An Approximation Result For SDE With Non-Lipschitz Coefficients

Consider the following stochastic differential equationwhere σ_i(x) : R→R and b(x) : R → R are Borel measurable functions, {W~i(t),i = 1,2,...} is an infinite sequence of independent standard Brown motion defined on the classical Wiener space. Here we mainly give an approximation result for SDE with non-Lipschitz conditions and the speed of convergence in L~p. Setting t_n =[2~nt]/2n, we construct an approximation sequence such thatSuppose that σ(x) and b(x) satisfy the following non-Lipschitz conditions:where 0 < η < 1/e, j = 1,2, and ρ_j,η are concave functions on R_+:Then for any 0 < t < T, we obtainHere the constants Cp and CPit may change in different occasions.Moreover, we also consider the following multi-dimensional Stratonavich SDE b% Xt)dt, i = l,...,d,where a : R+ x Rd h^ Rd x Rd and b : R+ x Rd h^ Rd are Borel measurable functions. Assuming that a is non-degenerate and bounded, also has bounded continuous derivative, we prove that there is a unique strong solution for the above multi-dimensional Stratonavich SDE.