Weak existence of the squared Bessel process and CIR process with skew reflection on a deterministic time dependent curve

由布朗运动驱动的Cox-Ingersoll-Ross过程与平方Bessel过程：相互作用与相变

Driven by Brownian motion Cox-Ingersoll-Ross and squared Bessel processes: interaction and phase transition

本文研究了由布朗运动驱动的CIR过程与Bessel过程的异同，证明平方Bessel过程是CIR过程的相变并探讨了两者的随机稳定性差异。

Squared Bessel processes of positive and negative dimension embedded in Brownian local times

The Ray--Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + μ\ell$ of a reflecting Brownian motion $|B|$ by a multiple $μ$ of its local time process $\ell$ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and negative. Here, we embed squared Bessel processes of all real dimensions directly in the local time process of $B…

Cox-Ingersoll-Ross 过程与 Bessel 过程熵作为时间及相关参数函数的研究

Entropies of Cox-Ingersoll-Ross and Bessel processes as functions of time and of related parameters

本文研究 CIR 过程与平方 Bessel 过程各类熵的长时间渐近行为，基于非中心卡方分布推导熵存在条件，并证明 CIR 熵收敛到平稳分布、平方 Bessel 熵的发散或有限性及两过程熵的收敛关系。

A limit theorem for singular stochastic differential equations

We study the weak limits of solutions to SDEs \[dX_n(t)=a_n\bigl(X_n(t)\bigr)\,dt+dW(t),\] where the sequence $\{a_n\}$ converges in some sense to $(c_- 1\mkern-4.5mu\mathrm{l}_{x<0}+c_+ 1\mkern-4.5mu\mathrm{l}_{x>0})/x+γδ_0$. Here $δ_0$ is the Dirac delta function concentrated at zero. A limit of $\{X_n\}$ may be a Bessel process, a skew Bessel process, or a mixture of Bessel processes.

平方Bessel过程在非线性期望下

Squared Bessel processes under nonlinear expectation

本文研究了在非线性期望下平方G-Bessel过程的定义、路径性质及拉普拉斯变换的上下界，并证明其时间-空间变换后为G'-CIR过程。

Time and place of the maximum for one-dimensional diffusion bridges and meanders

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and ge…

Bessel processes and hyperbolic Brownian motions stopped at different random times

Iterated Bessel processes R^γ(t), t>0, γ>0 and their counterparts on hyperbolic spaces, i.e. hyperbolic Brownian motions B^{hp}(t), t>0 are examined and their probability laws derived. The higher-order partial differential equations governing the distributions of I_R(t)=_1R^γ(_2R^γ(t)), t>0 and J_R(t) =_1R^γ(|_2R^γ(t)|^2), t>0 are obtained and discussed. Processes of the form R^γ(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are examined and numerous probability laws derived, including the Student law, the arcsin laws (also their asymmetric versions), the Lamperti distribution of the r…

Cox-Ingersoll-Ross过程和具有可访问边界的Bessel过程的强收敛性及时间正则性

Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries

本文研究了CIR过程和Bessel过程在可访问边界下的强收敛性及时间正则性，提出了一种隐式平方根欧拉方法，并证明其在L^p距离下的强收敛性。

Exact Simulation of Bessel Diffusions

We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and/or change of measure. All these diffusions are broadly used in mathematical finance for modelling asset prices, market indices, and interest rates. We show how the probability distributions of a squared Bessel bridge and a squared Bessel process with or withou…

Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes

The Bessel process with parameter $D>1$ and the Dyson model of interacting Brownian motions with coupling constant $β>0$ are extended to the processes in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES$^{(D)}$, and the elliptic Dyson model, eDYS$^{(β)}$, respectively. Both are realized on the circumference of a circle $[0, 2 πr)$ with radius $r >0$ as temporally inhomogeneous processes defined in a finite time interval $[0, t_*), t_* < \infty$. Transformations of them to Schrö…

平方贝塞尔过程的直接反演法及其应用

Direct Inversion for Chi-Square Sampling, the Squared Bessel Process, and Applications

提出基于二维切比雪夫展开的直接反演方法模拟平方贝塞尔过程及非中心卡方变量，解决了小自由度计算难题，并高效应用于CIR过程模拟。