Stochastic differential equations with singular coefficients on the straight line

Abstract

AbstractConsider the following stochastic differential equation (SDE): $$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$ X t = x + ∫ 0 t b ( s , X s ) d s + ∫ 0 t σ ( s , X s ) d B s , 0 ≤ t ≤ T , x ∈ R , where $\{B_{s}\}_{0\leq s\leq T}$ { B s } 0 ≤ s ≤ T is a 1-dimensional standard Brownian motion on $[0,T]$ [ 0 , T ] . Suppose that $q\in (1,\infty ]$ q ∈ ( 1 , ∞ ] , $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) , $b=b_{1}+b_{2}$ b = b 1 + b 2 , $b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$ b 1 ∈ L q ( 0 , T ; L p ( R ) ) such that $1/p+2/q<1$ 1 / p + 2 / q < 1 and $b_{2}$ b 2 is bounded measurable, with $\sigma \in L^{\infty }(0,T;{\mathcal{C}}_{u}(\mathbb{R}))$ σ ∈ L ∞ ( 0 , T ; C u ( R ) ) there being a real number $\delta >0$ δ > 0 such that $\sigma ^{2}\geq \delta $ σ 2 ≥ δ . Then there exists a weak solution to the above equation. Moreover, (i) if $\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$ σ ∈ C ( [ 0 , T ] ; C u ( R ) ) , all weak solutions have the same probability law on 1-dimensional classical Wiener space on $[0,T]$ [ 0 , T ] and there is a density associated with the above SDE; (ii) if $b_{2}=0$ b 2 = 0 , $p\in [2,\infty )$ p ∈ [ 2 , ∞ ) and $\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$ σ ∈ L 2 ( 0 , T ; C b 1 / 2 ( R ) ) , the pathwise uniqueness holds.