The gap distribution of directions in some Schottky groups
Xin Zhang
Journal of Modern Dynamics 2017, 11(1): 477-499 doi: 10.3934/jmd.2017019

We prove the existence and some properties of the limiting gap distribution for the directions of some Schottky group orbits in the Poincaré disk. A key feature is that the fundamental domains for these groups have infinite area.

keywords: Fuchsian groups of the second kind Patterson-Sullivan measure gap distribution
Robust optimal investment and reinsurance of an insurer under Jump-diffusion models
Xin Zhang Hui Meng Jie Xiong Yang Shen
Mathematical Control & Related Fields 2018, 8(0): 1-18 doi: 10.3934/mcrf.2019003

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

keywords: Robust control proportional reinsurance stochastic differential game HJBI equation jump-diffusion model
Subspace trust-region algorithm with conic model for unconstrained optimization
Xin Zhang Jie Wen Qin Ni
Numerical Algebra, Control & Optimization 2013, 3(2): 223-234 doi: 10.3934/naco.2013.3.223
In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.
keywords: conic model trust region method Unconstrained optimization subspace method global convergence.

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