The present paper proposes a new framework for describing the stock price dynamics. In the traditional geometric Brownian motion model and its variants, volatility plays a vital role. The modern studies of asset pricing expand around volatility, trying to improve the understanding of it and remove the gap between the theory and market data. Unlike this, we propose to replace volatility with trading volume in stock pricing models. This pricing strategy is based on two hypotheses: A price-volume relation with an idea borrowed from fluid flows and a white-noise hypothesis for the price rate of change (ROC) that is verified via statistic testing on actual market data. The new framework can be easily adopted to local volume and stochastic volume models for the option pricing problem, which will point out a new possible direction for this central problem in quantitative finance.

In this paper, we consider signal recovery in both noiseless and noisy cases via weighted $ℓ_p \ (0 < p ≤ 1)$ minimization when some partial support information on the signals is available. The uniform sufficient condition based on restricted isometry property (RIP) of order $tk$ for any given constant $t>d$ ($d≥1$ is determined by the prior support information) guarantees the recovery of all $k$-sparse signals with partial support information. The new uniform RIP conditions extend the state-of-the-art results for weighted $ℓ_p$-minimization in the literature to a complete regime, which fill the gap for any given constant $t > 2d$ on the RIP parameter, and include the existing optimal conditions for the $ℓ_p$-minimization and the weighted $ℓ_1$-minimization as special cases.

We introduce a two-step numerical scheme for reconstructing the shape of a triangle by its Dirichlet spectrum. With the help of the asymptotic behavior of the heat trace, the first step is to determine the area, the perimeter, and the sum of the reciprocals of the angles of the triangle. The shape is then reconstructed, in the second step, by an application of the Newton’s iterative method or the Levenberg-Marquardt algorithm for solving a nonlinear system of equations on the angles. Numerically, we have used only finitely many eigenvalues to reconstruct the triangles. To our best knowledge, this is the first numerical simulation for the classical inverse spectrum problem in the plane. In addition, we give a counter example to show that, even if we have infinitely many eigenvalues, the shape of a quadrilateral may not be heard.

Body attitude coordination plays an important role in multi-airplane synchronization. In this paper, we study the flocking dynamics of a modified model for body attitude coordination. In contrast to the original body attitude alignment models in Degond et al. (Math. Models Methods Appl. Sci., 27(6): 1005–1049, 2017) and Ha et al. (Discrete Contin. Dyn. Syst., 40(4): 2037–2060, 2020), we introduce the velocity alignment term and assume the velocity of each agent is variable. More precisely, the adjoint coefficient will vary with the linked individual changes. In this case, synchronization would include the body attitude alignment and velocity alignment. It will generate a new collective behaviour which is called body attitude flocking. As results, we present two sufficient frameworks leading to the body attitude flocking by technique estimates. Also, we show the finite-in-time stability of the system which is valid on any finite time interval. In addition, we formally derive a kinetic model of the model for body attitude coordination using the BBGKY hierarchy. We prove the well-posedness of the kinetic equation and show a rigorous justification for the mean-field limit of our model. Moreover, we present a sufficient condition for asymptotic flocking in the kinetic model. Finally, we also give the numerical simulations to verify our analysis results.

In this paper, we propose a novel Lagrange multiplier approach, named zero-factor (ZF) approach to solve a series of gradient flow problems. The numerical schemes based on the new algorithm are unconditionally energy stable with the original energy and do not require any extra assumption conditions. We also prove that the ZF schemes with specific zero factors lead to the popular SAV-type method. To reduce the computation cost and improve the accuracy and consistency, we propose a zero-factor approach with relaxation, which we named the relaxed zero-factor (RZF) method, to design unconditional energy stable schemes for gradient flows. The RZF schemes can be proved to be unconditionally energy stable with respect to a modified energy that is closer to the original energy, and provide a very simple calculation process. The variation of the introduced zero factor is highly consistent with the nonlinear free energy which implies that the introduced ZF method is a very efficient way to capture the sharp dissipation of nonlinear free energy. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

This paper considers the case of a firm’s dynamic pricing problem for a nonperishable product experiencing surging demand caused by rare events modelled by a marked point process. The firm aims to maximize its running revenue by selecting an optimal price process for the product until its inventory is depleted. Using the dynamic program and inspired by the viscosity solution technique, we solve the resulting integro-differential Hamilton-Jacobi-Bellman (HJB) equation and prove that the value function is its unique classical solution. We also establish structural properties for our problem and find that the optimal price always decreases with initial inventory level in the absence of surging demand. However, with surging demand, we find that the optimal price could increase rather than decrease at the initial inventory level.

In the previous paper [CSIAM Trans. Appl. Math. 2 (2021), 1–55], the authors proposed a theoretical framework for the analysis of RNA velocity, which is a promising concept in scRNA-seq data analysis to reveal the cell state-transition dynamical processes underlying snapshot data. The current paper is devoted to the algorithmic study of some key components in RNA velocity workflow. Four important points are addressed in this paper: (1) We construct a rational time-scale fixation method which can determine the global gene-shared latent time for cells. (2) We present an uncertainty quantification strategy for the inferred parameters obtained through the EM algorithm. (3) We establish the optimal criterion for the choice of velocity kernel bandwidth with respect to the sample size in the downstream analysis and discuss its implications. (4) We propose a temporal distance estimation approach between two cell clusters along the cellular development path. Some illustrative numerical tests are also carried out to verify our analysis. These results are intended to provide tools and insights in further development of RNA velocity type methods in the future.