Maximal $$L^q$$-Regularity for Parabolic Hamilton–Jacobi Equations and Applications to Mean Field Games

In this paper we investigate maximal $$L^q$$ L q -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided.

Life Insurance and Subsistence Consumption with an Exponential Utility

In this paper, we derive an explicit solution to the utility maximization problem of an individual with mortality risk and subsistence consumption constraint. We adopt an exponential utility for the individual’s consumption and the martingale and duality method is employed. From the explicit solution, we exhibit how the mortality intensity and subsistence consumption constraint affect, separately and together, portfolio, consumption and life insurance purchase.

Exact boundary null controllability for a coupled system of plate equations with variable coefficients

<p style='text-indent:20px;'>This paper studies a coupled system of plate equations with variable coefficients, subject to the clamped boundary conditions. By the Riemannian geometry approach, the duality method, the multiplier technique and a compact perturbation method, we establish exact boundary null controllability of the system under verifiable assumptions.</p>

Portfolio Optimization with Asset-Liability Ratio Regulation Constraints

This paper considers both a top regulation bound and a bottom regulation bound imposed on the asset-liability ratio at the regulatory time T to reduce risks of abnormal high-speed growth of asset price within a short period of time (or high investment leverage), and to mitigate risks of low assets’ return (or a sharp fall). Applying the stochastic optimal control technique, a Hamilton–Jacobi–Bellman (HJB) equation is derived. Then, the effective investment strategy and the minimum variance are obtained explicitly by using the Lagrange duality method. Moreover, some numerical examples are provided to verify the effectiveness of our results.

APPLICATION OF GEOMETROTHERMODYNAMICS TO THE SYSTEM WITH ZERO SOUND DESCRIBED BY THE METHOD OF HOLOGRAPHIC DUALITY

In the framework of the method of geometrothermodynamics, in present work, we studied the properties of equilibrium manifold of the system with zero-sound predicted by the holographic duality method. The results are invariant under the Legendre transformations, i.e. independent of the choice of thermodynamic potential. For the systems under consideration, the corresponding metrics, determinants of metrics and scalar curvatures are calculated, and their properties are also described. Using the holographic approach, a new type of quantum liquid was discovered. The heat capacity of the liquid obtained in this work at low temperatures depends on the temperature ∼ T6. Entropy, which depends on temperature and baryom density, was taken as the thermodynamic potential. 3-dimensional obtained that clearly show at which values of thermodynamic variables scalar curvatures tend to infinity or to zero, which indicates possible phase transitions and possible compensation of interactions by quantum effects, respectively. It is shown that both variants of metrics in this case lead to the same conclusion regarding the location of possible phase transition lines in the considered holographic system with zero sound.