arXiv subjects

Paweł M. Morkisz

Publications and source records attributed to Paweł M. Morkisz.

8 recordsLinked to original sources

arXiv2026

Error of randomized Milstein scheme for scalar SDEs with noisy information about coefficients and Wiener process

We investigate the strong approximation of scalar stochastic differential equations when the available standard information about the drift coefficient, the diffusion coefficient, the derivative of the diffusion coefficient, and the observed Wiener path is corrupted by noise. The precision of the drift, diffusion-information, and Wiener-path observations is described by three nonnegative parameters $δ_1,δ_2,δ_3$, where $δ_2$ controls both the noisy diffusion coefficient and the separate noisy derivative oracle required in the Milstein correction. We analyze a randomized Milstein scheme based only on this noisy information and prove, for $r\geq 2$, that its $L^r$-error is bounded by $C(n^{-\min\{γ_1+1/2,γ_2\}}+δ_1+δ_2+δ_3)$, where $n$ is the number of time steps and $γ_1,γ_2$ are the temporal Hölder exponents of the coefficients. We also prove a matching minimax lower bound in the randomized standard-information model considered in the paper. In particular, the Wiener-path contribution proportional to $δ_3$ is unavoidable, and the noisy randomized Milstein scheme is minimax order-optimal.

math.NA↗

arXiv2023

Deep learning-based estimation of time-dependent parameters in Markov models with application to nonlinear regression and SDEs

We present a novel deep learning method for estimating time-dependent parameters in Markov processes through discrete sampling. Departing from conventional machine learning, our approach reframes parameter approximation as an optimization problem using the maximum likelihood approach. Experimental validation focuses on parameter estimation in multivariate regression and stochastic differential equations (SDEs). Theoretical results show that the real solution is close to SDE with parameters approximated using our neural network-derived under specific conditions. Our work contributes to SDE-based model parameter estimation, offering a versatile tool for diverse fields.

stat.ML↗

arXiv2023

Euler scheme for approximation of solution of nonlinear ODEs under inexact information

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local Hölder continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.

math.NA↗

arXiv2023

On the randomized Euler algorithm under inexact information

This paper focuses on analyzing the error of the randomized Euler algorithm when only noisy information about the coefficients of the underlying stochastic differential equation (SDE) and the driving Wiener process is available. Two classes of disturbed Wiener process are considered, and the dependence of the algorithm's error on the regularity of the disturbing functions is investigated. The paper also presents results from numerical experiments to support the theoretical findings.

math.NA↗

arXiv2022

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function $f$. Namely, we assume that $f$ is globally of at most linear growth, satisfies globally one-side Lipschitz condition but it is only locally Hölder continuous. We provide a detailed error analysis of the Euler algorithm under such nonstandard regularity conditions. Moreover, we report results of numerical experiments.

math.NA↗

arXiv2020

Randomized Runge-Kutta method -- stability and convergence under inexact information

We deal with optimal approximation of solutions of ODEs under local Lipschitz condition and inexact discrete information about the right-hand side functions. We show that the randomized two-stage Runge-Kutta scheme is the optimal method among all randomized algorithms based on standard noisy information. We perform numerical experiments that confirm our theoretical findings. Moreover, for the optimal algorithm we rigorously investigate properties of regions of absolute stability.

math.NA↗

arXiv2019

Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information

We deal with pointwise approximation of solutions of scalar stochastic differential equations in the presence of informational noise about underlying drift and diffusion coefficients. We define a randomized derivative-free version of Milstein algorithm $\mathcal{\bar A}^{df-RM}_n$ and investigate its error. We also study lower bounds on the error of an arbitrary algorithm. It turns out that in some case the scheme $\mathcal{\bar A}^{df-RM}_n$ is the optimal one. Finally, in order to test the algorithm $\mathcal{\bar A}^{df-RM}_n$ in practice, we report performed numerical experiments.

math.NA↗

arXiv2018

Optimal approximation of stochastic integrals in analytic noise model

We study approximate stochastic Itô integration of processes belonging to a class of progressively measurable stochastic processes that are Hölder continuous in the $r$th mean. Inspired by increasingly popularity of computations with low precision (used on Graphics Processing Units -- GPUs and standard Computer Processing Units -- CPU for significant speedup), we introduce a suitable analytic noise model of standard noisy information about $X$ and $W$. In this model we show that the upper bounds on the error of the Riemann-Maruyama quadrature are proportional to $n^{-\varrho}+δ_1+δ_2$, where $n$ is a number of noisy evaluations of $X$ and $W$, $\varrho\in (0,1]$ is a Hölder exponent of $X$, and $δ_1,δ_2\geq 0$ are precision parameters for values of $X$ and $W$, respectively. Moreover, we show that the error of any algorithm based on at most $n$ noisy evaluations of $X$ and $W$ is at least $C(n^{-\varrho}+δ_1)$. Finally, we report numerical experiments performed on both CPU and GPU, that confirm our theoretical findings, together with some computational performance comparison between those two architectures.

math.NA↗