被积函数不连续时由多分形布朗运动驱动的积分的离散化

Tempered Fractional Multistable Motion and Tempered Multifractional Stable Motion

This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it tempered multifractional stable motion}. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise Hölder exponent, Hölder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These proc…

Wavelet series representation for multifractional multistable Riemann-Liouville process

The main goal of this paper is to construct a wavelet-type random series representation for a random field $X$, defined by a multistable stochastic integral, which generates a multifractional multistable Riemann-Liouville (mmRL) process $Y$. Such a representation provides, among other things, an efficient method of simulation of paths of $Y$. In order to obtain it, we expand in the Haar basis the integrand associated with $X$ and we use some fundamental properties of multistable stochastic integrals. Then, thanks to the Abel's summation rule and the Doob's maximal inequality for discrete subma…

On sharp rate of convergence for discretisation of integrals driven by fractional Brownian motions and related processes with discontinuous integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using mu…

关于带间断被积函数和局部时间的分数布朗运动随机积分离散化误差

Discretisation error for stochastic integrals with respect to the fractional Brownian motion with discontinuous integrands and local times

本文研究分数布朗运动随机积分的等距黎曼近似误差，证明归一化误差在L²拓扑下收敛到局部时间泛函，并给出收敛速率，补充了近期进展并提供局部时间模拟新方法。

Conditional Monte Carlo: A Change-of-Variables Approach

Conditional Monte Carlo (CMC) has been widely used for sensitivity estimation with discontinuous integrands as a standard simulation technique. A major limitation of using CMC in this context is that finding conditioning variables to ensure continuity and tractability of the resulting conditional expectation is often problem dependent and may be difficult. In this paper, we attempt to circumvent this difficulty by proposing a change-of-variables approach to CMC, leading to efficient sensitivity estimators under mild conditions that are satisfied by a wide class of discontinuous integrands. The…

随机淬火蒙特卡洛与重要性采样用于超快速增长函数及其在金融中的应用

Randomized Quasi-Monte Carlo and Importance Sampling for Super-Fast Growing Functions with Applications to Finance

本文提出了一种结合RQMC与重要性采样的方法，用于处理超快速增长函数的金融衍生品定价问题，通过轻尾条件和预积分技术提升了计算效率和收敛性。