scholarly journals Revisiting Feller Diffusion: Derivation and Simulation

2019 ◽  
pp. 1-15
Author(s):  
Ranjiva Munasinghe ◽  
Leslie Kanthan ◽  
Pathum Kossinna
Keyword(s):  
Fourier Transform ◽  
Probability Density Function ◽  
Probability Density ◽  
Method Of Characteristics ◽  
Sample Path ◽  
Planck Equation ◽  
Simulation Algorithm ◽  
Quantitative Finance ◽  
Potential Applications ◽  
The Fourier Transform

We propose a simpler derivation of the probability density function of Feller Diffusion by using the Fourier Transform on the associated Fokker-Planck equation and then solving the resulting equation via the Method of Characteristics. We use the derived probability density to formulate an exact simulation algorithm whereby a sample path increment is drawn directly from the density. We then proceed to use the simulation to verify key statistical properties of the process such as the moments and the martingale property. The simulation is also used to confirm properties related to hitting time probabilities. We also mention potential applications of the simulation in the setting of quantitative finance.

The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Caibin Zeng ◽  
YangQuan Chen ◽  
Qigui Yang
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Anomalous Diffusion ◽  
Planck Equation ◽  
Viscosity Coefficient ◽  
Infinite Variance ◽  
Nonlinear Stochastic Differential Equations ◽  
The Fourier Transform ◽  
Fractional Ito Formula

AbstractThis paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

scholarly journals Fourier-Transform-Based Surface Measurement and Reconstruction of Human Face Using the Projection of Monochromatic Structured Light

Sensors ◽  
2021 ◽  
Vol 21 (7) ◽  
pp. 2529
Author(s):  
Bingquan Chen ◽  
Hongsheng Li ◽  
Jun Yue ◽  
Peng Shi
Keyword(s):  
Fourier Transform ◽  
Structured Light ◽  
Surface Measurement ◽  
Human Face ◽  
Image Processing Algorithm ◽  
Optical Filtering ◽  
Linearly Polarized ◽  
Potential Applications ◽  
Filtering Technique ◽  
The Fourier Transform

This work presents a new approach of surface measurement of human face via the combination of the projection of monochromatic structured light, the optical filtering technique, the polarization technique and the Fourier-transform-based image-processing algorithm. The theoretical analyses and experimental results carried out in this study showed that the monochromatic feature of projected fringe pattern generated using our designed laser-beam-based optical system ensures the use of optical filtering technique for removing the effect of background illumination; the linearly-polarized characteristic makes it possible to employ a polarizer for eliminating the noised signal contributed by multiply-scattered photons; and the high-contrast sinusoidal fringes of the projected structured light provide the condition for accurate reconstruction using one-shot measurement based on Fourier transform profilometry. The proposed method with the portable and stable optical setup may have potential applications of indoor medical scan of human face and outdoor facial recognition without strict requirements of a dark environment and a stable object being observed.

Shape-function-based non-uniform Fourier transforms for seismic modeling with irregular grids

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Fang Ouyang ◽  
Jianguo Zhao ◽  
Shikun Dai ◽  
Longwei Chen ◽  
Shangxu Wang
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Shape Function ◽  
High Accuracy ◽  
Irregular Sampling ◽  
Shape Functions ◽  
Lower Accuracy ◽  
Potential Applications ◽  
The Fourier Transform ◽  
Seismic Models

Multi-dimensional Fourier transform on an irregular grid is a useful tool for various seismic forward problems caused by complex media and wavefield distributions. Using a shape-function-based strategy, we develop four different algorithms for 1D and 2D non-uniform Fourier transforms, including two high-accuracy Fourier transforms (LSF-FT and QSF-FT) and two non-uniform fast Fourier transforms (LSF-NUFFT and QSF-NUFFT), respectively based on linear and quadratic shape functions. The main advantage of incorporating shape functions into the Fourier transform is that triangular elements can be used to mesh any complex wavefield distribution in the 2D case. These algorithms, therefore, can be used in conjunction with any irregular sampling strategies. The accuracy and efficiency of the four non-uniform Fourier transforms are investigated and compared by applying them in the frequency-domain seismic wave modeling. All algorithms are compared with exact solutions. Numerical tests show that the quadratic shape-function-based algorithms are more accurate than those based on linear shape function. Moreover, LSF-FT/QSF-FT exhibits higher accuracy but much slower calculation speed, while LSF-NUFFT/QSF-NUFFT is highly efficient but has lower accuracy at near-source points. In contrast, a combination of these algorithms by using QSF-FT at near-source points and LSF-NUFFT/QSF-NUFFT at others, achieves satisfactory efficiency and high accuracy at all points. Although our tests are restricted to seismic models, these improved non-uniform fast Fourier transform algorithms may also have potential applications in other geophysical problems, such as forward modeling in complex gravity and magnetic models.

Probability-Based Optimal Path Planning for Two-Wheeled Mobile Robots

2015 ◽  
Author(s):  
Jaeyeon Lee ◽  
Wooram Park
Keyword(s):  
Probability Density Function ◽  
Path Planning ◽  
Mobile Robots ◽  
Probability Density ◽  
Density Function ◽  
Deterministic Model ◽  
Optimal Path ◽  
Planck Equation ◽  
Practical Implementation ◽  
Wheeled Mobile Robots

Most dynamic systems show uncertainty in their behavior. Therefore, a deterministic model is not sufficient to predict the stochastic behavior of such systems. Alternatively, a stochastic model can be used for better analysis and simulation. By numerically integrating the stochastic differential equation or solving the Fokker-Planck equation, we can obtain a probability density function of the motion of the system. Based on this probability density function, the path-of-probability (POP) method for path planning has been developed and verified in simulation. However, there are rooms for more improvements and its practical implementation has not been performed yet. This paper concerns formulation, simulation and practical implementation of the path-of-probability for two-wheeled mobile robots. In this framework, we define a new cost function which measures the averaged targeting error using root-mean-square (RMS), and iteratively minimize it to find an optimal path with the lowest targeting error. The proposed algorithm is implemented and tested with a two-wheeled mobile robot for performance verification.

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