Stochastic differential equations with singular coefficients on the straight line

Rongrong Tian,Jinlong Wei,Liang Ding

doi:10.1186/s13662-020-03097-8

Abstract

Consider the following stochastic differential equation (SDE): Xt=x+∫0tb(s,Xs)ds+∫0tσ(s,Xs)dBs,0≤t≤T,x∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ X_{t}=x+ \\int _{0}^{t}b(s,X_{s})\\,ds+ \\int _{0}^{t}\\sigma (s,X_{s}) \\,dB_{s}, \\quad 0\\leq t\\leq T, x\\in \\mathbb{R}, $$\\end{document} where {B_{s}}_{0leq sleq T} is a 1-dimensional standard Brownian motion on [0,T]. Suppose that qin (1,infty ], pin (1,infty ), b=b_{1}+b_{2}, b_{1}in L^{q}(0,T;L^{p}(mathbb{R})) such that 1/p+2/q<1 and b_{2} is bounded measurable, with sigma in L^{infty }(0,T;{mathcal{C}}_{u}(mathbb{R})) there being a real number delta >0 such that sigma ^{2}geq delta . Then there exists a weak solution to the above equation. Moreover, (i) if sigma in mathcal{C}([0,T];mathcal{C}_{u}(mathbb{R})), all weak solutions have the same probability law on 1-dimensional classical Wiener space on [0,T] and there is a density associated with the above SDE; (ii) if b_{2}=0, pin [2,infty ) and sigma in L^{2}(0,T;{mathcal{C}}_{b}^{1/2}({mathbb{R}})), the pathwise uniqueness holds.

Highlights

Introduction and main resultsConsider the following stochastic differential equation (SDE) in Rd: dXt = b(t, Xt) dt + σ (t, Xt) dBt, 0 < t ≤ T, X0 = x ∈ Rd, (1.1)where T > 0 is a given real number, b : [0, T] × Rd −→ Rd, σ : [0, T] × Rd −→ Rd×k are Borel measurable functions and {Bt}0≤t≤T is a k-dimensional standard Brownian motion defined on a given stochastic basis (, F, {Ft}0≤t≤T, P).The fundamental theory for (1.1) is developed mainly by Itô and furnishes a very important tool to construct diffusion process
(i) if σ ∈ C([0, T]; Cu(R)), all weak solutions have the same probability law on 1-dimensional classical Wiener space on [0, T] and there is a density associated with the above SDE; (ii) if b2 = 0, p ∈ [2, ∞) and σ ∈ L2(0, T; Cb1/2(R)), the pathwise uniqueness holds
Where T > 0 is a given real number, b : [0, T] × Rd −→ Rd, σ : [0, T] × Rd −→ Rd×k are Borel measurable functions and {Bt}0≤t≤T is a k-dimensional standard Brownian motion defined on a given stochastic basis (, F, {Ft}0≤t≤T, P)

Summary

Introduction

(i) if σ ∈ C([0, T]; Cu(R)), all weak solutions have the same probability law on 1-dimensional classical Wiener space on [0, T] and there is a density associated with the above SDE; (ii) if b2 = 0, p ∈ [2, ∞) and σ ∈ L2(0, T; Cb1/2(R)), the pathwise uniqueness holds. (2020) 2020:637 uniqueness if b is only bounded measurable but σ (t, ·) is Lipschitz continuous uniformly in t ∈ [0, T].

Results

Conclusion