ABSTRACT In this paper, we focus on exploring an approximate solution for pricing arithmetic Asian option (AAO) in continuous form under the mixed fractional Brownian motion (MFBM). To achieve this, we use the AAO in discrete form and show that the error analysis for the approximation between discrete and continuous forms is of order . Since the discrete form follows the reciprocal gamma distribution for sufficiently large monitoring dates, we can extract the approximate analytical solution along with its sensitivity parameters like and , and show that the proposed solution is valid for a 95% confidence interval. Additionally, to enhance the accuracy of the option of AAO in continuous form, we use the reciprocal gamma distribution as well as the geometric Asian option (GAO) in discrete form as control variates. The results show that employing both control variates acts more efficiently than using each of them solely.

ABSTRACT A new system of delayed differential equations for tumor‐immune system interactions is proposed and studied. The system describes the interactions between tumor cells and the immune system at the most aggressive phase of cancer, where tumor cells have developed mechanisms from earlier stages to evade immune responses. The model incorporates adoptive cellular immunotherapy as the only effective treatment, and accounts for the complex influence of interleukins. To render the model more biologically realistic, it also includes a time delay representing the lag between the interaction of tumor cells and activated immune cells and the subsequent effectiveness of the immune response. These aspects, captured together, reflect the biological novelty of our approach. We first establish the well‐posedness of the system, proving the global existence of the solution and its positivity. Then, we focus on the equilibrium states of the system, their asymptotic stability, and how the delay affects the stability of the equilibrium states and the behavior of the solution. We investigate the global stability of a tumor equilibrium state with biological relevance, and estimate the length of the delay that preserves the stability of a tumor‐free equilibrium. These results ensure the mathematical consistency of the model and provide a solid theoretical basis for analyzing how delayed immune responses can influence the long‐term behavior of tumor progression. Finally, we conduct some numerical simulations of the delayed system to illustrate the theoretical analysis and investigate the impact of the delay. We interpret the results and explore their biological implications in the context of cancer progression.

ABSTRACT Asymptotic stability of dynamical systems with delays on time scales were investigated, which not only realizes the research of continuous and discrete problems within a unified framework, but also provides a new perspective and effective mathematical tools for studying hybrid systems that contain both continuous and discrete processes. Furthermore, combined with the improved criterion for the stability of dynamical systems on time scales, the event‐trigger‐based consensus for multi‐agent systems with time delays on time scales are explored. And it is proved that Zeno behavior does not exist. Moreover, The leader‐following security consensus problems of nonlinear multi‐agent systems with time delays on time scales under DoS attacks are studied. Numerical examples are provided to demonstrate the effectiveness of proposed criteria.

ABSTRACT The bifurcation of a fractional‐order neural network (FONN) with three different time delays is studied in this paper. The dynamical behaviors of the developed network with different conditions are firstly explored once time delay can be viewed as zero, and the corresponding bifurcation critical values are calculated. Then, the bifurcations of three systems with time delays are examined. By adopting Cramer's rule, one of the time delays is selected as a bifurcation parameter and more accurate bifurcation conditions are figured out. The notable highlight is to detect the solution to bifurcation problems compared with the transformation strategy with respect to time delays. Finally, numerical simulations are implemented to validate the correctness of the theoretical analysis.

ABSTRACT This paper studies an investment‐reinsurance contract between an insurer and a reinsurer with asymmetric bargaining power. We assume that the surplus of the insurer follows a jump‐diffusion process. To reduce the risk of claims, the insurer can purchase proportional reinsurance, with the reinsurance premium calculated based on the expected value principle. The surplus of the insurer and the reinsurer can be allocated to a financial market consisting of a risk‐free asset and a risky asset, respectively. The price processes of the insurer's and reinsurer's risky assets satisfy different square root factor processes. To consider the benefits of both the insurer and the reinsurer, the optimization problem is formulated as an asymmetric Nash bargaining game. To maximize the weighted product of the expected exponential utility of the terminal wealth of both parties, explicit expressions for the Pareto‐optimal strategy and the corresponding value function are derived by employing stochastic control techniques and the Hamilton‐Jacobi‐Bellman (HJB) equation. In addition, we provide equilibrium strategies under several special cases, including cases where only the insurer or the reinsurer is considered, as well as models under the CEV (Constant Elasticity of Variance) model and the Heston stochastic volatility model. Finally, numerical examples illustrate the impact of bargaining power on optimal strategy.

ABSTRACT A semi‐analytical calculation of the magnetic flux leakage owing to the presence of a spheroidal void inclusion in a ferromagnetic half‐space is presented. A decomposition strategy to a number of sub‐solutions that are associated with the different interfaces of the geometry, proposed in previous works, has been adopted. The fact that each sub‐solution is expressed in a conformal to the interface coordinate system results in a partially sparse linear system of equations, which are produced by mode‐matching at the interfaces, thus providing a very fast computation of the modal expansion coefficients and making the resulting scheme suitable for use with parametric inversion algorithms. The results of the method are compared against numerical simulations, carried out using a commercial finite elements package.

ABSTRACT This paper investigates the robust finite‐time control for uncertain discrete‐time singular systems with nonlinear repeated scalars and Markovian jump parameters. By employing the implicit function theorem and Lyapunov function approaches, sufficient conditions are derived to guarantee the regularity, causality, uniqueness of solutions in the neighborhood of the origin, singular stochastic finite‐time stabilization, and singular stochastic finite‐time stabilization for the addressed systems. According to the descriptor system approach, solvability criteria for these control problems are also established, which can be effectively converted into linear matrix inequalities. Additionally, the correctness and feasibility of the proposed method are demonstrated by numerical examples.

ABSTRACT This work presents an efficient numerical scheme for a class of third‐kind Volterra delay integral equations (VDIEs) and integro‐differential equations with proportional delays (DVIDEs), based on the Bernstein approximation method. By exploiting the structure of Bernstein polynomials, the problems are reformulated as tractable systems of algebraic equations. The proposed method is thoroughly explained, along with its application to equations and an estimation of the associated error. The developed approach demonstrates significant efficiency, stability, and minimal computational memory demands. We thoroughly confirm the method's accuracy and reliability by testing it with various numerical examples. Furthermore, the accuracy of the scheme is validated by comparing the numerical results with analytical solutions, as well as with the moving least squares and radial basis function collocation methods, demonstrating consistency with theoretical error estimates.

ABSTRACT The main objective of this paper is to consider a small time large deviation principle for the three dimensional stochastic planetary geostrophic equations of large‐scale ocean circulation driven by multiplicative noise, which involves not only the study of small noise, but also the investigation of the effect of the small, but highly nonlinear term. To obtain the small time large deviation principle, we will show that the probability that the solution stays outside an energy ball is exponentially small so that we can restrict the solution to a sufficiently large energy ball, the main idea is similar as in [15,21].

ABSTRACT T‐spherical fuzzy sets (TSFSs) provide a robust and flexible framework for modeling uncertainty by aggregating membership, nonmembership, and neutrality degrees using t‐norms. This study introduces a new family of set‐theoretic weighted similarity and distance measures specifically designed for TSFSs, which should pique curiosity about their innovative approach. This study demonstrates that numerous traditional measurements fail to satisfy fundamental axiomatic properties. Conversely, our suggested measurements demonstrate unwavering reliability and efficacy over a wide range of scenarios. To validate the practical utility of these measures, we integrate them into a deep learning classification framework. Finally, we apply the proposed similarity measurement to deep learning classification of 3D CT scans. It can improve classification accuracy by effectively handling uncertain features in medical imaging data.