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1. Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

   https://doi.org/10.1051/cocv/2021027  

   Wang, Hanxiao; Yong, Jiongmin (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   Buttazzo, G.; Casas, E.; de Teresa, L.; Glowinski, R.; Leugering, G.; Trélat, E.; Zhang, X. (Ed.)

   An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value Z ( r , r ) of Z (⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented. 

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2. Quadrilateral Mesh Generation III : Optimizing Singularity Configuration Based on Abel-Jacobi Theory

   Zheng, Xiaopeng; Zhu, Yiming; Chen, Wei; Lei, Na; Luo, Zhongxuan; Gu, Xianfeng (April 2022, Computer methods in applied mechanics and engineering)

   null (Ed.)

   This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel-Jacobi theory of algebraic \textcolor{red}{curves}. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh. In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel-Jacobi map for a given divisor; optimize the divisor to satisfy the Abel-Jacobi condition by integer programming; compute \textcolor{red}{a} flat Riemannian metric with cone singularities at the divisor by Ricci flow; \textcolor{red}{isometrically} immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh. In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh. The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries. 

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3. A finite horizon optimal stochastic impulse control problem with a decision lag

   Li, C.; Yong, J. (April 2021, Dynamics of continuous discrete and impulsive systems)

   null (Ed.)

   This paper studies an optimal stochastic impulse control problem in a finite time horizon with a decision lag, by which we mean that after an impulse is made, a fixed number units of time has to be elapsed before the next impulse is allowed to be made. The continuity of the value function is proved. A suitable version of dynamic programming principle is established, which takes into account the dependence of state process on the elapsed time. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is derived, which exhibits some special feature of the problem. The value function of this optimal impulse control problem is characterized as the unique viscosity solution to the corresponding HJB equation. An optimal impulse control is constructed provided the value function is given. Moreover, a limiting case with the waiting time approaching 0 is discussed. 

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4. Extended Mean Field Type Control Theory: A Master Equation Approach With Some Applications

   Bensoussan, Alain; Kim, Joohyun; Yam, Sheung (July 2025, Journal of optimization theory and applications)

   The primary objective of this article is to present a general framework for users and applications of the master equation approach in extended mean field type control, for- mulated with a McKean-Vlasov stochastic differential equation that depends on the law of both the control and state variables. This control problem has recently gained significant attention and has been extensively studied at the level of the Bellman equa- tion. Here, we extend the analysis to the master equation and derive the corresponding Hamilton-Jacobi-Bellman equation. A key novelty of our approach is that we do not directly rely on the Fokker-Planck equation, which surprisingly leads to a significant simplification. We provide a concise theoretical presentation with proofs, as the stan- dard theory of stochastic control is not directly applicable. In the current work, the solution is constructed using an ansatz-based approach to dynamic programming via the master equation.We illustrate this method with a practical example. All proofs are presented in a self-contained manner. This paper offers a structured presentation of the extended mean field type control problem, serving as a valuable toolbox for users who are less focused on mathematical intricacies but seek a general framework for application. 

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5. Linear operators, the Hurwitz zeta function and Dirichlet L-functions

   https://doi.org/10.1016/j.jnt.2020.05.018  

   Bianco_Prado, Bernardo; Klinger-Logan, Kim (December 2020, Journal of Number Theory)

   At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other L-functions does converge. 

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