JIMO
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
MBE
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
JIMO
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.

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