LIBOR market model with stochastic volatility
Lixin Wu Fan Zhang
Journal of Industrial & Management Optimization 2006, 2(2): 199-227 doi: 10.3934/jimo.2006.2.199
In this paper we extend the standard LIBOR market model to accommodate the pronounced phenomenon of implied volatility smiles/skews. We adopt a multiplicative stochastic factor to the volatility functions of all relevant forward rates. The stochastic factor follows a square-root diffusion process, and it can be correlated to the forward rates. For any swap rate, we derive an approximate process under its corresponding forward swap measure. By utilizing the analytical tractability of the approximate process, we develop a closed-form formula for swaptions in term of Fourier transforms. Extensive numerical tests are carried out to support the swaptions formula. The extended model captures the downward volatility skews by taking negative correlations between forward rates and their volatilities, which is consistent with empirical findings.
keywords: stochastic volatility LIBOR model Fast Fourier transform (FFT). square-root process swaptions
Some new results on multi-dimension Knapsack problem
Yuzhong Zhang Fan Zhang Maocheng Cai
Journal of Industrial & Management Optimization 2005, 1(3): 315-321 doi: 10.3934/jimo.2005.1.315
We claim a conclusion on Multi-Dimensional Knapsack Problem (MKP), which extends an important proposition by Dantzig firstly, then address to a special case of this problem, and constitute a polynomial algorithm, extending Zukerman et al's work.
keywords: integer programming Knapsack problem polynomial time algorithm approximation algorithm.
Modelling Population Growth with Delayed Nonlocal Reaction in 2-Dimensions
Dong Liang Jianhong Wu Fan Zhang
Mathematical Biosciences & Engineering 2005, 2(1): 111-132 doi: 10.3934/mbe.2005.2.111
In this paper, we consider the population growth of a single species living in a two-dimensional spatial domain. New reaction-diffusion equation models with delayed nonlocal reaction are developed in two-dimensional bounded domains combining different boundary conditions. The important feature of the models is the reflection of the joint effect of the diffusion dynamics and the nonlocal maturation delayed effect. We consider and analyze numerical solutions of the mature population dynamics with some well-known birth functions. In particular, we observe and study the occurrences of asymptotically stable steady state solutions and periodic waves for the two-dimensional problems with nonlocal delayed reaction. We also investigate numerically the effects of various parameters on the period, the peak and the shape of the periodic wave as well as the shape of the asymptotically stable steady state solution.
keywords: time delay numerical analysis. 2-d reaction-diffusion non-local reac- tion population growth
Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays
Shihe Xu Meng Bai Fangwei Zhang
Discrete & Continuous Dynamical Systems - B 2017 doi: 10.3934/dcdsb.2017213

In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case $c$ (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as $t\to ∞$. The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions.

keywords: Tumor growth free boundary problem global existence and uniqueness asymptotic behavior stability
On the limit quasi-shadowing property
Fang Zhang Yunhua Zhou
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2861-2879 doi: 10.3934/dcds.2017123

In this paper, we study the limit quasi-shadowing property for diffeomorphisms. We prove that any quasi-partially hyperbolic pseudoorbit $\{x_{i},n_{i}\}_{i∈ \mathbb{Z}}$ can be $\mathcal{L}^p$-, limit and asymptotic quasi-shadowed by a points sequence $\{y_{k}\}_{k∈ \mathbb{Z}}$. We also investigate the $\mathcal{L}^p$-, limit and asymptotic quasi-shadowing properties for partially hyperbolic diffeomorphisms which are dynamically coherent.

keywords: limit quasi-shadowing quasi-partially hyperbolic pseudoorbit partial hyperbolicity dynamical coherence
Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm
Steven D. Galbraith Ping Wang Fangguo Zhang
Advances in Mathematics of Communications 2017, 11(3): 453-469 doi: 10.3934/amc.2017038

The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step giant-step algorithm (BSGS) or Pollard rho. Montgomery's simultaneous modular inversion can also be used to speed up Pollard rho when running many walks in parallel. We generalize these ideas and exploit the fact that for any two elliptic curve points X and Y, we can efficiently get X-Y when we compute X+Y. We apply these ideas to speed up the baby-step giant-step algorithm. Compared to the previous methods, the new methods can achieve a significant speedup for computing elliptic curve discrete logarithms in small groups or small intervals.

Another contribution of our paper is to give an analysis of the average-case running time of Bernstein and Lange's "grumpy giants and a baby" algorithm, and also to consider this algorithm in the case of groups with efficient inversion.

Our conclusion is that, in the fully-optimised context, both the interleaved BSGS and grumpy-giants algorithms have superior average-case running time compared with Pollard rho. Furthermore, for the discrete logarithm problem in an interval, the interleaved BSGS algorithm is considerably faster than the Pollard kangaroo or Gaudry-Schost methods.

keywords: Baby-step giant-step algorithm elliptic curve discrete logarithm negation map

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