Modern Stochastics: Theory and Applications

We introduce a dynamic cutting approach for the numerical approximation of Lévy-driven stochastic differential equations. The key idea is to remove small jumps according to a time-dependent threshold, so that the retained jumps form a time-inhomogeneous compound Poisson process. We derive ${L^{p}}$-strong convergence rates for the adjusted Euler scheme. Numerical experiments at matched computational cost compare the dynamic cutting scheme with the classical Asmussen–Rosiński truncation and demonstrate consistently smaller strong errors when the time-dependent jump coefficient has a singularity at $t=0$.

Let ${B^{H}}$ be a fractional Brownian motion with Hurst index $\frac{1}{2}\lt H\lt 1$. In this paper, we consider the time fractional functional differential equation of the form where $\frac{3}{2}-H\lt \gamma \lt 1$, ${^{C}}{D_{t}^{\gamma }}$ denotes the Caputo derivative, and ${x_{t}}\in {\mathcal{C}_{r}}=\mathcal{C}([-r,0],\mathbb{R})$ with ${x_{t}}(u)=x(t+u)$, $u\in [-r,0]$. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.

Multidimensional generalized backward stochastic differential equations (GBSDEs) are studied within a general filtration that supports a Brownian motion under weak assumptions on the associated data. The existence and uniqueness of solutions in ${\mathbb{L}^{p}}$ for $p\in (1,2)$ are established. The results apply to generators that are stochastic monotone in the y-variable, stochastic Lipschitz in the z-variable, and satisfy a general stochastic linear growth condition.

In this paper, new closed form formulae for moments of the (generalized) Student’s t-distribution are derived in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of the Student’s t-distribution can be written in terms of the familiar Gamma function, Kummer’s confluent hypergeometric function, and the hypergeometric function. This work aims to provide a concise and unified treatment of the moments for this important distribution.