Paweł Przybyłowicz

Andrzej Kałuża, Paweł M. Morkisz, Paweł Przybyłowicz

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.

Paweł Przybyłowicz

The paper deals with strong global approximation of SDEs driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.

Andrzej Kałuża, Paweł M. Morkisz, Bartłomiej Mulewicz et al.

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.