Time-Changed Bessel Processes and Credit Risk

Tilted stable subordinators, Gamma time changes and Occupation Time of rays by Bessel Spiders

We exhibit, in the form of some identities in law, some connections between tilted stable subordinators, time-changed by independent Gamma processes and the occupation times of Bessel spiders, or their bridges. These identities in law are then explained thanks to excursion theory.

带常数漂移的分数阶 Bessel 过程：谱分析与排队应用

Fractional Bessel Process with Constant Drift: Spectral Analysis and Queueing Applications

本文通过稳定从属过程逆过程的时间变换构造带常数负漂移的分数阶 Bessel 过程，推导其转移密度的显式谱表示及平稳分布、相关性结构等解析性质，并在排队论中给出具体应用。

Explicit construction of a dynamic Bessel bridge of dimension 3

Given a deterministically time-changed Brownian motion $Z$ starting from 1, whose time-change $V(t)$ satisfies $V(t) > t$ for all $t > 0$, we perform an explicit construction of a process $X$ which is Brownian motion in its own filtration and that hits zero for the first time at $V(τ)$, where $τ:= \inf\{t>0: Z_t =0\}$. We also provide the semimartingale decomposition of $X$ under the filtration jointly generated by $X$ and $Z$. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process $X$ may be viewed as the analogue of a 3-dimension…

具有随机漂移和不同布朗时钟的广义计数过程

Generalized Counting Process with Random Drift and Different Brownian Clocks

本文引入了具有确定性和随机漂移的广义计数过程（GCP），推导了其概率律和分数阶微分方程，并研究了在多种布朗时钟及不完全伽马函数相关从属过程下的时间变更GCP及其分数阶积分。

Optimizing a variable-rate diffusion to hit an infinitesimal target at a set time

I consider a stochastic optimization problem for a time-changed Bessel process whose diffusion rate is constrained to be between two positive values $r_{1}<r_{2}$. The problem is to find an optimal adapted strategy for the choice of diffusion rate in order to maximize the chance of hitting an infinitesimal region around the origin at a set time in the future. More precisely, the parameter associated with "the chance of hitting the origin" is the exponent for a singularity induced at the origin of the final time probability density. I show that the optimal exponent solves a transcendental equat…

Large volatility-stabilized markets

We investigate the behavior of systems of interacting diffusion processes, known as volatility-stabilized market models in the mathematical finance literature, when the number of diffusions tends to infinity. We show that, after an appropriate rescaling of the time parameter, the empirical measure of the system converges to the solution of a degenerate parabolic partial differential equation. A stochastic representation of the latter in terms of one-dimensional distributions of a time-changed squared Bessel process allows us to give an explicit description of the limit.