## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

CPAA

We consider a class of nearly integrable
Hamiltonian systems with Hamiltonian being
$H(\theta,I,u,v)=h(I)+\frac{1}{2}\sum_{j=1}^{m}\Omega_j(u_j^2-v_j^2)+f(\theta,I,u,v)$. By introducing external parameter and KAM methods,
we prove that, if the frequency
mapping has nonzero Brouwer topological degree at some Diophantine
frequency, the hyperbolic invariant torus with this frequency
persists under small perturbations.

DCDS

We consider the valuation of a block of perpetual ESOs and the optimal
exercise decision for an employee endowed with them and with trading restrictions.
A fluid model is proposed to characterize the exercise process.
The objective is to maximize the overall discount
returns for the employee through exercising the options over time.
The optimal value function is defined as the grant-date fair value of the block of options, and is then
shown by the dynamic programming principle to be a continuous constrained viscosity solution
to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully nonlinear second order elliptic partial differential equation (PDE)
in the plane.
We prove the comparison principle and the uniqueness.
The numerical simulation is discussed and the corresponding optimal decision turns out to be a threshold-style strategy. These results provide an appropriate method to estimate the cost of the ESOs for the company and
also offer favorable suggestions on selecting right moments
to exercise the options over time for the employee.

## Year of publication

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