2017/10/14 by Peng Jin, Jin, Peng
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Primary: 60H10 #Probability (math.PR) #Secondary: 60J60 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1710.05227
openalex publication_date 2017/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study weak solutions for the following type of stochastic differential equation dXt=dWt+b(t, Xt)dt, t≥ s, Xs=x, where b: [0,∞) × ℝd → ℝd is a measurable drift, W=(Wt)t ≥ 0 is a d-dimensional Brownian motion and (s,x)∈ [0,∞) × ℝd is the starting point. A solution X=(Xt)t ≥ s for the above SDE is called a Brownian motion with time-dependent drift b starting from (s,x). Under the assumption that |b| belongs to the forward-Kato class F Kd-1α for some α∈ (0,1/2), we prove that the above SDE has a unique weak solution for every starting point (s,x)∈ [0,∞) × ℝd.