arXiv subjects
Publications and source records attributed to Ali Fathi.
Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module such that the set of associated prime ideals of the quotient module $M/L$ is finite for all submodules $L$ of $M$. In this paper, it is shown that there is a finitely generated submodule $N$ of $M$ such that the set of associated primes of $M/N$ and the support of $M/N$ are equal.
A corporate bond trader in a typical sell side institution such as a bank provides liquidity to the market participants by buying/selling securities and maintaining an inventory. Upon receiving a request for a buy/sell price quote (RFQ), the trader provides a quote by adding a spread over a \textit{prevalent market price}. For illiquid bonds, the market price is harder to observe, and traders often resort to available benchmark bond prices (such as MarketAxess, Bloomberg, etc.). In \cite{Bergault2023ModelingLI}, the concept of \textit{Fair Transfer Price} for an illiquid corporate bond was introduced which is derived from an infinite horizon stochastic optimal control problem (for maximizing the trader's expected P\&L, regularized by the quadratic variation). In this paper, we consider the same optimization objective, however, we approach the estimation of an optimal bid-ask spread quoting strategy in a data driven manner and show that it can be learned using Reinforcement Learning. Furthermore, we perform extensive outcome analysis to examine the reasonableness of the trained agent's behavior.
Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\mathfrak a$, is non-zero. It is known that $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we give a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.
We consider two data-driven approaches to hedging, Reinforcement Learning and Deep Trajectory-based Stochastic Optimal Control, under a stepwise mean-variance objective. We compare their performance for a European call option in the presence of transaction costs under discrete trading schedules. We do this for a setting where stock prices follow Black-Scholes-Merton dynamics and the "book-keeping" price for the option is given by the Black-Scholes-Merton model with the same parameters. This simulated data setting provides a "sanitized" lab environment with simple enough features where we can conduct a detailed study of strengths, features, issues, and limitations of these two approaches. However, the formulation is model free and could allow any other setting with available book-keeping prices. We consider this study as a first step to develop, test, and validate autonomous hedging agents, and we provide blueprints for such efforts that address various concerns and requirements.
Let $R$ be a commutative Noetherian ring, $\mathfrak a$ a proper ideal of $R$ and $N$ a non-zero finitely generated $R$-module with $N\neq \mathfrak a N$. Let $d$ (respectively $c$) be the smallest (respectively greatest) non-negative integer $i$ such that the local cohomology $\operatorname{H}^i_{\mathfrak a}(N)$ is non-zero. In this paper, we provide sharp bounds under inclusion for the annihilators of the local cohomology modules $\operatorname{H}^d_{\mathfrak a}(N)$, $\operatorname{H}^c_{\mathfrak a}(N)$ and these annihilators are computed in certain cases. Also, we construct a counterexample to Lynch's conjecture.
Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of integers $i$ such that the generalized local homology $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is not finitely generated as an $\widehat{R}^{\mathfrak{a}}$-module, where $\widehat{R}^{\mathfrak{a}}$ denotes the $\mathfrak{a}$-adic completion of $R$. In particular, if $R$ is a complete semi-local ring, then $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is a finitely generated $\widehat{R}^{\mathfrak{a}}$-module for all non-negative integers $i$ if and only if $(0:_N\mathfrak{a}+\textrm{Ann}(M))$ has finite length.
This paper is the report of the problem proposed for the !Optimizer 2021 competition, and the solutions of the gold medalist team, i.e., the Panda team. The competition was held in two stages, the research and development stage and a two-week contest stage, consisting of five rounds, and seven teams succeeded in finishing both stages to the end. In this joint report of the winner team Panda and the problem design committee coordinated by Mojtaba Tefagh, we first explain each of the five rounds and then provide the solutions proposed by our team (Panda) to fulfill the required tasks in the fastest and most accurate way. Afterward, some preprocessing and data manipulating ideas used to enhance the algorithms would be presented. All codes are written in the Julia language, which showed a better performance than Python on optimization problems in our comparisons during the R&D stage, and are publicly available in the Github repository: https://github.com/Optimizer-Competition-Panda
Deep Reinforcement Learning (DRL) has become an appealing solution to algorithmic trading such as high frequency trading of stocks and cyptocurrencies. However, DRL have been shown to be susceptible to adversarial attacks. It follows that algorithmic trading DRL agents may also be compromised by such adversarial techniques, leading to policy manipulation. In this paper, we develop a threat model for deep trading policies, and propose two attack techniques for manipulating the performance of such policies at test-time. Furthermore, we demonstrate the effectiveness of the proposed attacks against benchmark and real-world DQN trading agents.
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a non-negative integer. Let $M$ and $N$ be two finitely generated $R$-modules. In certain cases, we give some bounds under inclusion for the annihilators of $\operatorname{Ext}^t_R(M, N)$ and $\operatorname{H}^t_{\mathfrak a}(M)$ in terms of minimal primary decomposition of the zero submodule of $M$ which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.
This paper is a note on new directions and methodologies for validation and explanation of Machine Learning (ML) models employed for retail credit scoring in finance. Our proposed framework draws motivation from the field of Artificial Intelligence (AI) security and adversarial ML where the need for certifying the performance of the ML algorithms in the face of their overwhelming complexity poses a need for rethinking the traditional notions of model architecture selection, sensitivity analysis and stress testing. Our point of view is that the phenomenon of adversarial perturbations when detached from the AI security domain, has purely algorithmic roots and fall within the scope of model risk assessment. We propose a model criticism and explanation framework based on adversarially generated counterfactual examples for tabular data. A counterfactual example to a given instance in this context is defined as a synthetically generated data point sampled from the estimated data distribution which is treated differently by a model. The counterfactual examples can be used to provide a black-box instance-level explanation of the model behaviour as well as studying the regions in the input space where the model performance deteriorates. Adversarial example generating algorithms are extensively studied in the image and natural language processing (NLP) domains. However, most financial data come in tabular format and naive application of the existing techniques on this class of datasets generates unrealistic samples. In this paper, we propose a counterfactual example generation method capable of handling tabular data including discrete and categorical variables. Our proposed algorithm uses a gradient-free optimization based on genetic algorithms and therefore is applicable to any classification model.
This paper studies an application of machine learning in extracting features from the historical market implied corporate bond yields. We consider an example of a hypothetical illiquid fixed income market. After choosing a surrogate liquid market, we apply the Denoising Autoencoder (DAE) algorithm to learn the features of the missing yield parameters from the historical data of the instruments traded in the chosen liquid market. The DAE algorithm is then challenged by two "point-in-time" inpainting algorithms taken from the image processing and computer vision domain. It is observed that, when tested on unobserved rate surfaces, the DAE algorithm exhibits superior performance thanks to the features it has learned from the historical shapes of yield curves.
This paper studies the application of machine learning in extracting the market implied features from historical risk neutral corporate bond yields. We consider the example of a hypothetical illiquid fixed income market. After choosing a surrogate liquid market, we apply the Denoising Autoencoder algorithm from the field of computer vision and pattern recognition to learn the features of the missing yield parameters from the historically implied data of the instruments traded in the chosen liquid market. The results of the trained machine learning algorithm are compared with the outputs of a point in- time 2 dimensional interpolation algorithm known as the Thin Plate Spline. Finally, the performances of the two algorithms are compared.
The spectral eta function for certain families of Dirac operators on noncommutative $3$-torus is considered and the regularity at zero is proved. By using variational techniques, we show that $\eta_{D}(0)$ is a conformal invariant. By studying the Laurent expansion at zero of $\text{TR} (|D|^{-z})$, the conformal invariance of $\zeta'_{|D|}(0)$ for noncommutative $3$-torus is proved. Finally, for the coupled Dirac operator, a local formula for the variation $\partial_A\eta_{D+A}(0)$ is derived which is the analogue of the so called induced Chern-Simons term in quantum field theory literature.
We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen's original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes' algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation $\delta_{w}\delta_{\bar{w}}\log\det(\Delta)$.
Let $\frak a$, $\frak b$ be ideals of a commutative Noetherian ring $R$ and let $M$, $N$ be finite $R$-modules. The concept of an $\frak a$-filter grade of $\frak b$ on $M$ is introduced and several characterizations and properties of this notion are given. Then, using the above characterizations, we obtain some results on generalized local cohomology modules $H^i_{\frak a}(M, N)$. In particular, first we determine the least integer $i$ for which $H^i_{\frak a}(M, N)$ is not Artinian. Then we prove that $H^i_{\frak a}(M, N)$ is Artinian for all $i\in\mathbb N_0$ if and only if $\dim{R}/({\frak a+Ann M+Ann N})=0$. Also, we establish the Nagel-Schenzel formula for generalized local cohomology modules. Finally, in a certain case, the set of attached primes of $H^i_{\frak a}(M, N)$ is determined and a comparison between this set and the set of attached primes of $H^i_{\frak a}(N)$ is given.