Vulnerable European and American Options in a Market Model with Optional Hazard Process

We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for an vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability

Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). We work with a generic filtration \FF for which the martingale representation property is assumed to hold with respect to a square-integrable martingale $M$ and the goal of this work is of twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver $A$. Second, we study extended penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a particular optimal stopping problem or a Dynkin game in which the set of admissible exercise time is constrained to the right support of the measure generated by $A$

Pairs Trading: An Optimal Selling Rule with Constraints

The focus of this paper is on identifying the most effective selling strategy for pairs trading of stocks. In pairs trading, a long position is held in one stock while a short position is held in another. The goal is to determine the optimal time to sell the long position and repurchase the short position in order to close the pairs position. The paper presents an optimal pairs-trading selling rule with trading constraints. In particular, the underlying stock prices evolve according to a two dimensional geometric Brownian motion and the trading permission process is given in terms of a two-state {trading allowed, trading not allowed} Markov chain. It is shown that the optimal policy can be determined by a threshold curve which is obtained by solving the associated HJB equations (quasi-variational inequalities). A closed form solution is obtained. A verification theorem is provided. Numerical experiments are also reported to demonstrate the optimal policies and value functions

Navigational Drift Analysis for Visual Odometry

Visual odometry estimates a robot's ego-motion with cameras installed on itself. With the advantages brought by camera being a sensor, visual odometry has been widely adopted in robotics and navigation fields. Drift (or error accumulation) from relative motion concatenation is an intrinsic problem of visual odometry in long-range navigation, as visual odometry is a sensor based on relative measurements. General error analysis using ``mean'' and ``covariance'' of positional error in each axis is not fully capable to describe the behavior of drift. Moreover, no theoretic drift analysis is available for performance evaluation and algorithms comparison. Drift distribution is established in the paper, as a function of the covariance matrix from positional error propagation model. To validate the drift model, experiment with a specific setting is conducted

TrTr: A Versatile Pre-Trained Large Traffic Model based on Transformer for Capturing Trajectory Diversity in Vehicle Population

Understanding trajectory diversity is a fundamental aspect of addressing practical traffic tasks. However, capturing the diversity of trajectories presents challenges, particularly with traditional machine learning and recurrent neural networks due to the requirement of large-scale parameters. The emerging Transformer technology, renowned for its parallel computation capabilities enabling the utilization of models with hundreds of millions of parameters, offers a promising solution. In this study, we apply the Transformer architecture to traffic tasks, aiming to learn the diversity of trajectories within vehicle populations. We analyze the Transformer's attention mechanism and its adaptability to the goals of traffic tasks, and subsequently, design specific pre-training tasks. To achieve this, we create a data structure tailored to the attention mechanism and introduce a set of noises that correspond to spatio-temporal demands, which are incorporated into the structured data during the pre-training process. The designed pre-training model demonstrates excellent performance in capturing the spatial distribution of the vehicle population, with no instances of vehicle overlap and an RMSE of 0.6059 when compared to the ground truth values. In the context of time series prediction, approximately 95% of the predicted trajectories' speeds closely align with the true speeds, within a deviation of 7.5144m/s. Furthermore, in the stability test, the model exhibits robustness by continuously predicting a time series ten times longer than the input sequence, delivering smooth trajectories and showcasing diverse driving behaviors. The pre-trained model also provides a good basis for downstream fine-tuning tasks. The number of parameters of our model is over 50 million.Comment: 16 pages, 6 figures, under reviewed by Transportation Research Board Annual Meeting, work in updat

Early Screening of Children With Autism Spectrum Disorder Based on Electroencephalogram Signal Feature Selection With L1-Norm Regularization

Early screening is vital and helpful for implementing intensive intervention and rehabilitation therapy for children with autism spectrum disorder (ASD). Research has shown that electroencephalogram (EEG) signals can reflect abnormal brain function of children with ASD, and screening with EEG signals has the characteristics of good real-time performance and high sensitivity. However, the existing EEG screening algorithms mostly focus on the data analysis in the resting state, and the extracted EEG features have some disadvantages such as weak representation capacity and information redundancy. In this study, we utilized the event-related potential (ERP) technique to acquire the EEG data of the subjects under positive and negative emotional stimulation and proposed an EEG Feature Selection Algorithm based on L1-norm regularization to perform screening of autism. The proposed EEG Feature Selection Algorithm includes the following steps: (1) extracting 20 EEG features from the raw data, (2) classification with support vector machine, (3) selecting appropriate EEG feature with L1-norm regularization according to the classification performance. The experimental results show that the accuracy for screening of children with ASD can reach 93.8% and 87.5% under positive and negative emotional stimulation and the proposed algorithm can effectively eliminate redundant features and improve screening accuracy