Numerical Methods for SDEs with Additive Noise and Distributional Drift

Explore numerical methods for stochastic differential equations (SDEs) with distributional drift in this mathematical lecture. Gain a comprehensive overview of recent results on strong and weak error rates for Euler-type schemes applied to SDEs where the drift belongs to negative fractional Sobolev spaces or negative Besov spaces with regularity index in (-1/2, 0). Begin by reviewing various notions of solutions for this class of SDEs before diving into the numerical aspects. Examine a two-step Euler-type scheme that has been successfully applied to different settings, including additive Brownian noise and additive fractional Brownian noise, with detailed derivations of strong error bounds for both linear SDE cases and nonlinear McKean equation cases. Discover an alternative Euler-type scheme designed for alpha-stable additive noise scenarios, including bounds for density error rates that connect to weak error analysis in linear cases. Conclude with numerical results that demonstrate the practical applications of these theoretical developments. This presentation was recorded during the thematic meeting "New trends of stochastic nonlinear systems: well-posedness, dynamics and numerics" at the Centre International de Rencontres Mathématiques in Marseille, France.