Lq(Lp)-theory of stochastic differential equations

By Pengcheng Xia, Longjie Xie, Xicheng Zhang, Guohuan Zhao. 2020

62 citations recorded by the source corpus.

Abstract 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 R d : d X t = b ( t , X t ) d t + σ ( t , X t ) d W t , X 0 = x , where σ is bounded, uniformly continuous and nondegenerate, b ∈ L ˜ q 1 p 1 and ∇ σ ∈ L ˜ q 2 p 2 for some p i , q i ∈ [ 2 , ∞ ) with d p i + 2 q i 1 , i = 1 , 2 , where L ˜ q i p i , i = 1 , 2 are some localized spaces of L q i ( R + ; L p i ( R d ) ) . Moreover, in the endpoint case b ∈ L ˜ ∞ d ; uni ⊂ L ˜ ∞ d , we also show the weak well-posedness.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/361d60e03796fc31226b195f6295c71a35d57d78

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Citations in the corpus

Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance (2020, methodology): “To this end, we make a Zvonkin type transform as in [13], [26] and [25].”
Well-posedness of distribution dependent SDEs with singular drifts (2018, background): “The following lemma list some easy properties of H̃ q (see [35] and [30]).”