Related papers: Optional splitting formula in a progressively enla…

We model the price of a stock via a Lang\'{e}vin equation with multi-dimensional fluctuations coupled in the price and in time. We generalize previous models in that we assume that the fluctuations conditioned on the time step are compound…

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ in $\Omega\times(0,\infty)$, $u=0$ in…

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable…

This paper proposes a new estimation procedure for the ambiguity function of a non-stationary time series. The stochastic properties of the empirical ambiguity function calculated from a single sample in time are derived. Different…

We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then…

There has been considerable recent study in "sub-diffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. There are many ways to look at this newer approach and one…

We examine the continuous-time counterpart of mirror descent, namely mirror flow, on classification problems which are linearly separable. Such problems are minimised `at infinity' and have many possible solutions; we study which solution…

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic…

As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear…

A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq…

Let $\mathbb{F}\subset \mathbb{G}$ be two filtrations and $S$ be a $\mathbb{F}$ semimartingale possessing a $\mathbb{F}$ local martingale deflator. Consider $\tau$ a $\mathbb{G}$ stopping time. We study the problem whether $S^{\tau-}$ or…

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a…

In this paper, we consider option pricing in a framework of the fractional Heston-type model with $H>1/2$. As it is impossible to obtain an explicit formula for the expectation $\mathbb E f(S_T)$ in this case, where $S_T$ is the asset price…

A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length $\Delta T$ and a history part, where the local part is…

We propose and analyze a generalized splitting method to sample approximately from a distribution conditional on the occurrence of a rare event. This has important applications in a variety of contexts in operations research, engineering,…

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu,…

In this article, a three-time levels compact scheme is proposed to solve the partial integro-differential equation governing the option prices under jump-diffusion models. In the proposed compact scheme, the second derivative approximation…

We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance…

Sample splitting is widely used in statistical applications, including classically in classification and more recently for inference post model selection. Motivating by problems in the study of diet, physical activity, and health, we…

Consider $K$ processes, each generating a sequence of identical and independent random variables. The probability measures of these processes have random parameters that must be estimated. Specifically, they share a parameter $\theta$…