There has been a renewed interest in exponential concentration inequalities for stochastic processes in probability and statistics over the last three decades. De la Peña established a nice exponential inequality for a discrete time locally square integrable martingale. In this paper, we obtain de la Peña’s inequalities for a stochastic integral of multivariate point processes. The proof is primarily based on Doléans–Dade exponential formula and the optional stopping theorem. As an application, we obtain an exponential inequality for block counting process in $\Lambda -$coalescent.

This article studies the theoretical properties of the numerical scheme for backward stochastic differential equations, extending the relevant results of Briand et al. with more general assumptions. To be more precise, the Brown motion will be approximated using the sum of a sequence of martingale differences or a sequence of i.i.d. Gaussian variables instead of the i.i.d. Bernoulli sequence. We cope with an adaptation problem of $Y^n$ by defining a new process ${\widehat{Y}}^n$; then, we can obtain the Donsker-type theorem for numerical solutions using a similar method to Briand et al.

伊藤积分是随机分析和金融数学中的基础性内容之一。为了在实际金融研究中使用伊藤积分 ，将理论上的伊藤积分离散化是必不可少的步骤。本文考虑在随机时间对伊藤积分进行离散化，研究离散化后产生的误差的稳定收敛. 在一定的条件下, 我们得到伊藤积分离散化误差的稳定收敛极限是一个新的伊藤积分.