Numerical approximations of irregular differential equations

Summary

Recent advances in the study of regularization by noise phenomena have pushed forward our understanding of numerical approximations for stochastic differential equations with irregular coefficients which are discontinuous or even distributions. However, the current results only treat equations whose coefficients are uniformly bounded which leave out interesting cases that one encounters in practice. This Phd project aims to develop further these advances in numerical approximations of differential equations with unbounded coefficients, and therefore will address practical implementations.

Consider, for instance, a multidimensional stochastic differential equation driven by Brownian motion with a drift and its corresponding Euler-Maruyama scheme. We are interest in the optimal convergence rate of the scheme in situations when the drift is a measurable function of time and space. Note that we do not assume any continuity property on the drift.

When the drift is bounded, the recent article “Quantifying a convergence theorem of Gyöngy and Krylov” by Konstantinos Dareiotis, Máté Gerencsér, Khoa Lê obtains a strong convergence rate of order ½, which is the same as in the classical case when the drift is Lipschitz continuous. An extensions of this result to the case of integrable drift is reported in “Taming singular stochastic differential equations: A numerical method” by Khoa Lê and Chengcheng Ling.

The current challenging problem would be extending these results to equations with growing drifts (for example, unbounded measurable drifts).

Similar problems for stochastic partial differential equations could be also considered.

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