arXiv subjects

Verena Schwarz

Publications and source records attributed to Verena Schwarz.

7 recordsLinked to original sources

arXiv2026

Approximation of the Lévy-driven stochastic heat equation on the sphere

The stochastic heat equation on the sphere driven by an additive square-integra\-ble Lévy process is approximated by a spectral method in space and forward and backward Euler--Maruyama schemes in time. New regularity results are proven for its solution. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. For a given regularity of the initial condition and two different settings of regularity for the driving noise, strong convergence rates for the spectral approximation and for the Euler--Maruyama methods are proven. Moreover, weak rates of up to twice the strong rates are shown. Numerical simulations confirm the theoretical results.

math.PR↗

arXiv2026

Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift

We present an adaptive approximation scheme for jump-diffusion SDEs with discontinuous drift and (possibly) degenerate diffusion. This transformation-based doubly-adaptive quasi-Milstein scheme is the first scheme that has strong convergence rate $1$ in terms of the number of evaluations of the driving noise processes in $L^p$ for $p\in[1,\infty)$. To obtain our result, we prove that under slightly stronger assumptions, which are still weaker than those in the existing literature, a related doubly-adaptive quasi-Milstein scheme has convergence order $1$. This scheme is doubly-adaptive in the sense that it is jump-adapted, i.e. all jump times of the Poisson noise are grid points, and it includes an adaptive step-size strategy to account for the discontinuities of the drift.

math.NA↗

arXiv2025

A Skorohod measurable universal functional representation of solutions to semimartingale SDEs

In this paper we show the existence of a universal Skorohod measurable functional representation for a large class of semimartingale-driven stochastic differential equations. For this we prove that paths of the strong solutions of stochastic differential equations can be written as measurable functions of the paths of their driving processes into the space of all càdlàg functions equipped with the Borel sigma-field generated by all open sets with respect to the Skorohod metric. This result can be applied to calculate Malliavin derivatives for SDEs driven by pure-jump Lévy processes with drift.

math.PR↗

arXiv2025

Milstein-type schemes for McKean-Vlasov SDEs driven by Brownian motion and Poisson random measure (with super-linear coefficients)

In this work, we present a general Milstein-type scheme for McKean-Vlasov stochastic differential equations (SDEs) driven by Brownian motion and Poisson random measure and the associated system of interacting particles where drift, diffusion and jump coefficients may grow super-linearly in the state variable and linearly in the measure component. The strong rate of $\mathcal{L}^2$-convergence of the proposed scheme is shown to be arbitrarily close to one under appropriate regularity assumptions on the coefficients. For the derivation of the Milstein scheme and to show its strong rate of convergence, we provide an Itô formula for the interacting particle system connected with the McKean-Vlasov SDE driven by Brownian motion and Poisson random measure. Moreover, we use the notion of Lions derivative to examine our results. The two-fold challenges arising due to the presence of the empirical measure and super-linearity of the jump coefficient are resolved by identifying and exploiting an appropriate coercivity-type condition.

math.PR↗

arXiv2023

A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient

We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme we prove $L^p$-convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than anything known before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4 - in a special case even order 1. Order 3/4 is conjectured to be optimal.

math.NA↗

arXiv2023

Randomized Milstein algorithm for approximation of solutions of jump-diffusion SDEs

We investigate the error of the randomized Milstein algorithm for solving scalar jump-diffusion stochastic differential equations. We provide a complete error analysis under substantially weaker assumptions than known in the literature. In case the jump-commutativity condition is satisfied, we prove optimality of the randomized Milstein algorithm by proving a matching lower bound. Moreover, we give some insight into the multidimensional case by investigating the optimal convergence rate for the approximation of jump-diffusion type Lévys' areas. Finally, we report numerical experiments that support our theoretical findings.

math.NA↗

arXiv2023

Lower error bounds and optimality of approximation for jump-diffusion SDEs with discontinuous drift

In this note we prove sharp lower error bounds for numerical methods for jump-diffusion stochastic differential equations (SDEs) with discontinuous drift. We study the approximation of jump-diffusion SDEs with non-adaptive as well as jump-adapted approximation schemes and provide lower error bounds of order $3/4$ for both classes of approximation schemes. This yields optimality of the transformation-based jump-adapted quasi-Milstein scheme.

math.NA↗