TY - JOUR
T1 - Lq(Lp)-theory of stochastic differential equations
AU - Xia, Pengcheng
AU - Xie, Longjie
AU - Zhang, Xicheng
AU - Zhao, Guohuan
N1 - Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/8
Y1 - 2020/8
N2 - In this paper we show the weak differentiability of the unique strong solution with respect to the starting point x as well as Bismut–Elworthy–Li's derivative formula for the following stochastic differential equation in Rd: dXt=b(t,Xt)dt+σ(t,Xt)dWt,X0=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜q1p1 and ∇σ∈L˜q2p2 for some pi,qi∈[2,∞) with [Formula presented], i=1,2, where L˜qipi,i=1,2 are some localized spaces of Lqi(R+;Lpi(Rd)). Moreover, in the endpoint case b∈L˜∞d;uni⊂L˜∞d, we also show the weak well-posedness.
AB - In this paper we show the weak differentiability of the unique strong solution with respect to the starting point x as well as Bismut–Elworthy–Li's derivative formula for the following stochastic differential equation in Rd: dXt=b(t,Xt)dt+σ(t,Xt)dWt,X0=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜q1p1 and ∇σ∈L˜q2p2 for some pi,qi∈[2,∞) with [Formula presented], i=1,2, where L˜qipi,i=1,2 are some localized spaces of Lqi(R+;Lpi(Rd)). Moreover, in the endpoint case b∈L˜∞d;uni⊂L˜∞d, we also show the weak well-posedness.
KW - Krylov's estimate, L(L)-estimates
KW - Zvonkin's transformation
KW - duality
UR - https://www.scopus.com/pages/publications/85082024085
U2 - 10.1016/j.spa.2020.03.004
DO - 10.1016/j.spa.2020.03.004
M3 - Article
AN - SCOPUS:85082024085
SN - 0304-4149
VL - 130
SP - 5188
EP - 5211
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
IS - 8
ER -