Based on stochastic curvilinear integrals in the Cairoli-Walsh sense and in the Itô-Udrişte sense, we develop an original theory regarding the multitime stochastic differential systems. The first group of the original results refer to the complete integrable stochastic differential systems, the path independent stochastic curvilinear integral, the Itô-Udrişte stochastic calculus rules, examples of path independent processes, and volumetric processes. The second group of original results include the multitime Itô-Udrişte product formula, first stochastic integrals and adjoint multitime stochastic Pfaff systems. Thirdly, we formulate and we prove a multitime maximum principle for optimal control problems based on stochastic curvilinear integral actions subject to multitime Itô-Udrişte process constraints. Our theory requires the Lagrangian and the Hamiltonian as stochastic 1-forms.

Multitime Hamilton-Jacobi-Bellman PDE, Divergence, Multitime dynamic programming, Multitime maximum principle,

The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.

The subject of this paper is to analyse the Mathematical Principia of Economic 3D Black Holes in Roegenian economics. In detail, we study two main problems: (i) mathematical origin of economic 3D black holes; and (ii) entropy and internal political stability depending on national income and the total investment, for economic Reissner–Nordström (RN) 3D black hole. To solve these problems, it was necessary to jump from macroeconomic side to microeconomic side (a substantial approach as they are so different), to complete the thermodynamics–economics dictionary with new entities, and to introduce the flow between two macroeconomic systems. The main contribution is about introducing and studying the Schwarzschild-type metric on an economic 4D system, together with Rindler coordinates, Einstein 4D partial differential equations (PDEs), and economic RN 3D black holes. In addition, we introduce some economic Ricci type flows or waves, for further research.

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