mathematics)
• Title: Optimality of classical or periodic barrier strategies for Lévy processes
• Abstract:
We revisit the stochastic control problem in two cases with Lévy processes that minimize running and controlling costs. Existing studies have shown the optimality of classical or periodic barrier strategies when driven by Brownian motion or Lévy processes with one-sided jumps. Under the assumption that we can be controlled at any time or only at Poissonian dividend-decision times, we show the optimality of classical or periodic barrier strategies for a general class of Lévy processes.

### Math-Fi seminar on 14 Jul.

2022.07.13 Wed up
• Date: 14 Jul. (Thu.)
• Place: On the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Takuji Arai (Keio University)
• Title: Constrained optimal stopping under a regime-switching model
• Abstract:
We investigate an optimal stopping problem for the expected value of a discounted payoff on a regime-switching geometric Brownian motion under two constraints on the possible stopping times: only at exogenous random times and only during a specific regime. The main objectives are to show that an optimal stopping time exists as a threshold type under some boundary conditions and to derive expressions of the value functions and the optimal threshold. To this end, we solve the corresponding variational inequality and show that its solution coincides with the value functions. Some numerical results are also introduced. Furthermore, we investigate some asymptotic behaviors. This talk is based on joint work with Masahiko Takenaka.

### Math-Fi seminar on 7 Jul.

2022.07.06 Wed up
• Date: 7 Jul. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Pierre Bras (Sorbonne Université)
• Title: Asymptotics for the total variation distance between an SDE and its Euler-Maruyama scheme in small time
• Abstract:
We give bounds for the total variation distance between the law of an SDE and the law of its one-step Euler-Maruyama scheme as $t \to 0$. The case of the total variation is more complex to deal with than the classic case of Wasserstein ($L^p$) distances. We show that this distance is of order $t^{1/3}$, and more generally of order $t^{r/(2r+1)}$ for any $r \in \mathbb{N}$. Improving the bounds from $1/3$ to $r/(2r+1)$ relies on a weighted multi-level Richardson-Romberg extrapolation which consists in linear combination annealing the terms of a Taylor expansion, up to some order. This method was introduced for bias reduction in practical problems, but is used here for theoretical purposes.