Strong existence and uniqueness of solutions of SDEs with time dependent Kato class coefficients

2020/10/22 by Saisai Yang, Tusheng Zhang, Yang, Saisai +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Physics Problems #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2010.11467

Abstract

Consider stochastic differential equations (SDEs) in \Rd: dX_t=dW_t+b(t,X_t)\d t, where W is a Brownian motion, b(⋅, ⋅) is a measurable vector field. It is known that if |b|²(⋅, ⋅)=|b|²(⋅) belongs to the Kato class \K_d,2, then there is a weak solution to the SDE. In this article we show that if |b|² belongs to the Kato class \K_d,\a for some \a ∈ (0,2) (\a can be arbitrarily close to 2), then there exists a unique strong solution to the stochastic differential equations, extending the results in the existing literature as demonstrated by examples. Furthermore, we allow the drift to be time-dependent. The new regularity estimates we established for the solutions of parabolic equations with Kato class coefficients play a crucial role.

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